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Albert EINSTEIN 2. and Relativity
for Kids
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thsoni
His Life and Ideas with 21 Activities
and Thought Experiments
Ft
Jerome Pohlen
TWO PILLARS OF WISDOM
The man loved mysterious Nature as a lover loves his distant beloved. In his day there did not exist the dull specialization that stares with self-conceit through horn-rimmed glasses and destroys poetry.
—ALBERT EINSTEIN, ON MICHAEL FARADAY
“About Max Planck’s studies on radiation, misgivings of a fundamental nature have arisen in my mind, so that I am reading his article with mixed feelings.” So Einstein wrote to his Dolly from Milan in April of 1901, a scant four months after Planck’s “act of desperation” in Berlin had saved his own reputation but failed to alert the physics community to the storm ahead. In the same letter Einstein ruefully admits, “soon I will have honored all physicists from the North Sea to the southern tip of Italy with my [job inquiry].” Emboldened by his first published article, which had appeared in the prestigious journal Annalen Der Physik, Einstein had sent a slew of postcards requesting an assistant’s position to the well-known physicists and chemists of Europe. None of these missives bore fruit, and as far as we know few of them were even graced with a reply. Although Einstein was convinced that Weber was behind the rejections, Einstein’s indifferent final academic record and his failure to receive the pro forma job offer from the Poly would likely have been enough.
Despite these disappointments he was scraping together a living through part-time jobs and private lessons and forging ahead with his independent thinking about the current state of theoretical physics. For much of this time he would be separated from his fiancée, but writing to her frequently. In his very next letter to Maric he continues discussing Planck: “Maybe his newest theory is more general. I intend to have a go at it.” A little later in the letter he comments, “I have also somewhat changed my idea about the nature of the latent heat in solids, because my views on the nature of radiation have again sunk back into the sea of haziness. Perhaps the future will bring something more sensible.” His last words were prescient; his views on radiation would emerge from haziness to enlarge the Planck radiation theory in a revolutionary manner, while the latent (or specific) heat of solids, a seemingly mundane topic, would provide Planck’s theory with the radical physical interpretation that it currently lacked. But before this could occur, Einstein needed to plumb deeply into thermodynamics, Planck’s specialty, and the newer atomistic discipline of statistical mechanics, which attempted to explain and extend the laws of thermodynamics. His main scientific motivation at the time was not to unravel the puzzles of relative motion. Einstein’s famous insight, that resolving these puzzles would require a major reshaping of our conceptions of time and space, would not occur to him for four more years. Rather, his primary scientific focus from his student days was “to find facts which would attest to the existence of atoms of a definite size.” Proving the existence of atoms and understanding the physical laws governing their behavior was the original quest of the Valiant Swabian.
The atomic world was the frontier of physics at the beginning of the twentieth century. The disciplines of what is now called classical physics had all developed without a need to delve too deeply into the question of the nature of the microscopic constituents of matter. That situation had now changed. If physics was going to progress, it would be essential to understand the fundamental origin of electromagnetic phenomena, of heat flow, of the properties of solids (e.g., electrical conductivity, thermal conduction and insulation, transparency, hardness), and the physical laws leading to chemical reactions. The answers to these questions would only be found by understanding the makeup of the atom and the physical interactions between atoms and molecules.
Modern physics had begun with the work of Sir Isaac Newton in the second half of the seventeenth century. He introduced a new paradigm for the motion of objects (masses) in space: first by the bold assertion that the natural state of motion of a solid body is to move at a constant speed in a straight line (Newton’s First Law), and then by the statement that the state of motion changes in a predictable manner when “forces” are acting on the body (Newton’s Second Law). If one knew the force and mass of the body, the Second Law would determine the instantaneous acceleration of the mass, a, via the relation F/m = a. What it meant to speak of the instantaneousrate of change of any quantity wasn’t (and isn’t) obvious, but Newton cleared this up by means of a mathematical innovation, the invention of calculus. From this point forward, mechanics came to mean the study of the motion of masses under the influence of forces described by elaborations of Newton’s Second Law, which could now be written as a “differential equation” using calculus.
For this law to be useful, scientists would need to have a mathematical representation of the forces in nature, the F on the left-hand side of F = ma. The forces of nature cannot be deduced; they can only be hypothesized (okay, guessed) and tested for whether their consequences make sense and agree with experimental measurements. No amount of mathematical legerdemain can get around that. Newton’s Second Law was an empty tautology unless one had an independent mathematical expression for the forces that mattered in a given situation.
Newton gained eternal fame by divining the big one, the one we all know from infancy: the force of gravity. His universal law of gravitation stated that two masses are attracted to each other along the line between their centers, and the strength of that attraction is proportional to the product of their masses and inversely proportional to the square of the distance between them. Of course this attraction is very weak between normal-size masses like two people, but between the earth and a person or the earth and the sun it is a big deal. From this law of gravitation and his Second Law, Newton was able to calculate all kinds of solid-body motion: the orbits of the planets in the solar system, the relation between the moon and tides, the trajectories of cannonballs. Thus Newton had published the first major section of the “book of Nature,” which was, Galileo famously declared, “written in the language of mathematics.”
Along with the stunning mathematical insights of Newton and their vast practical applications came an ontology, a view of what the fundamental categories of nature were, and how events in the world were related. As Einstein put it in his autobiographical notes, “In the beginning—if such a thing existed—God created Newton’s laws of motion together with the necessary masses and forces. That is all. Anything further is the result of suitable mathematical methods through deduction. What the nineteenth century achieved on this basis … must arouse the admiration of any receptive man … we must not therefore be surprised that … all the physicists of the last century saw in classical mechanics a firm and empirical basis for all … of natural science.”
At the core of the Newtonian view of nature was the concept of rigid determinism, majestically expressed by the Marquis de Laplace:
We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at any given moment knew all of the forces that animate nature and the mutual positions of the beings that compose it, if this intellect were vast enough to submit these data to analysis, could condense into a single formula the movements of the greatest bodies of the universe and that of the lightest atom; for such an intellect nothing could be uncertain and the future just like the past would be present before its eyes.
This Marquis Pierre Simon de Laplace was one of the great masters of classical mechanics in the nineteenth century and became known as the “French Newton.” He was willing to literally put his neck on the line for his natural philosophy. When he presented his five-volume study of celestial mechanics to Napoleon, he was greeted with the intimidating question: “Monsieur Laplace, they tell me you have written this large book on the system of the universe and have never even mentioned its Creator.” Laplace, normally quite politic in his dealings with influential men, in this case drew himself up and replied bluntly, “I have no need of that hypothesis.”
While the relation between mass and the force of gravity was the only fundamental law discovered by Newton, he and other physicists knew that there must be other forces with associated laws, for example, the pressure exerted by a gas (pressure is force per unit area), which surely must have a microscopic origin. Near the end of the eighteenth century Charles Augustin de Coulomb, using a sensitive instrument known as the torsion balance, definitively measured another type of force, also of invisible origin: the electrical force. Coulomb and others determined that, in addition to mass, there is another important property of matter, electrical charge, and that two charged bodies exert forces on each other in a manner similar to the way gravity works in Newton’s Second Law, that is, proportional to the product of their charges and inversely proportional to the square of the distance between them. However, there is a major difference between this electrostatic force and gravity; charges come in two types, positive and negative. Opposite charges attract each other, but like charges repel. Matter is usually electrically neutral (that is, made up of an equal number of positive and negative charges) or very nearly neutral, so two chunks of matter don’t usually exert much long-range electrical force on each another. Therefore, despite the fact that the electrical force is much stronger than the gravitational force (when appropriately compared), it doesn’t have the same kind of macroscopic effects as gravity.
Early in the nineteenth century it became clear that the story was even more complicated. Moving charges (i.e., electrical currents) create yet another force, known to the ancients but not understood as related to electricity: magnetism. Primarily through the work of the English experimental physicist Michael Faraday, it became clear that electricity and magnetism were intimately related because, for example, magnetic fields could be used to create electrical currents. Exploiting this principle, discovered in 1831 and now known as Faraday’s law, Faraday was able to build the first electrical generator (he had earlier made the first functioning electric motor). Faraday’s discovery would lead to an expansion of the classical ontology of physics, because it implied that electrical charges and currents gave rise to unseen electric and magnetic fields, which permeated space and were not associated with matter at all but rather represented a potential to exert a force on charged matter. These were the “unseen forces” that moved the compass needle, which had fascinated Einstein as a child. Besides masses, forces, and charges, there were now fields as well.
Faraday had risen from the status of a lowly bookbinder’s apprentice to become Fullerian Professor of Chemistry at the Royal Institution (during his life he rejected a knighthood and twice declined the presidency of the Royal Society). When asked by the four-time prime minister William Gladstone the value of electricity, he is said to have quipped,1 “One day sir, you may tax it.” He had little formal mathematical education and showed by experiment that his ideas were correct but did not formulate them into a rigorous theory.
That task was left to the Scottish physicist/mathematician James Clerk Maxwell. Maxwell was a deeply religious man, related to minor nobility, who showed an Einsteinian fascination with natural phenomena from a young age. As early as age three he would wander around the family estate asking how things worked, or as he put it, “What’s the go o’ that?” He is widely regarded as the third-greatest physicist of all time, after Newton and Einstein, although he is surely much less known to the public. He wrote his first important scientific paper at the age of sixteen and attended Cambridge University, where he excelled and became a Fellow shortly after graduation. One of his contemporaries wrote of him, “He was the one acknowledged genius … it was certain that he would be one of that small but sacred band to whom it would be given to enlarge the bounds of human knowledge.” At the age of twenty-three Maxwell expressed his philosophy of science in terms that prefigure similar sentiments of both Planck and Einstein:
Happy is the man who can recognize in the work of today a connected portion of the work of life, and an embodiment of the work of Eternity. The foundations of his confidence are unchangeable, for he has been made a partaker of Infinity. He strenuously works out his daily enterprises, because the present is given to him for a possession.
Maxwell had a full beard and a certain reserved presence that was hard to warm up to (very unlike Einstein, the mensch); however, he was a loyal friend and an almost saintly husband—in all, a man of character and integrity. Despite his diffidence, he possessed a rapier-like wit, which he would only occasionally display, as in the following. In his forties, having “retired” to his Scottish country estate for health and personal reasons, he was convinced to return to England to head the new Cavendish Laboratory at Cambridge; he did a superb job and became an important administrative figure in British science. In this capacity he was asked to explain to Queen Victoria the importance of creating a very high vacuum. He described the encounter thus:
I was sent for to London to be ready to explain to the Queen why Otto von Guerike devoted himself to the discovery of nothing, and to show her the two hemispheres in which he kept it … and how after 200 years W. Crookes has come much nearer to nothing and has sealed it up in a glass globe for public inspection. Her majesty however let us off very easily and did not make much ado about nothing, as she had much heavy work cut out for her all the rest of the day.
The young Maxwell came to know the much older Faraday personally as well as through his work and realized that his experimental discoveries, which Faraday had framed qualitatively, could be cast into a set of equations that describe all electromagnetic phenomena in four compact formulas, now universally known as Maxwell’s equations. Like Newton’s Second Law these are four differential equations, not describing masses and forces but rather electrical fields, magnetic fields, electrical charges and currents. If Maxwell had used only Faraday’s law and the previously known laws of electrostatics and magnetism, he would have found similar equations but with a disturbing asymmetry between the role of the electric and magnetic fields. Maxwell decided in 1861 that these two fields were different expressions of the same unified force, and had the brilliant insight to add a new term to one of the equations describing the magnetic field, which had the effect of making the full set of equations perfectly symmetric in regions of space where there were no electrical charges or currents (as in vacuum). Thus he essentially added a major clause to the laws of electromagnetism. The new term gave rise to new effects, called “displacement currents,” which were verified experimentally. They also made the equations structurally perfect. Boltzmann, quoting Goethe, said of Maxwell’s equations, “was it God that wrote those lines?”
FIGURE 4.1. James Clerk Maxwell at roughly the age at which he proposed the fundamental laws of classical electromagnetism. Courtesy of the Master and Fellows of Trinity College Cambridge.
Having added his new contribution to the electromagnetic laws, Maxwell made a historic discovery: electric and magnetic fields could propagate through the vacuum in the form of a wave that carried energy and could exert both electric and magnetic forces. In physics the term wave refers to a disturbance in a medium (e.g., water or air) that oscillates in time and typically is extended, at any single instant, over a large region of space. In this case the strength of the disturbance is measured by the strength of the electric field, so that if an electric charge sat at one point in space the electric field would push the charge alternately up and then down, like a rubber ball bobbing on surface waves propagating through water. Moreover, if you moved along with the wave, like a surfer, the field would always push you in one direction, just as the surfboard stays at the leading edge of a water wave (for a while).
Maxwell showed that the distance between crests of the electromagnetic waves could be made arbitrarily large or small; that is, any wavelength was possible. Thus he discovered what we now call the electromagnetic spectrum, extending, for example, from radio waves having a wavelength of a meter, to thermal radiation (as we saw earlier) at ten millionths of a meter, visible light at half a millionth of a meter, and on to x-rays at ten billionths of a meter. This was a spectacular finding; but the epiphany, the earthshaking revelation, was the speed of the waves: all of them traveled at the same speed, the speed of light! Suddenly disparate phenomena involving man-made electrical devices, natural electric and magnetic phenomena, color, and vision were unified into one phenomenon, the propagation of electromagnetic waves at 186,000 miles per second.
The beauty and significance of this discovery has awed physicists ever since. One of the greatest modern theoretical physicists, Richard Feynman, wrote of this event: “From the long view of the history of mankind … the most significant event of the nineteenth century will be judged as Maxwell’s discovery of the laws of electrodynamics. The American Civil War will pale into provincial insignificance in comparison with this important scientific event of the same decade.” Maxwell himself, with typical understatement, wrote to a friend in 1865, “I have also a paper afloat, with an electromagnetic theory of light, which, until I am convinced of the contrary, I hold to be great guns.”
Maxwell would go on to make other major contributions to physics, specifically with his statistical theory of gases, which will be of great relevance below, but he was not recognized as a transcendent figure during his lifetime. He died of abdominal cancer in 1879 at the age of forty-eight, still at the peak of his scientific powers. While in hindsight we view Maxwell as poorly rewarded in his time for his genius and service to society (he was never knighted, for example), Maxwell did not see it that way. On his deathbed he told his doctor, “I have been thinking how very gently I have always been dealt with. I have never had a violent shove in my life. The only desire which I can have is, like David, to serve my generation by the will of God and then fall asleep.”
Maxwell’s achievement particularly captivated Einstein. Maxwell, Faraday, and Newton were the three physicists whose picture he had on the wall in his study later in life. Of Maxwell he wrote, “[the purely mechanical worldpicture was upset by] the great revolution forever linked with the names of Faraday, Maxwell and Hertz. The lion’s share of this revolution was Maxwell’s … since Maxwell’s time physical reality has been thought of as represented by continuous fields…. this change in the conception of reality is the most profound and fruitful that physics has experienced since the time of Newton.” Elsewhere he said, “Imagine his feelings when the differential equations he had formulated proved to him that electromagnetic fields spread in the form of … waves and with the speed of light”; and, “to few men in the world has such an experience been vouchsafed.”
Maxwell had completed the second pillar of classical physics, what we now call classical electrodynamics, to go along with the first pillar, classical mechanics. But neither his nor Newton’s equations in themselves answered the fundamental question: what is the universe made of? One knew that there were masses and charges and forces and fields, but what were the building blocks of the everyday world? The enormous challenge was to extend these physical laws down to this conjectured “atomic” scale. Were there new, microscale forces not detectable at everyday dimensions? Did Newton’s and Maxwell’s laws still hold there? Were atoms little billiard balls with mass and electrical charge obeying classical mechanics and electrodynamics? Were there atoms at all, or were they just “theoretical constructs,” as many physicists and chemists maintained until the end of the nineteenth century?
At the time of Maxwell there was no way to probe the internal structure of atoms or molecules directly. As Maxwell put it, “No one has ever seen or handled a single molecule. Molecular science therefore … cannot be subjected to direct experiment.” However physicists, led by Maxwell and Boltzmann, were beginning to use the atomic concept to explain in great depth the macroscopic behavior of gases. In doing so they were inferring properties of atoms and their interactions. This was the work that Einstein never forgave Herr Weber, his erstwhile mentor, for ignoring. It is here that Einstein first put his shoulder to the wheel.
1 This wonderful incident may well be apocryphal, as there is no contemporaneous account of it.
THE PERFECT INSTRUMENTS OF THE CREATOR
“The Boltzmann is magnificent,” Einstein wrote to Maric in September of 1900. “I am firmly convinced that the principles of his theory are right, … that in the case of gases we are really dealing with discrete particles of definite finite size which are moving according to certain conditions … the hypothetical forces between molecules are not an essential component of the theory, as the whole energy is kinetic. This is a step forward in the dynamical explanation of physical phenomena.” Einstein was reading Boltzmann’s Lectures on the Theory of Gases. The Viennese physicist Ludwig Boltzmann and Maxwell had developed a theory of gases in the 1860s with much the same content, but with the difference that Boltzmann wrote long, difficult-to-decode treatises, while Maxwell’s work was much more succinct. Maxwell commented on this drily: “By the study of Boltzmann I have been unable to understand him. He was unable to understand me on account of my shortness, and his length was and is an equal stumbling block to me.” Einstein, despite the enthusiasm he expressed to his fiancée in 1900, was later to warn students, “Boltzmann … is not easy reading. There are very many great physicists who do not understand it.” It is likely that Einstein had no access to Maxwell’s work on gases in 1900, and as he did not read English until much later in life, he would not have been able to benefit from it anyway (in contrast, Maxwell’s electrodynamics was available to Einstein in German textbooks).
Maxwell beautifully described the scientific advance he had made in atomic theory in an address to the Royal Society in 1873 titled, simply, “Molecules.”
An atom is a body which cannot be cut in two. A molecule is the smallest possible portion of a particular substance. The mind of man has perplexed itself with many hard questions…. [Among them] do atoms exist, or is matter infinitely divisible? …
According to Democritus and the atomic school, we must answer in the negative. After a certain number of sub-divisions, [a piece of matter] would be divided into a number of parts each of which is incapable of further sub-division. We should thus, in imagination, arrive at the atom, which, as its name literally signifies, cannot be cut in two. This is the atomic doctrine of Democritus, Epicurus, and Lucretius, and, I may add, of your lecturer.
Maxwell goes on to describe how chemists had already learned that the smallest amount of water is a molecule made up of two “molecules” of hydrogen and one “molecule” of oxygen (here he has decided, somewhat confusingly, to use molecule to refer to both atoms and molecules). Then he arrives at his current research.
Our business this evening is to describe some researches in molecular science, and in particular to place before you any definite information which has been obtained respecting the molecules themselves. The old atomic theory, as described by Lucretius and revived in modern times, asserts that the molecules of all bodies are in motion, even when the body itself appears to be at rest…. In liquids and gases, … the molecules are not confined within any definite limits, but work their way through the whole mass, even when that mass is not disturbed by any visible motion…. Now the recent progress of molecular science began with the study of the mechanical effect of the impact of these moving molecules when they strike against any solid body. Of course these flying molecules must beat against whatever is placed among them, and the constant succession of these strokes is, according to our theory, the sole cause of what is called the pressure of air and other gases.
This simple picture, that gas pressure arises from the collisions of enormous numbers of molecules with the walls of the container, along with simple ideas of classical mechanics, allows Maxwell to derive Boyle’s law, that the pressure of the gas is proportional to its density. It also allows him to understand the observation that the ratio of volumes of any two gases depends only on the ratio of temperatures of the gases. The relation of temperature to volume of a gas is critical: in this view absolute temperature (what we now call the kelvin scale) is related to molecular motion and is proportional to the average of the square of the molecular velocity in a gas. Since the energy of motion for any mass, called kinetic energy, is just one-half its mass times the square of the velocity, this also means that for a gas its energy is just proportional to temperature. As Einstein had noted in his letter to Maric, in the Maxwell-Boltzmann theory, the entire energy of a gas is the kinetic energy of moving molecules. This principle of the Maxwell-Boltzmann theory, that the energy of each molecule is proportional to the temperature, applies even in the solid state, in which the molecules vibrate back and forth around fixed positions instead of moving freely throughout the substance. This property of the theory would perplex Einstein later, when he was trying to make sense of Planck’s radiation law.
“The most important consequence which flows from [our theory],” Maxwell continues, “is that a cubic centimetre of every gas at standard temperature and pressure contains the same number of molecules.” This fact about gases was conjectured by the Italian scientist Amadeo Avogadro in 1811. In 1865 Josef Loschmidt, a professor in Vienna and later a colleague of Boltzmann, had estimated this actual number, which is very large: 2.6 × 1019, or roughly five billion squared. (This “Loschmidt number” is closely related to Avogadro’s number, which is the number of molecules in a mole of any gas—both Einstein and Planck were very interested in accurately determining these numbers). With all this information about gas properties, it was possible for Maxwell to determine the average velocity of a molecule in air. He found it to be roughly one thousand miles per hour. He described the implications most picturesquely:
If all these molecules were flying in the same direction, they would constitute a wind blowing at the rate of seventeen miles a minute, and the only wind which approaches this velocity is that which proceeds from the mouth of a cannon. How, then, are you and I able to stand here? Only because the molecules happen to be flying in different directions, so that those which strike against our backs enable us to support the storm which is beating against our faces. Indeed, if this molecular bombardment were to cease, even for an instant, our veins would swell, our breath would leave us, and we should, literally, expire…. If we wish to form a mental representation of what is going on among the molecules in calm air, we cannot do better than observe a swarm of bees, when every individual bee is flying furiously, first in one direction, and then in another, while the swarm as a whole … remains at rest.
Maxwell goes on to describe how his own experiments and others have determined that the molecules in a gas are continually colliding with one another, moving only about ten times their diameter before changing direction again through a collision, leading to a kind of random motion called diffusion. Because of this constant changing of direction, the actual distance moved from the starting point during a given time is much less than if the molecule were moving in a straight line. This explained why, when Maxwell took the lid off a vial of ammonia in the lecture, its characteristic odor was not immediately detected in the far reaches of the lecture hall. The same kind of diffusion occurs in liquids such as water, but much more slowly. Maxwell then throws off a poetic but profound comment: “Lucretius … tells us to look at a sunbeam shining through a darkened room … and to observe the motes which chase each other in all directions…. This motion of the visible motes … is but a result of the far more complicated motion of the invisible atoms which knock the motes about.” Exactly this process occurs to small particles suspended in a liquid but visible under a microscope, so-called Brownian motion. In one of his four masterpieces of 1905 Einstein would actually take the suggestion of Lucretius and Maxwell seriously and, by careful analysis, turn this into a precise method for determining Avogadro’s number! Experiments by the French physicist Jean Perrin would confirm Einstein’s predictions and determine that number very precisely; as a result Perrin received the Nobel Prize for Physics in 1926, long after his work had permanently put to rest doubts about the existence of atoms.
The kind of complex, essentially random motion characteristic of gas molecules gave rise to a new way of doing physics, described by Maxwell in the same lecture. “The modern atomists have therefore adopted a method which I believe new in the department of mathematical physics, though it has long been in use in the section of statistics.” Thus was born the discipline of statistical mechanics. Maxwell could only assume that the invisible molecules obeyed Newtonian mechanics; he had no reason to doubt this. But in describing what would happen in a gas, he realized that one must inevitably encounter the weak point in Laplace’s grandiose dictum. Laplace had imagined an intellect that “at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed.” Maxwell realized that getting all the necessary information and using it to predict the future was an absurd proposition. “The equations of dynamics completely express the laws of the historical [Laplacian] method as applied to matter, but the application of these equations implies a perfect knowledge of all the data … but the smallest portion of matter which we can subject to experiment consists of millions of molecules, not one of which becomes individually sensible to us … so that we are obliged to abandon the historical method and to adopt the statistical method of dealing with a large group of molecules.” Maxwell’s point is that for all practical purposes one doesn’t want to know what each molecule is doing anyway; for example, to find the pressure exerted by a gas one needs only to know the average number of molecules hitting the wall of a container per second, and how much momentum (mass times velocity) they transfer to the wall.
This was the key insight of Maxwell and Boltzmann: to predict the physical properties of a large aggregation of molecules, one needed only to find their average behavior, assuming they were behaving as randomly as allowed by the laws of physics. Calculating these properties was relatively easy for a gas, where most of the time the molecules are not in close contact; for liquids and solids it was much harder and in certain cases still challenges the physicists of the twenty-first century. Tied up with this insight was a new understanding of the laws of thermodynamics. The First Law says that heat is a form of energy, and that the total energy (heat plus mechanical) always stays the same (is “conserved”) even when one form is being changed into the other. For example, when a car is moving at 60 miles per hour, it has a lot of mechanical energy, specifically kinetic energy, ½mv2, where m is the mass of the car and v is its speed (60 mph in this case). When you slam on the brakes, that kinetic energy doesn’t disappear; it is turned into heat in your brakes and tires, due to friction. From the point of view of statistical mechanics, that heat is just mechanical energy transmitted to the molecules of the road and tires, distributed in some complicated and apparently random manner among them. So heat is just random, microscopic mechanical energy, stored in various forms in the atoms and molecules of gases, liquids, and solids.
This view sheds light on the Second Law, which states that disorder always increases and is measured by a quantity called entropy. This law now can be interpreted as saying that in any process where something changes (e.g., the car coming to a stop), you can never perfectly “reorganize” all the energy that goes into the random motion of molecules. It is always too hard to retrieve all of it in a useful form. Before the car stopped, all its molecules (in addition to some random motion due to its non-zero temperature) were moving together in the same direction at 60 mph, providing a kinetic energy that could be used to do useful work, such as dragging a heavy object against friction. As the car stops, that energy is transformed into the less usable form of heat. It is not that we can’t turn heat back into usable energy (e.g., use it to get the car moving again); it is just that we can’t do it perfectly. We could run some water over the hot brake discs of our stopped car, which could generate steam, which could turn a turbine, and, presto, we would get back some useful mechanical energy. This of course is not the best-designed heat engine one could imagine. But the Second Law says that no matter how carefully or cleverly you design an engine to turn heat into useful mechanical energy, you will always find that you have to put more heat energy in than you get back.
To make this all precise and tractable in a mathematical theory, the German physicist Rudolph Clausius, while a professor at our familiar Zurich Poly in 1865, introduced the notion of entropy, which is a measure of how much the microscopic disorder increases in every process involving heat exchange. The word entropy was chosen from the Greek word for “transformation,” and indeed Clausius was guided by just the picture we have been painting: heat is the internal energy of atoms or molecules, which can be partially but never fully transformed to usable energy. Now, with their new statistical mechanics, Maxwell and Boltzmann were trying to make this idea of the internal energy of a trillion trillion rocking and rolling molecules precise, and in so doing come to understand entropy and the laws of thermodynamics on the basis of atomic theory. This program was so controversial that even by the end of the century, thirty years later, Planck, the thermodynamicist par excellence, was reluctant to adopt it. It was only his quantum conundrum that forced him to overcome his scruples, as we will see.
The key point is that the statistical mechanics of Maxwell and Boltzmann was still Newtonian mechanics, just applied to a system so complicated that one imagines it behaving like a massive game of chance, in which each molecular collision with a wall or with another molecule is like a coin being tossed (heads you go to the right, tails you go to the left). The worldview is the same as that of Newton and Laplace; only the method is different. Maxwell, had he lived another two decades, might have begun to recognize the leaks springing in this optimistic vessel, since the basic inconsistency in this view appeared at the intersection of his two great inventions, the theory of electromagnetic radiation and the statistical theory of matter. However, that was not to be; he would pass away a mere five years after his spectacular lecture on molecular science, having spent those final years occupied by his administrative duties. At the end of that same lecture, having anticipated the next twenty-five years of physical theory, the devout Maxwell makes one of the great historical appeals for intelligent design:
Natural causes, as we know, are at work, which tend to modify, if they do not at length destroy, all the arrangements and dimensions of the earth and the whole solar system. But … the molecules out of which these systems are built … remain unbroken and unworn.
They continue this day as they were created, perfect in number and measure and weight, and from the ineffaceable characters impressed on them we may learn that those aspirations after accuracy in measurement, truth in statement, and justice in action, which we reckon among our noblest attributes as men, are ours because they are essential constituents of the image of Him Who in the beginning created, not only the heaven and the earth, but the materials of which heaven and earth consist.
The next century would demonstrate in many ways, culminating in the awesome demonstration of August 1945, that atoms are not as indestructible as Maxwell had supposed. And Einstein would be the first to understand, through his most famous equation, E = mc2, just how much energy would be released when the perfect instruments of the Creator were disassembled.DIFFICULT COUNTING
The tomb of Ludwig Boltzmann in Vienna is engraved with a very short and simple-looking equation, which, ironically, he never wrote down during his lifetime:
S = k log W.
S is the universal symbol for entropy, k is a fundamental constant of nature known as Boltzmann’s constant, and log W is the logarithm1 of a number, W, relating to the physical system of interest, a number that Boltzmann called the number of “complexions.” (The number W can be so devilishly hard to calculate for many physical systems of interest that the greatest mathematical physicists of the twenty-first century, and the most powerful computers as well, are helpless to determine it.) This equation was the lever for setting the quantum revolution in motion. It would form the basis for Planck’s derivation of his radiation law and for Einstein’s first insights into the quantum nature of light.
Clausius had introduced the concept of entropy to explain heat flow, but he had no idea how a physicist could calculate this quantity from any fundamental mechanical theory (presumably an atomic theory). It was the statistical mechanics of Boltzmann, Maxwell, Gibbs (and Einstein) that gave the recipe. The recipe is deceptively simple sounding. It has two parts: (1) Whatever can happen, will happen. (2) No atom (or molecule) is special.
FIGURE 7.1. The epitaph on the tomb of Ludwig Boltzmann, S = k log W, expressing the fundamental equation of entropy, which he had discovered. Image courtesy Daderot.
Imagine shooting gas molecules one by one through a small hole into a box where they bounce around. Mentally divide the box into two equal parts with an imaginary partition. Any part of the box is equally accessible to the molecules (whatever can happen, will), and there is no reason for any molecule to be at one place at a given time versus another (no molecule is special). Suppose for the moment you can actually see the molecules directly. With one molecule in the box you look in periodically, and you find it roughly half the time on the left side of the box and half the time on the right side. Now add a second molecule, wait a bit, and start looking again. Roughly one-fourth of the time both molecules are on the left, one-fourth of the time both are on the right, and half the time one is on the left and one is on the right. Why is the last case more likely than the first two? Because there are two ways that you can get the last case (molecule 1 on right, molecule 2 on left; molecule 2 on right, molecule 1 on left) but only one way you can get the first two cases. This is just like tossing two coins and finding that one heads and one tails happens roughly twice as often as heads-heads and tails-tails. We can now get fancy and define three “states” for the two-molecule gas: in state one, both are on the left; state two, both on the right; and state three, one on the left and one on the right. For the first two states Boltzmann’s W = 1 (there is only one way to get these states), but for the third state2 W = 2. The entropy of this third state is then larger than that of the other two states, according to Boltzmann’s formula (we need not delve into the mysterious properties of the logarithm function to reach this conclusion). In actuality a physicist would specify the states in more detail than just which half of the box the molecules are in, but the underlying concept and method is exactly the same.
Now imagine we have a few trillion trillion molecules in the box (as indeed we usually do). There is still only one way to have all of them on one side; however, W, the number of ways of having about half on the right side and half on the left side, is unspeakably large. We literally have no words, no analogies, for numbers of this magnitude. As a feeble attempt, imagine the following: take all the atoms in the universe and, in one second, clone each of them, so as to create a second “universe.” Now repeat this every second, creating 4, 8, 16, and so on “universes.” Do this every second for the entire age of our current universe. Add up all of the atoms in all these universes and you will arrive at a number that is incredibly big, all right, but still this number is negligibly small compared with the number of states of high entropy of one liter of gas. These high entropy states, in which the gas molecules are roughly equally distributed in each half, have (not surprisingly) enormously higher entropy than the states in which the molecules are all or mostly on one side of the box.
Suppose that we go to a lot of trouble and evacuate the gas from the box, put an airtight partition in the middle, and put the gas back in on the left side, so that we set the system up in this very improbable (low entropy) state. From Maxwell we know that the gas molecules are flying around at 1,000 mph, colliding with one another and the walls and thus creating pressure on the walls. If we then remove the partition, the gas will rapidly fill the entire box again, approximately equally in each half. The entropy of the system will have increased. After the molecules spread themselves out roughly equally in the box, the molecules will still be colliding and moving around, but on average there will be roughly equal numbers on each side of the box. Intuitively this situation is the most disordered state (you don’t need to make special efforts to achieve it), and according to Boltzmann’s principle, this is the state of maximum entropy. This is the atomic explanation for the Second Law of thermodynamics, that entropy always increases or stays the same. Whenever we try to generate useful work from heat, we are essentially trying to create order out of this molecular chaos, and we are fighting against the laws of probability.
Consider further the previous example, where we have opened a partition and let the gas fill the entire box instead of just half the box. Now it is possible that if we wait long enough, all the collisions could work out just right, and all the gas molecules could reconvene on the left side. Is it worth waiting for this to occur? Not really. One can easily calculate that if all the states are equally likely and if we have only forty gas molecules in the box, it would take about the age of the universe for this to happen. With a trillion trillion molecules in the box? As they say in New York: fuggedaboutit.
This was the subtle point that Einstein missed in trying to prove that the Second Law of thermodynamics, the increase of entropy, is an absolute law. It isn’t absolute; entropy is allowed to decrease. Just don’t bet on it.
In fact, this overwhelmingly probable increase of entropy is how we determine the direction of time. Imagine that the gas in our box is colored and hence visible as it expands; if we saw a movie of the gas contracting back into half of the box, we would immediately assume that the movie was being run backward. Because the arrow of time is so fundamental, it was natural for physicists to assume that the increase of entropy was an absolute law of nature and not just a very, very, very, very, very … likely occurrence. When the young Einstein made this mistake, he was in good company; Boltzmann also got this wrong until his critics pointed it out to him. However, the canny Scot, Maxwell, was not fooled, and described the situation with colorful imagery: “the Second Law of Thermodynamics has the same degree of truth as the statement that if you throw a tumblerful of water into the sea, you cannot get the same tumblerful of water out again.”
Maxwell invented an imaginary creature, dubbed “Maxwell’s demon,” to illustrate this point further. His demons were “lively beings incapable of doing work” (i.e., of adding energy to the gas). He imagined these miniature sprites hovering in the gas, which is uniformly distributed in the box, but now a partition is added to the box, in which the demon has cleverly fashioned a frictionless trapdoor. Whenever the demon sees a gas molecule coming at him from the right to left with high velocity, he lets it through, then closes the trapdoor before any molecules can escape from the left to right. In this way over time he groups the faster molecules on the left and the slower ones on the right. But for a gas the temperature is proportional to the average energy, so by doing this the demon has heated the left side and cooled the right side, without putting any energy in. In other words the demon has created a refrigerator on the right (and a heater on the left), neither of which require any fuel (Einstein definitely would have rejected this patent). And why did Maxwell create his demons? Not as a serious proposal for an invention. Instead his intention was “to show that the Second Law of Thermodynamics has only a statistical certainty.” Maxwell’s demons, spawned around 1870, did not fancy the trip across the Channel, and so the true meaning of the Second Law was not understood in Europe for several more decades.
While Einstein did not recognize the “demonic” exception to the Second Law when he was reinventing Gibbs’s statistical mechanics in 1903–4, he did very much focus on what he called “Boltzmann’s principle,” the mathematical epitaph, S = k log W, mentioned above. In his third paper on statistical mechanics in 1904, he states, “I derive an expression for the entropy of a system, which is completely analogous to the expression found by Boltzmann for ideal gases [S = k log W] and assumed by Planck in his theory of radiation” (italics added). Later in that same paper he explicitly applies his results to Planck’s thermal radiation law, although in a manner that doesn’t yet refer to the quantum concept. This made Einstein the first physicist to extend the use of Planck’s law and to accept that statistical mechanics, which had previously been used only to describe gases, could also explain the properties of electromagnetic radiation. Radiation at this point was conceived to be a purely wave phenomenon, having nothing in common with the aggregate of particles (molecules) that make up a gas. Einstein now analyzed thermal radiation using his statistical methods, and he was beginning to see the problems with Planck’s “desperate” solution.
So what had Planck actually assumed about radiation, and how had he used Boltzmann’s principle to justify the formula that he had initially guessed by fitting the data? Planck had not been as bold as Einstein; he did not apply statistical mechanics to radiation but rather to the matter that exchanged energy with radiation. The Planck radiation law is, strictly speaking, only completely correct for what physicists call a “blackbody.” We all learn in school that the color white is a mixture of all colors and that black is the absence of color. A perfectly black body absorbs all light that falls upon it; hence no light of any color is directly reflected from it, and it appears black. In contrast, a surface that looks blue to us absorbs most of the red and yellow light incident on it and reflects the blue to our eye. But does the black object actually emit no light? Well, yes and no. It doesn’t emit any visiblelight, but it does send out a lot of electromagnetic radiation; however, as we learned earlier, if the object is at room temperature, the radiation is mainly at infrared wavelengths, which we can’t see. As already mentioned, the radiation law is precisely the rule for how much EM radiation of a given wavelength a blackbody emits at a given temperature.
To test this ideal behavior, physicists had to find a perfectly black body, not just for visible radiation but for all possible wavelengths. Unfortunately all real materials reflect EM radiation at some wavelengths, so soot, oil, burnt toast, and the other obvious candidates don’t actually do the job. So the experimenters came up with a clever idea: instead of using the surface of a material, they would use the inside of a kind of furnace with a small hole. Any radiation that went in through the hole would bounce around, being reflected many times, but eventually it would be absorbed before escaping. Thus any light coming out of the hole must have been emitted from the walls and would be representative of a perfect blackbody.
It was this kind of ideal black box or “radiation cavity” that Planck analyzed between 1895 and 1900. And one of his first ideas was to transfer his ignorance of the blackbody law from radiation to matter. He assumed that the walls of the cavity were made of molecules that would vibrate at a certain frequency in response to the EM radiation that fell upon them. Then by a clever argument he related the density of the energy of EM radiation at a given frequency (his goal), to the average energy of the vibrating molecules at the same frequency.3 He thus no longer had to deal with Maxwell’s equations describing the electromagnetic waves; he could assume that Newton’s laws held for the molecular vibrations, and he could use statistical mechanics. However, instead of doing the obvious thing and calculating the average energy of a molecule from statistical mechanics à la Boltzmann, he chose to find the entropy of the molecules.
He did this for a strange historical reason. When he began studying blackbody radiation in 1895, he hoped to find the missing principle that would restore the perfection of the Second Law and make the increase of entropy and hence the arrow of time an absolute principle of nature. While he had been convinced that the equations of matter allowed for entropy to decrease (although very rarely!), he hoped that those of Maxwell would prevent this from ever happening. This turned out to be a vain hope, as Boltzmann himself was able to demonstrate to Planck. However, having committed himself to the study of the entropy of radiation, and since the actual radiation law was still not definitively known, Planck continued his investigations. He knew that if he could find the average entropy of thermal radiation, it was related by straightforward mathematical steps to the average energy density, and hence to the correct radiation law.
At first Planck made an incorrect argument, which did not rely on Boltzmann’s principle. This led him to what was then called the Planck-Wien4 radiation law, and the embarrassing retraction in October 1900 after that law was ruled out by the experiments of his friends Rubens and Kurlbaum. At that point, guided by their experimental results and his mathematical intuition, as we saw earlier, he guessed the right form of the radiation law. Now, working backward from his apparently correct guess for the energy density of the radiation, he could figure out what the corresponding mathematical expression for the entropy of the radiation had to be. So this distinguished physicist was in a position oddly familiar to novice physics students, who might find the correct answer to a problem listed in the solutions at the back of their textbook but can’t quite figure out how to get that answer based on the principles they are supposed to have learned.
Faced with this quandary, for the first time in his career Planck resorted to Boltzmann’s principle. By accepting and using that principle (the formula S = k log W), he now had an approach to justify his empirical guess from the fundamental laws of statistical physics. What he needed to do was to count the possible states of molecular vibration, W, and show that when plugged into Boltzmann’s formula, it gave the answer that he “knew” was correct.
The mathematical problem he faced can be posed as follows. Planck assumed that all the molecules in the walls of the blackbody cavity had a fixed total amount of energy, which we can think of as a quantity of liquid, such as ten gallons of milk. For simplicity, imagine that there are one hundred molecules in the walls and that each molecule corresponds to a container that can hold up to the entire ten gallons. The question is how many ways can the ten gallons be shared among the hundred containers? If milk (and energy) are assumed to be continuous, infinitely divisible quantities, then the obvious answer is an infinite number of ways. But this didn’t deter Planck. The number of places you can put a gas molecule in a box is also infinite, but Boltzmann had found that his answer for the entropy of a system didn’t depend in any important way on how he divided the box into smaller boxes. So Planck essentially put little tick marks on the molecular energy containers, saying, for our imaginary example, that milk could only be distributed one fluid ounce at a time. Now he could go ahead and calculate the finite number of ways the milk could be shared and how that number depended on both the total amount of milk (the energy), the number of containers (molecules), and the size of the tick marks (the minimum “quantum” of energy). He was expecting that, as for Boltzmann’s gas calculation, nothing crucial would depend on the size of the tick marks. He was mistaken.
Try as he might, if he let the spacing of the tick marks get smaller and smaller, the calculation yielded the wrong entropy and the wrong radiation law. Finally he was forced to the conclusion that there must be some smallest spacing of the tick marks, that is, that energy could only be distributed among the molecules in some smallest “quantized” unit. Since there was absolutely zero justification for this final hypothesis, it is clear why Planck called it “an act of desperation.” To his credit, however, Planck did not shy away from stating clearly his unprecedented conclusion in his famous lecture of December 14, 1900, on the blackbody law:
We consider, however—this is the most essential point of the whole calculation—[the energy] E to be composed of a very definite number of equal parts and use thereto the constant of nature h = 6.55 × 10−27 erg-sec. This constant, multiplied by the frequency ν … gives us the energy element, ε.
Now we can understand fully this cryptic statement. The “definite number of equal parts” were the “tick marks,” that is, the minimum quantum of molecular vibrational energy, ε. Moreover it was clear from other considerations that in order to get the right radiation law, this minimum energy must be proportional to the frequency at which the molecules vibrated; thus he was forced to the conclusion that ε = hν. Here the Greek letter ν stands for the vibration frequency, and h (as Planck says) is a new constant of nature, undreamt of in our previous natural philosophies. Finally, because the radiation law was measured experimentally, he could go to the data and quickly figure out the actual value of the constant h (quoted above), which is now known as Planck’s constant and is the signature of all things quantum.
Planck later said that the radiation law had to be justified “no matter how high the cost.” Although he didn’t emphasize it at the time, the cost was very high. Planck’s little, technical fudge, if taken seriously, said something very, very strange about forces and motion at the atomic scale. It said that the Newtonian picture could not be right. For all intents and purposes, Planck had described molecules as little balls on springs, which stored energy by being compressed, and when the springs vibrated the energy was transferred back and forth between this stored (potential) energy and the kinetic energy of motion of the molecules, but in such a way that the sum of these energies, the total energy, was conserved. This much is standard Newtonian physics.
But in Newtonian physics the initial amount of total energy can vary continuously; all you need to do is compress the spring a little more, and it will have a little more energy. The fact that it can have any amount of energy (between some limits) appears intuitively to be related to the very fact that space is continuous. Nothing in Newtonian physics could explain quantized amounts of energy, the idea that the spring could only be compressed, say, precisely 1 or 2 or 3 or … inches but nothing in between. This was like imagining a car that can only go 0, 10, 20, … miles per hour and nothing in between. The obvious question is: how does it get from 0 to 10 miles per hour without passing through the intermediate values as it accelerates?
There was nothing innocent about Planck’s explanation of the radiation law. If it were the real explanation, it was a time bomb hidden in a thicket of algebra, which would explode with earth-shattering implications. Atoms and molecules were not little Newtonian billiard balls; they obeyed completely different and counterintuitive laws.
But Planck did not insist that his quantum hypothesis was a statement about the real mechanics of actual molecules. In fact he dropped a small hint in his lecture that perhaps energy is not really quantized. He denoted the total energy of his molecules as E and stated, “dividing E by ε we get the number P of energy elements which must be divided over the N resonators [molecules]. If this ratio is not an integer, we take for P an integer in the neighborhood” (italics added). But if molecular vibrations were really quantized, then E/ε would have to be a whole number! Planck was hedging his bets, signaling that one didn’t have to take this crazy energy element too seriously. Planck thought the constant of nature he had discovered, h, was very important, but there is no evidence that he believed his derivation invalidated Newtonian mechanics on the atomic scale.
Why not? Theoretical physics is a tricky business; sometimes one can get the right answer with assumptions that are wrong, or at least with stronger assumptions than one really needs. Perhaps another line of argument would occur to Planck, one that would preserve the welcome constant h but dispense with the uncomfortable assumption of the energy quantum, ε. Perhaps this weird, apparent quantization of energy only involved the interaction of radiation with matter but not mechanics per se. After all, there had been no obvious evidence of Planck’s constant in other areas of physics. It could be a new embarrassment if he trumpeted this energy quantum as a breakthrough in atomic physics and it turned out not to be so. No, best to play it safe, thought Planck; no need to cry wolf.
So, remarkably, Planck said nothing more in print for five full years about his great discovery, and the strange assumption buried in his derivation remained almost unnoticed. Except in Bern. There the unknown patent clerk’s searching investigations into the foundations of statistical mechanics were placing Planck’s Rube Goldberg mechanism on the witness stand and returning a verdict: not innocent.
1 Experts will know that in this equation the base of the logarithm is not the usual base 10 version, but is what is called the natural logarithm. The difference is not essential for understanding the meaning of entropy.
2 In chapters 24–25 we will learn that under certain circumstances the method for counting the states of quantum gases can differ from this classical reasoning. However, Boltzmann’s equation for entropy still holds, just with a different counting method for W.
3 Planck did not call his vibrating entities molecules but used the term “resonators” instead to emphasize that they were idealized microscopic oscillators and that he was not committing himself to any atomic theory. At this point the composition of the atom, with a compact nucleus and electrons bound to it, was not known, although, as we saw from Maxwell, the concept of atoms and molecules was widely accepted by the leading statistical physicists of the time.
4 Recall that after Planck came up with his new radiation law, which agreed with experiment, his name became attached to the new correct law and was dropped from the older law, now referred to as simply Wien’s law.
THOSE FABULOUS MOLECULES
One of the great open questions in the history of science is how Einstein came to the core idea of his paradigm-shifting paper of 1905. No, not his paper on special relativity or his paper proposing the famous equation E = mc2. Einstein was asked over and over again how he had developed the key insights leading to the special and general theories of relativity, and he answered with various charming anecdotes that have become part of his legend. As far as we know, he never went on record as to how he came up with the basic conception for his first paper of the annus mirabilis, a radical alternative to Maxwell’s theory of electromagnetic waves, which is the only one of his discoveries that he himself labeled as “revolutionary.” He says nothing directly about how he arrived at his first work on quantum theory in either his contemporary correspondence or in the papers preceding it. However, there are a few clues in the historical record, and these suggest that the key insight was his realization that the Planck radiation law was absolutely incompatible with statistical mechanics, at least in the form developed by Maxwell, Boltzmann, and Gibbs. This understanding likely matured during the year 1904 and early in 1905, when he was living a comfortable married life with Mileva and, as he was unknown to the wider physics community, his scientific correspondence was quite thin, leaving few traces of his profound ruminations.
As already mentioned, by 1903 Einstein had settled into his routine in Bern, working six days a week at the patent office, giving private lessons, and nonetheless finding time to pursue research in fundamental physics. Later he would refer to this period as “those happy Bernese years.” With his charisma and joie de vivre he had very quickly acquired a group of comrades who would share this idyllic interlude with him. The first of these new companions was a Romanian philosophy student, Maurice Solovine, who showed up at his flat in response to Einstein’s earnest advertisement for private physics lessons. A typical Einsteinian episode ensued. Following an enthusiastic invitation to enter his humble abode, Solovine was immediately “struck by the extraordinary brilliance of his large eyes.” Two and a half hours passed in a twinkling as the men discussed science and philosophy, and by the next session Einstein, having quite forgotten the original profit motive, declared physics lessons too much of a bother and proposed instead that they should meet freely to discuss ideas of all sorts. Very soon they added to their ranks another young aspiring intellectual, Conrad Habicht, a mathematics student, who had attended the Poly a bit ahead of Einstein and whose acquaintance Einstein had made during his vagabond years after graduation.
Habicht had the most jovial and high-spirited relationship with Einstein of all his peers; their letters to each other are rife with playful sarcasm. Together with Solovine, the two men founded a reading and discussion group, which they satirically dubbed the “Olympia Academy.” Habicht graciously allowed Einstein the esteemed position of president, complete with a commemorative (cartoon) bust and a grandiloquent dedication in Latin, celebrating his unerring command of “those fabulous molecules.” It was also Habicht who dubbed our Valiant Swabian “Albert Ritter von Steissbein,” which loosely translates as “Knight of the Tailbone,” presenting him with an engraved tin plate bearing this title. Far from being offended, Einstein and Mileva “laughed so much they thought they would die,” and henceforth Albert occasionally signed letters to Habicht with this sobriquet. The heraldic crest above his bust is aptly chosen: a link of sausages, one of the few foodstuffs the Olympians could afford to eat at their august gatherings.
Despite the evident joviality of the meetings, the members, along with occasional guests such as the attentive but silent Mileva, took their studies very seriously, and Einstein acquired many of his lasting philosophical views during the two years of meetings. The group would convene at the apartment of one of the members, and over a frugal repast would debate the meaning and merits of the assigned works, which included philosophy (David Hume and John Stuart Mill), history and philosophy of science (Henri Poincaré and Ernst Mach), and occasionally great literature (Don Quixote and Antigone).
FIGURE 8.1. (a) Hand-drawn cartoon by Maurice Solovine celebrating Einstein as President of the Olympia Academy, with his bust garlanded in hanging sausages. (b) Satirical inscription in Latin that accompanied the cartoon. It translates as “The man of Hechingen, expert in the noble arts, versed in all literary forms – leading the age towards learning, a man perfectly and clearly erudite, imbued with exquisite, subtle and elegant knowledge, steeped in the revolutionary science of the cosmos, bursting with knowledge of natural things, a man with the greatest peace of mind and marvelous family virtue, never shrinking from civic duties, the powerful guide to those fabulous receptive molecules, infallible high priest of the poor in spirit.” Courtesy the Albert Einstein Archive.
Solovine recalled that these gatherings brimmed over with merriment, although woe to him who would slight the gravity of the occasion. On one memorable night, Solovine, having skipped out on a meeting at the last minute to attend a concert, returned to his apartment to find his bed piled high with furniture and household items, his room enveloped in thick cigar smoke, and a scolding note in Latin pinned to his wall. Einstein remembered these get-togethers with the greatest fondness. In 1953 he wrote to Solovine:
FIGURE 8.2. The principals of the Olympia Academy circa 1903; on the left is Conrad Habicht, center is Maurice Solovine, and on the right is Einstein. ETH-Bibliothek Zurich, Image Archive.
In your short active life dear Academy, you took delight, with childlike joy, in all that was clear and intelligent. Your members created you to make fun of the long-established sister academies. How well your mockery hit the mark I have learned to appreciate through long years of careful observation…. Even if we have become somewhat decrepit, a glimmer of your bright, vivifying radiance still lights our lonely pilgrimage…. To you our fidelity and devotion until the last learned gasp!
A. E.—now corresponding member
Besides Habicht and Solovine, Einstein had another intellectual confidant in Michele Besso. He had met Besso, who was six years older and already a graduate of the Poly, while he was a student in Zurich. Besso, of middle-class Jewish origin, was an engineer, who had applied for and obtained a position in the patent office in Bern at Einstein’s suggestion. Einstein had also introduced him to his eventual wife, Anna Winteler, who was a daughter of Einstein’s host family in Aarau. Besso and Einstein became lifelong and intimate friends. If Einstein occasionally exhibited the absentmindedness of a starry-eyed dreamer, he was an absolute model of Swiss efficiency compared with the impractical Besso, whose boss had once pronounced him “completely useless and almost unbalanced.” Einstein, while granting that in many respects Michele was “an awful schlemiel,” nonetheless enjoyed and profited from their exchanges: “[Besso] has an extraordinarily fine mind whose working, though disorderly, I watch with great delight.” Later, when reflecting upon his achievements of 1905, he said that he “could not have found a better sounding-board in the whole of Europe.” Besso and Einstein walked home together daily during that period, and Einstein shared with him his developing ideas about physics. Although Besso is often mentioned as the first to hear about relativity theory, apparently the main subject of their conversations was something else: Einstein’s new hypothesis about the nature of light.
This hypothesis surely would have been presented to the Olympia Academy; however, by the end of 1904, Habicht had obtained a post as a mathematics and physics teacher in Schiers, quite a distance from Bern, and in 1905 Solovine moved from Bern to Paris, where he worked on a journal, the Revue Philosophique. Hence the academy was out of session as Einstein was producing his string of hits. In May of 1905 he sent a typical jocular missive to Habicht, whose absence he had felt:
Dear Habicht, such a solemn air of silence has descended between us that I almost feel as if I am committing a sacrilege when I break it now with some inconsequential babble….
So what are you up to you frozen whale, you smoked, dried, canned piece of soul…? Why have you still not sent me your dissertation? Don’t you know that I am one of the 1½ fellows that would read it with interest and pleasure, you wretched man?
I promise you four papers in return, the first … deals with radiation and the energy properties of light and is very revolutionary…. The fourth paper is still only a rough draft at this point, and is an electrodynamics of moving bodies which employs a modification of the theory of space and time.
This lively discourse indicates Einstein’s contemporaneous valuation of his great works of 1905; they were all exciting, and he was proud of them, but one was actually revolutionary, the one on the quantum properties of light. This paper, titled “On a Heuristic Point of View Concerning the Production and Transformation of Light,” grew out of his last work on statistical mechanics, in which he had focused for the first time on Planck’s theory of radiation, the theory that required the critical but unappreciated introduction of a smallest energy element, ε = hν.
Recall that Planck had avoided the question of whether he was allowed to apply statistical mechanics to radiation by relating the radiation energy, which he wanted to calculate, to the average energy of vibrating molecules in the blackbody. But instead of directly calculating this energy, he took the odd detour of calculating the entropy of the molecules, and, with full knowledge of the answer he had to get in order to agree with experiment, he wrestled the entropy into the correct form, using the outré assumption of a minimal energy element. Given Einstein’s focus on statistical mechanics, which, unaware of Gibbs’s prior work, he thought he was extending in a novel manner beyond gases, he would naturally focus on the place at which Planck took his detour. Why not just calculate the average energy of a molecule?
In this context, what is needed is the average “thermal energy” of the molecule. If we imagine the molecule as consisting of several atoms connected by chemical bonds, which can vibrate just like macroscopic springs, and if it were possible to direct energy to a particular isolated molecule, then according to the classical view one can tune the energy to any desired value (until it gets so large that the molecule breaks apart). However, Planck had appended the ad hoc restriction that each vibration could only have certain energy values, constrained to be a whole number times hν (although it wasn’t really clear he believed wholeheartedly in this constraint). Nonetheless, in either view, the energy of any specific molecule was something that could and would vary in time, whatever the temperature of the surroundings. But each molecule is typically in contact with other molecules through collisions (in the gaseous state) or mutual vibrations in the solid state, so that if the material’s temperature is fixed, each molecule has a definite average energy that does depend on the temperature of the surrounding environment and is termed the thermal energy.
Several times in his papers on statistical mechanics Einstein had already done a calculation for the kinetic part of the average thermal energy, the contribution of molecular velocity to the total thermal energy. It was a trivial generalization of those calculations to consider a molecule that vibrates back and forth and hence also has potential energy. Potential energy is just that—energy stored by doing work against a force. Continuing the spring analogy, when a mass on a spring is pulled and then clamped at a greater extension, potential energy is stored in the mass, which can be released when the mass is unpinned and oscillates back and forth. When a molecule vibrates, its chemical bonds are compressed and stretched like a spring, alternately storing and releasing potential energy. In this case, according to Maxwell/Boltzmann/Einstein statistical mechanics, the average potential and kinetic energies are equal and have a simple relation to temperature, Emol = kT, where k is Boltzmann’s constant (the same one that comes into the formula for the entropy, S = k log W), and T is the temperature.1 Note that this average thermal energy is independent of the frequency.
It may initially seem counterintuitive that two identical masses connected to springs that vibrate at very different frequencies (i.e., have different degrees of stiffness) can have on average the same energy. The high-frequency vibrating mass will oscillate faster, and thus will have higher kinetic energy, which, for an oscillator, also must be equal on average to its potential energy. Thus it should just have more total energy, right? No, because the two springs will also vibrate different distances from their unstretched positions. The relation Emol = kT tells us that the high-frequency molecule must make smaller-amplitude vibrations than the low-frequency molecule, in just such a way that the average energy of the two is equal. This statement of classical statistical mechanics, that all vibrating structures have the same average energy, has a fancy name: the equipartition theorem; it implies that the total energy of the system is equally shared by each microscopic part. This theorem will be very important in Einstein’s reasoning going forward.
Since Einstein had by now developed a great understanding of classical statistical mechanics, he surely would have leaned initially toward the “obvious” equipartition formula: Emol = kT, which via Planck’s reasoning would immediately yield a hypothesis for the radiation law. However, as we shall soon see, using the hypothesis so obtained gives a paradoxical result, so Einstein at some point rejected this approach. Planck’s route, on the other hand, involved an obvious fudge, the ad hoc hypothesis of a minimal energy element, with no fundamental justification at all. This dead end is the point at which ordinary mortals would have thrown up their hands and given up. Instead, in one of the greatest demonstrations of flexible thinking ever, Einstein abandoned his beloved classical statistical mechanics and opened his mind to a new and bizarre possibility, the possibility that the hallowed Maxwell equations, whose perfection he had long admired, were not the final word on the nature of light.
1 Here and elsewhere I assume that temperature is measured using the absolute (kelvin) scale, where it is always a positive number.
PLANCK’S NOBEL NIGHTMARE
The two constants [h, k] … which occur in the equation for radiative entropy offer the possibility of establishing a system of units for length, mass, time and temperature which are independent of specific bodies or materials and which necessarily maintain their meaning for all time and for all civilizations, even those which are extraterrestrial and non-human.
—MAX PLANCK
It was the fall of 1908, and Svante Augustus Arrhenius was determined to see that Max Planck received the Nobel Prize for Physics that year. Arrhenius, a scientist of impressively broad and bold speculations, had recently returned from a tour of Europe, where he was received warmly as befitted the first Swedish winner of the newly minted Nobel prizes. Arrhenius had won the Chemistry Prize in 1903 (two years after the establishment of the awards) for his groundbreaking work on electrolytic chemistry. He was widely recognized as a founder of the discipline of physical chemistry, which works at the boundary of the fields of physics and chemistry. In 1905 he had been offered a professorship in Berlin but had turned it down to remain in Sweden and head the new Nobel Institute for Physical Research; after receiving the prize he would be a member of the Nobel Award Committee in Physics and a de facto member of the Chemistry Committee for the remainder of his life. As such he had enormous influence over who received these awards, and he did not hesitate to use that influence.
Arrhenius, like all his contemporaries, was blissfully unaware of the looming crisis in atomic physics, uncovered by the work of the young Einstein, who was now becoming known—not for challenging the Newtonian paradigm of continuous motion but instead for dismissing another Newtonian axiom, the concept of absolute time. While Einstein had quickly moved to the terra incognita of the nascent quantum theory, assuming that atoms existed and trying to figure out their laws of motion and their interactions with radiation, Arrhenius was still fighting the last war, the war to prove that atoms were real. The ensuing episode illustrated just how oblivious the scientific community was to the gathering storm.
Had Arrhenius known the story of the checkered career of the German/Swiss Jew, who was still not recognized formally by the conservative professoriate of Switzerland in 1908, he likely would have recognized a kindred spirit. Arrhenius grew up near Uppsala, Sweden, where his father was a surveyor for the University of Uppsala, the oldest and among the most prestigious of the Nordic universities. A science and math prodigy, he had matriculated at the university at age seventeen, and received his degree in two years, before moving on to graduate studies in physics. However, in a striking parallel to Einstein, he alienated the senior members of the faculty, Tobias Thalen (physics) and Per Theodor Cleve (chemistry), and left after three years to complete his doctorate at the new Physical Institute of the Academy of Sciences in Stockholm. Unfortunately for Arrhenius the new institute was not yet allowed to grant PhDs on its own. Thus when, in 1884, he produced a monumental 150-page work on the conductivity of electrolytic solutions, explaining, for example, the high conductivity of salt in water by its dissociation into ions, it was received with great skepticism by a committee consisting mainly of faculty whom he had spurned at Uppsala. In the end the thesis was approved with the lowest possible passing grade, non sine laude approbatur, (“accepted, not without praise”). Forty years later Arrhenius would recount bitterly that Cleve and Thalen even refused to offer him the customary congratulations after the doctoral ceremony, saying that they had decided to “sacrifice him.” Although this work and its extensions would eventually earn him the Nobel Prize, the grade it had received was so poor that he was at least nominally disqualified from pursuing an academic career in Sweden at the time.
Here, however, his story diverges from that of Einstein, for he boldly sent the devalued thesis to the leading lights of European chemistry and physics, Clausius (inventor of the concept of entropy), van ‘t Hoff in Amsterdam (who would be the first Nobel Laureate in Chemistry), and Ostwald in Riga (the ninth Nobel chemistry laureate). One of these men, Ostwald, immediately recognized its innovativeness, to the extent that he even traveled personally to Uppsala to offer Arrhenius a job at his own institution.1 Arrhenius did not cut a particularly impressive figure, according to Ostwald: “[Arrhenius] is somewhat corpulent with a red face and a short mustache, short hair; he reminds me more of a student of agriculture than a theoretical chemist with unusual ideas.” But a brilliant chemist he was, and eventually Arrhenius did move to Europe and trained with Ostwald, van ‘t Hoff, and even with Boltzmann before returning to Sweden to become the unquestioned leader of Swedish physical chemistry, and the person who defined the international scope of the Nobel prizes at their inception.
A decade later, at the turn of the century, there was still a major movement in chemistry and physics that regarded atoms as somewhat suspect heuristic entities, a movement led by Arrhenius’s former mentor, Wilhelm Ostwald. This school of thought was known as “energetics” and also had adherents in the Swedish physics community, which maintained an attitude of distrust toward theory in general and of “pronounced hostility toward atomismand toward atomic theory” in particular. Arrhenius had decided to put this movement to final rest and make 1908 the Nobel Year of the Atom. Max Planck would receive the physics prize for the manner in which his radiation law had led to an accurate determination of Avogadro’s number and the elementary unit of atomic charge, e. The chemistry prize would be awarded to the British physicist Ernest Rutherford, who had shown that atoms disintegrated (i.e., emitted doubly ionized helium atoms, known as alpha particles) during radioactive decay. In a very recent experiment with Geiger, Rutherford had deduced a value of the elementary charge from alpha particles in excellent agreement with that calculated by Planck using his radiation law, tying the two prizes neatly together.
The fact that Rutherford considered himself a physicist and would be very surprised to know that he had been reclassified a chemist2 did not deter Arrhenius from his plan. Arrhenius had nominated Rutherford for both the physics and chemistry prizes that year, but it is likely that he had planned all along to support Planck in the Physics Committee, of which he was a member. By the time of the crucial meeting on September 18, 1908, he knew that the Chemistry Committee (based on an internal report he had apparently ghostwritten) was committed to awarding the prize to Rutherford. Planck and Wien had been jointly nominated in physics for the theory of heat radiation by Ivar Fredholm, a Swedish mathematician and mathematical physicist, and Arrhenius swung his support to this nomination, but with the intention of splitting the ticket and engineering a prize for Planck alone.
Why did Arrhenius think that Planck alone should be recognized? Because at that time Arrhenius was not interested in the physical principles behind the law of thermal radiation3 so much as in its connection to the fundamental constants in molecular chemistry. This is an aspect of Planck’s work of 1900 that is barely mentioned in modern times, but at that time it overshadowed his radical quantum hypothesis. Planck’s radiation law depended on the two newly discovered physical constants that he introduced, h, the “quantum of action” (Planck’s constant), and k, Boltzmann’s constant (the constant associated with entropy through the equation S = klog W and thermal energy through the equipartition relation Emol = kT.) From a careful fit of blackbody radiation data one can extract quite precise values for both h and k, and Planck did so immediately after deriving his radiation law in 1900. The constant h appeared to him completely enigmatic and was not put to any immediate use, but the constant k, which only later became known as Boltzmann’s constant,4 was instantly recognized as providing a theoretical microscope for studying the atom.
In his December 1900 magnum opus Planck states, “To conclude, I may point to an important consequence of this theory which at the same time makes possible a further test of its reliability.” He goes on to show by straightforward steps that the Boltzmann constant satisfies the simple relationship k = R/Na, where R is the constant in the ideal gas law PV = RT for a mole of gas, and Na is Avogadro’s number (which has struck fear into so many beginning chemistry students), the number of atoms contained in a mole of any gas. This number was imperfectly known in 1900, whereas R was very well known. Hence by extracting k very precisely from the radiation law, Avogadro’s number could be determined to unprecedented precision. Planck found the value Na = 6.175 × 1023, which is within 2.5 percent of the currently accepted value 6.022 × 1023. Using the same information, he could determine the mass of a hydrogen atom, again with high accuracy. Finally, in a coup that must have impressed the physical chemist, he used considerations from electrolytic chemistry, Arrhenius’s own field, to find the elementary charge on a proton, obtaining a value within 2.5 percent of the modern value. In contrast the best-known value of e, the charge on an electron, measured by J. J. Thomson from electron studies, was off by 35 percent! Planck concluded his 1900 analysis with the confident declaration, “If the theory is at all correct, all of these relations should not be approximately, but absolutely valid. The accuracy of the calculations … is thus much better than all determinations up to now.”
Planck had always been fascinated by fundamental constants as expressions of the absolute and eternal in physics. Even before his work of 1900 he had realized that the radiation law involved two distinct and new fundamental constants. Fundamental constants allow one to define what are called absolute units, units of measurement relating to the basic laws of physics. For example, the speed of light, c, provides a natural unit of velocity, because no signal can travel faster than c and all relativistic phenomena become more and more important as this speed is approached. In the famous twin paradox of special relativity, your identical twin ages more and more slowly compared with you as her relative velocity approaches c. Planck pointed out that his two newly discovered constants, when combined with the speed of light and the gravitational constant, would allow fundamental units to be defined for all physical quantities (length, time, temperature, etc.). Transported by this revelation, the staid professor allowed his inner geek to emerge in print, rhapsodizing that these units would be valid for “all times and civilizations … even extraterrestrial ones.” Later, when Planck became embroiled in a philosophical debate with the Viennese philosopher-scientist Ernst Mach, Mach would lampoon his exuberance over fundamental units: “concern for a physics valid for all times and all peoples, including Martians, seems to me very premature and even almost comic.”
Nonetheless in 1900 it was these fundamental constants, which had emerged from his radiation law, that most excited Planck, and not his unexamined introduction of discontinuity into the laws of physics. To his disappointment, the rest of the physics community did not immediately appreciate even this aspect of his breakthrough. He later recounted:
I could derive some satisfaction from these results. But matters were viewed quite differently by other physicists. Such a calculation of an elementary electrical [charge] from measurements of thermal radiation was not even given serious consideration in some quarters. But I did not allow myself to become disturbed by such a lack of confidence in my constant k. Nevertheless, I only became completely certain on learning that Ernest Rutherford had obtained a [very similar] value by counting alpha particles.
This spectacular agreement between completely disparate physical phenomena, all pointing to a single consistent atomic picture of the world, had convinced Arrhenius that Planck alone should be recognized with the physics Nobel Prize in 1908. It was the connection to fundamental constants that distinguished Planck’s work from Wien’s in Arrhenius’s mind, and in his report to the Nobel Physics Committee he barely mentioned Planck’s derivation of the radiation law and completely omitted any mention of “quanta of energy.” Planck’s use of the constant k, he said, had “made it extremely plausible that the view that matter consists of molecules and atoms is correct…. No doubt this is the most important offspring of Planck’s magnificent work.”
Arrhenius’s enthusiasm did not sweep through the conservative Physics Committee unchallenged. Among its members was the distinguished experimentalist Knut Angström, who had actually done experiments on heat radiation and was aware of the experimental prehistory leading up to Planck’s “act of desperation.” With much justice he wrote, “it is very far from being that the theoretical works have guided the experimental ones, but rather that one could justly make a completely contrary statement.” However, there was a small problem with his argument that an experimenter should receive or share the prize: none had been nominated that year. Angström and the other skeptics on the Physics Committee were reluctantly convinced by Arrhenius to join the Planck bandwagon.
And so the modest, upright Planck (who had himself nominated Rutherford for the physics prize that year) might have received this honor, not because of a deep appreciation of the true significance of his work, elucidated by Einstein from 1905 to 1907, but rather because of a general ignorance of its full implications. After the Physics Section of the Swedish Academy had approved Planck as the awardee, rumors of the result quickly traveled around the continent, apparently reaching Planck himself, who stated to the press, “[if true] I presume that I owe this honor principally to my works in the area of heat radiation.” But the full Swedish Academy would still have to approve the recommendation of the Physics Committee, and in the interim between these votes something had changed the mood in Stockholm. The most famous theoretical physicist of his generation, the man Einstein admired the most, had finally spoken publicly on the Planck law, and his opinion would derail Arrhenius’s well-laid plans.
Hendrik Antoon Lorentz was born on July 18, 1853, in Arnhem, the Netherlands, to an unexceptional middle-class family. His extraordinary brilliance was recognized early, and by the age of twenty-four he was appointed to the newly created Chair of Theoretical Physics at the University of Leiden. He devoted his early years to the application and extension of Maxwell’s theory of radiation. In particular, while J. J. Thomson is credited with “discovering the electron” in 1897, Lorentz deduced its existence a year earlier, in 1896, from his analysis of light emitted from a gas in the presence of a magnetic field—the “Zeeman effect,” discovered by his former student and assistant Pieter Zeeman. He shared the Nobel Prize in Physics with Zeeman in 1902 for this work (the first theorist to be so honored) and went on to develop an elegant theory of the interaction of electrons with light, published in 1904. In related work, Lorentz came to the very edge of the special theory of relativity, coming up short only by his unwillingness to interpret relativistic effects as arising from the relative nature of time, as did Einstein in 1905. In fact Lorentz was troubled by Einstein’s approach, complaining, “Einstein simply postulateswhat we have deduced with some difficulty and not altogether satisfactorily, from the fundamental equations of the electromagnetic field.” Despite these misgivings, within a few years Lorentz became Einstein’s close confidant and scientific father figure, supporting and providing constructive criticism for all his major research.
We have already heard that Einstein regarded Lorentz as the most powerful thinker he had ever encountered, but his admiration ran deeper than this because of Lorentz’s elegant, generous, and kindly spirit. Late in his life Einstein wrote: “Everything which emanated from his supremely great mind was as clear and beautiful as a good work of art…. For me personally he meant more than all of the others I have met in my life’s journey. Just as he mastered physics and mathematical structures, so he mastered also himself—with ease and perfect serenity.” Einstein’s attitude toward Lorentz was, by 1908, shared by much of the European physics and mathematics community. His encyclopedic knowledge of many subdisciplines made his opinion the final word for many. “Whatever was accepted by Lorentz was accepted; and whatever was rejected by him was rejected or at best labeled controversial.” He, alone among all theorists, had preceded Einstein in noting Planck’s essential use of the “energy element” in his derivation of the radiation law as early as 1903, and had “wrestled continuously with this problem [of heat radiation]” in the years leading up to 1908.
Lorentz, with his unmatched mastery of both electromagnetic theory and the dynamics of electrons, set out to deduce the Planck radiation law rigorously, directly from the motion of charged electrons, which radiate and absorb radiation, without introducing fictitious molecules or “resonators” as Planck had done. But no matter how hard he tried, he succeeded only in rediscovering the low-frequency, approximate law of Rayleigh-Jeans (and Einstein), which led to absurd consequences (infinite energy) at high frequencies. In April of 1908 he announced his findings at the International Congress of Mathematicians in Rome.
The theory of Planck is the only one that would provide us with a formula in accord with experimental results; but we could accept it only with the stipulation that we completely rework our basic conception of electromagnetic phenomena…. However I must address myself to the question of how the Jeans theory, which involves no constants other than … [Boltzmann’s constant] k, can take into account the peak of the radiation curve which has been demonstrated by experiments [the absence of high-frequency thermal radiation predicted by Jeans]. The explanation given by Jeans—which is really the only one that can be given—is that the maximum is illusory; its existence is simply an indication that it has not been possible to realize a body that is black to short wavelengths…. Fortunately, we can hope that new experimental determinations of the radiation function will permit a choice between the two theories.
Lorentz, the most respected theoretical physicist of his generation, was leaning in favor of the Jeans “slow catastrophe” theory!
The German physics community was initially stunned, and then apoplectic. Two outstanding experimenters, Lummer and Pringsheim (whom Angström had favored for the 1908 Nobel Prize) wrote scathingly, “If we examine the Jeans-Lorentz formula, we see at first glance that it leads to completely impossible consequences which are in crass conflict not only with the results of all observations of radiation, but with everyday experience.” They continued with more than a hint of sarcasm: “We might therefore dismiss this formula without further examination were it not for the eminence and authority of the two theoretical physicists who defend it.” Wilhelm Wien expressed the outrage felt by many when he stated, “I was extremely disappointed by the lecture which Lorentz presented in Rome…. he came up with nothing more than the old theory of Jeans without adding any new viewpoint…. [That theory] is not worthy of discussion in the experimental field…. What purpose is served by submitting these questions to mathematicians, since they can provide no judgment in this matter? … In this instance Lorentz has not shown himself to be a leader of science” (italics added).
Subjected to this torrent of criticism, even the serene Lorentz must have had second thoughts. By June 1908, only two months later, he wrote a long letter to Wien, which contained an apology in the form of an embarrassing thought experiment. In the Rayleigh-Jeans theory (which Lorentz had rederived), the energy emitted as heat radiation at a given frequency is proportional to the temperature. A metal such as silver heated to 1200°K (roughly 900°C) will glow with blinding white light. When the temperature is reduced to room temperature (roughly 300°K), it should still be emitting one-quarter as much white light, according this theory. Hence, Lorentz continued, if the theory were right an unheated silver mirror would glow visibly in the dark! He concluded, “thus we should really dismiss the Jeans theory … and we are left with only the theory of Planck. Do not think that I do not respect it, quite the contrary, I admire it greatly for its boldness and success.”
Lorentz’s speedy retraction and his professed admiration for Planck’s theory, expressed privately, did not get as much publicity as his public criticism of it in Rome. A mathematician with substantial influence in the Swedish Academy, Gosta Mittag-Leffler, had gotten wind of Lorentz’s critique and used it to his advantage. He wished to ingratiate himself with the French clique, led by the great mathematician Poincaré, and to divert the physics prize to the second-choice candidate, the French pioneer of color photography Gabriel Lippmann. In this scheme he was aided by the original nominator of Planck and Wien, the mathematician Ivar Fredholm, who was unhappy with the elimination of Wien from the award. Fredholm wrote to Mittag-Leffler before the full academy vote was taken, criticizing the Physics Committee decision and stating specifically that Planck had based the derivation of his radiation law on “a completely new hypothesis, which can hardly be considered plausible, namely the hypothesis of the elementary quanta of energy.”
With the sudden realization that Planck’s law was radical and controversial, his nomination was soundly defeated in the general academy vote, and the prize was instead awarded to Lippmann. Mittag-Lefler sent a gleeful letter taking credit to a French mathematician: “It is I, along with Phragmen, who got the prize awarded to Lippmann. Arrhenius wanted to give it to Planck in Berlin, but his report, which he had somehow gotten accepted unanimously by the Committee, was so stupid that I was able to crush it easily.” The physics world was only beginning to grapple with the inevitability of the quantum revolution, proclaimed by the Valiant Swabian in his great works of 1905–7. Planck, Lorentz, and the other great scientists of Europe would have to look to this fearless interloper to lead them forward in the quest for a new microscopic mechanics. More than a decade would pass, and much would happen, before the Nobel committee would again be willing to consider giving the prize for an atomic theory.
1 There is some irony here in comparison with Einstein, who sixteen years later would write to the very same man asking for a job, leading to a famous letter from his father to Ostwald (behind Einstein’s back) essentially pleading with Ostwald to assent. No known answer was received, but in 1910 Ostwald became the first scientist to nominate Einstein for the Nobel Prize.
2 Rutherford, who had spent many years proving that elements transmuted during radioactive decay, later joked that “he had seen many transmutations in his time but none as quick as his own transmutation into a chemist.”
3 Arrhenius did have another reason to be interested in thermal radiation. He was the first scientist to recognize the role of CO2 in trapping heat radiation and warming the planet, and even suggested the possibility that human-generated industrial CO2 emissions would enhance this effect. He published this idea in that same year, 1908, arguing that it was a good thing and might forestall future ice ages.
4 It was initially called by many “Planck’s constant,” leading to no end of confusion until the conventions settled down, assigning h to Planck and k to Boltzmann.THE IMPORTANCE OF BEING NERNST
I visited Prof. Einstein in Zurich. It was for me an extremely stimulating and interesting meeting. I believe that, as regards the development of physics, we can be very happy to have such an original young thinker; a “Boltzmann redivivus [reborn].”
—WALTHER NERNST, MARCH 1910
Although in Salzburg it was clear to observers such as Max Born that “Einstein’s achievement received its seal[of approval] before the assembled world of scientists,” the achievement most recognized was his theory of relativity, which by then had been well confirmed by the fast-electron experiments of Alfred Bucherer in Bonn. As just noted, Einstein’s quantum hypotheses were regarded by all the leaders present in Salzburg as perhaps inventive, but certainly rash and premature. This impression was possibly facilitated by Einstein’s decision not to mention in his lecture his lesser-known work on the specific heat of solids. This work, showing that Planck’s distribution law of thermal radiation could also be applied to electrically neutral vibrations of atoms, clearly indicated the necessity of a non-Newtonian atomic mechanics, as opposed to treating Planck’s “energy element” as some anomaly associated only with the interaction of radiation and matter, as Planck had suggested. But there was one leader, not present at Salzburg, who was keenly aware of Einstein’s work on specific heat, and to whom it was a very big deal. His name was Walther Nernst.
Nernst was a physical chemist, like Arrhenius, and indeed the two met in their twenties while Arrhenius was studying in Germany in 1886 and became fast friends, as well as boisterous drinking companions. Arrhenius pronounced Nernst’s work on heat conduction “the best that any laboratory practitioner has done in a long time” and got him invited to work with his patron, Ostwald, in Leipzig. Like Arrhenius, Nernst was not much to look at. He was short of stature with a “fishlike mouth” and had a distinctive, high-pitched voice that was often employed in claiming priority for some idea or other, which he typically insisted had already appeared in his famous textbook on physical chemistry. When, in the early 1900s, Nernst became well known in Berlin society, a joke circulated about a superman whose brain God had created but whose body had been left to lesser craftsmen. The disappointing result was then brought to life by the Devil for amusement; you can guess the name of this golem.
Nernst himself had a distinctive form of sarcastic humor, which he expressed with a completely deadpan delivery. For example, surrounded by avid hikers including Planck, who climbed mountains well into his seventies, he would opine that he too had climbed a mountain once in his youth, and that would suffice for a lifetime. He was indeed a talented scientist; Einstein praised “his truly amazing scientific instinct combined both with a sovereign knowledge of an enormous volume of factual materials … and with a rare mastery of the experimental methods and tricks in which he excelled.” Despite what Einstein called his “childlike vanity,” he was, according to Nernst trainee Robert Millikan, “in the main, popular in the laboratory, despite the fact that in the academic world he nearly always had a quarrel on with somebody.” Millikan recounts that in 1912 he wrote a review chapter for Nernst’s famous textbook, dealing with the determination of the electric charge, e (the topic on which Millikan would eventually do Nobel-winning research). Nernst initially accepted the draft, but after Jean Perrin, a well-known French physicist and eventual Nobel laureate himself, annoyed Nernst at a conference, he demanded that Millikan expunge every mention of the man’s name from the chapter. Perrin’s offense was speaking too long in the lecture before Nernst was scheduled to speak, hence depriving Nernst of some of his allotted time. Such a combination of determination, charisma, and pettiness was Walther Nernst.
These qualities conspired to make Nernst one of the most successful scientific “operators” of the twentieth century, a person capable of convincing officials in government and industry of the importance of scientific work in general, and of his own contributions in particular. Soon these attributes would have a great influence on Einstein’s career and on the development of quantum theory. Nernst’s political savvy showed itself early in his career. At age twenty-four, in 1888, he developed an idea, originally due to the chemist van ‘t Hoff, into an important relation now known as the Nernst equation, which allows the useful energy of a chemical reaction to be estimated from electrochemical measurements, and he quickly parlayed this into an academic position at Göttingen. By 1895 he had procured a full professorship and his own physical chemistry institute there, obtained under the threat of leaving for a position in Munich. Around that time he began working on one of the key technological problems of the era: cheap, durable electric lighting. In fact electric lighting provided a primary motivation for the momentous research program on blackbody radiation by Rubens, Kurlbaum, and others that was reaching fruition at almost the same time.
By 1897 Nernst had developed and patented the “Nernst lamp,” based on a heated cerium oxide glass rod, which was superior in a number of ways to the incandescent metal filament bulb pioneered by Edison and others. Understanding that the development of the market for lighting was unpredictable, Nernst refused to accept royalties for his invention, which was licensed by the German firm AEG, instead insisting on a large lump sum payment up front. As perhaps Nernst had foreseen, his lamp shone brightly for a few years, magnificently lighting the German pavilion at the Paris Exhibition of 1900 and selling over four million units in the next decade, but eventually losing out to the less-expensive tungsten filament technology perfected by a former Nernst student, the chemist Irving Langmuir. Long before this denouement Nernst had visited Edison in American in 1898, and after being subjected to a lecture on the irrelevance of academic work by the aging inventor, he shouted into Edison’s ear trumpet, “How much did you get for your light-bulb patent?” Upon Edison’s reply that he got nothing, Nernst bellowed, “I got a million marks1 for mine! The trouble with you Edison is that you are not a good businessman!”
Because of his negotiating acumen, the commercial failure of his lamp was of little consequence;2 Nernst was now a wealthy man and a figure to be reckoned with in German society. In 1905 he accepted a professorship at the center of German science, the University of Berlin, where he became a close associate of Planck, whose earlier thermodynamic work had put the Nernst equation on a more sound theoretical footing. His distinguished colleague in Berlin, the organic chemist Emil Fischer, who already had receive the Nobel Prize in 1902, described him approvingly as “versatile, many-faceted, full of curiosity and enterprise.” His students referred to him jokingly as the Kommerzienrat, German for a successful businessman (in contrast to the usual Geheimrat, denoting a distinguished scholar). Even before moving, he had been decorated by the emperor and appointed privy councillor, and upon his arrival he was immediately admitted to the prestigious Prussian Academy of Sciences.
Around this time he made the most important scientific discovery of his life, a principle that is now known as the Third Law of thermodynamics. This principle, in modern terms, is expressed by the statement that the entropy of any system tends to zero as its temperature tends to absolute zero. Unlike the other two laws of thermodynamics, which don’t require quantum effects to make sense, this law is all about quantum “freezing.” According to Boltzmann, a system’s entropy depends on the number of its accessible states at a given temperature. We already have seen from Einstein’s work on vibrations in solids that when a solid gets very cold the atoms cannot vibrate, because they don’t have enough thermal energy to reach even the first excited quantum level, which is hυ higher in energy than the lowest (ground) state. In other words the system tends toward a single, unique ground state, and the entropy, by definition, is zero. Nernst’s law says this always happens for any system, no matter how complicated.
Why, you may ask, would a chemist with a practical and empirical bent like Nernst care deeply about this apparently abstract principle? Because for some years previously it had been realized that knowledge of a system’s low-temperature entropy behavior would allow one to use chemical reaction data over a limited range of temperatures and predict at other temperatures and pressures how much of each product a particular reaction would yield, a fundamental question in all of chemistry. Nernst quickly set out to calculate the “reaction constants” that followed from his principle and showed that they predicted well the results of various experiments. Already in 1910, when he was meeting Einstein for the first time, his approach had been crucially employed by his colleague Fritz Haber to facilitate the development of a chemical technique of immense importance for humanity, the process for removing nitrogen from the air to make ammonia for fertilizer (and explosives).
Characteristically, Nernst referred to the principle he had discovered as “his theorem,” which is exactly what it wasn’t. A theorem is something logically deduced from other accepted rules or axioms. Nernst’s “theorem” was a hypothesis based on analysis of data; it had no grounding in atomic theory at all, nor could it have had at that time. Systems that obey the equipartition principle of classical mechanics do not obey Nernst’s law; so his conjecture actually contradicted the currently accepted theory. We now know quantum ideas are essential for the validity of this principle. And that is where Einstein comes in.
Nernst surely realized that his “theorem” was going to require some microscopic underpinning to become a law. And there were absolutely no hints of a microscopic theory that would violate equipartition coming from the leading theorists of the time. Yes, there was that strange business about the radiation law, but it seemed, as Planck would surely have advised him, to relate mainly to matter in interaction with radiation. Thus one can imagine his excitement when he became aware of Einstein’s 1907 paper, explaining how applying Planck ideas to vibrations in solids predicted a violation of equipartition, and just the kind of violation (the freezing out of vibrations) that would justify Nernst’s “theorem.” No wonder that in early March 1910 Nernst made a special trip to Zurich to meet the wunderkind who had provided the first mathematical theory consistent with his historic conjecture.
Despite Einstein’s rise in status as an associate professor, very few of his colleagues and acquaintances had any idea that a major intellectual figure was among them. His informality with the students and his ever-present sense of humor hardly signaled an august personage; and his dress had not improved greatly from the time he used a runner from his dresser as an impromptu scarf. His first (and only) doctoral student, Hans Tanner, describes his “rather shabby attire, with trousers too short for him and an iron watch chain,” hastening to add that his lecturing style immediately “captured our hearts.” An associate professorship for a man with a family could not have supported much elegance anyway; an older colleague at the time wrote of Einstein, to none other than Arrhenius, “I am most interested in the associate professor at the university, A. Einstein, a still young, totally brilliant chap from whom one can learn a lot…. I believe he has a great future, at present he lives with his wife and children in very modest conditions. He certainly deserves a better fate.”
The general impression of Einstein’s status changed suddenly and dramatically after Nernst’s visit. George Hevesy, a young assistant at the Zurich Poly at that time, who went on to become a Nobel laureate in chemistry, recalls that Nernst’s visit “made Einstein famous. Einstein in 1909 was unknown in Zurich. Then Nernst came and people in Zurich said ‘that Einstein must be a clever fellow if the great Nernst comes all the way from Berlin to Zurich to talk to him.’ ”
Very little is known about the details of this visit except that Nernst filled Einstein in on the state-of-the-art measurements of specific heat as a function of temperature being done in his laboratory, and Einstein surely impressed Nernst with his profound understanding of thermodynamics and statistical mechanics, and with his thoughts on how the quantum hypothesis could clear up many issues. Nernst’s reaction is worth quoting at length:
I visited Prof. Einstein in Zurich. It was for me an extremely stimulating and interesting meeting. I believe that, as regards the development of physics, we can be very happy to have such an original young thinker; a “Boltzmann redivivus [reborn]”; the same certainty and speed of thought; great boldness in theory, which however cannot harm, since the most intimate contact with experiment is preserved. Einstein’s “quantum hypothesis” is probably among the most remarkable thought [constructions] ever; if it is correct then it indicates completely new paths both for the so-called “physics of the ether” and for all molecular theories; if it is false, well, then it will remain for all times “a beautiful memory.”
The most striking thing about this remarkable quotation is that Nernst, Max Planck’s close colleague and friend, refers to Einstein’s quantum hypothesis without mentioning Planck at all. It is clear to him that in the hands of Einstein, Planck’s ad hoc patch-up of radiation theory has become something very different: a vision of a completely new electromagnetic and molecular theory. In fact, there is no indication that Einstein’s 1907 proclamation of a sweeping quantum revolution in molecular mechanics was noted by anyone until Nernst took it up, apparently in late 1909 or early 1910. Not a single paper had been written relating to the quantum theory as applied to specific heat between early 1907, when Einstein’s work appeared, and February 17, 1910, when Nernst read his first paper on the subject to the Prussian Academy of Sciences, mentioning Einstein’s theory briefly at the end.3 With Nernst leading the charge this changed dramatically; ten such papers appeared in 1911, and over thirty total in the subsequent two years. Moreover it is very likely that, during or shortly after that visit, Nernst conceived the project of bringing Einstein to Berlin; a postcard from Nernst dated July 31, 1910, begins, “I have made inquiries regarding Einstein, but have not yet received any news.”
As for Einstein, who by now had been struggling fruitlessly for more than two years to explain light quanta by modifying Maxwell’s equations, the visit by Nernst was a great morale boost. A week after Nernst left he wrote to Laub: “For me the theory of quanta is a settled matter. My predictions regarding the specific heats are apparently being brilliantly confirmed. Nernst, who has just been here to see me, and Rubens are busily engaged in the experimental verification, so that we will soon know where we stand.” And three months later he wrote to Sommerfeld with further results: “It seems incontrovertible that energy of a periodical nature, wherever it occurs, always occurs in energy quanta that are multiples of hυ … [whether] as radiation or as oscillation of material [molecular] structures…. It now seems pretty certain that as regards the heat content, the molecules of solid substances behave essentially similar to Planck’s resonators. Nernst found the relationship confirmed in the case of silver and some other substances.”
But all this support for his heuristic ideas did not for one minute distract him from the underlying challenge: how do wave and particle properties manage to coexist in a full mathematical theory of quantum mechanics or electrodynamics? With characteristic wit he summarized his view in the same letter to Sommerfeld. “The crucial point in the whole question seems to me to be: ‘Can energy quanta and Huygens’ principle [of wave interference] be made compatible with each other?’ The appearances are against it but, as it seems, the Lord knew all the same how to get out of the tight spot.”
1 The equivalent of roughly 4.5 million 2008 dollars. Diane Barkan, in her biography of Nernst, states that the actual amount Nernst received is unverifiable, but the legend persists.
2 One consequence his invention did have, ironically, was the rupture of his friendship with Arrhenius. In 1897 in Stockholm Nernst demonstrated his lamp for Arrhenius, who laughed vigorously when it blew all the fuses in the hotel. From this small incident a lifelong feud nucleated, with the consequence that Nernst received the Nobel Prize only in 1921, after many lesser lights of chemistry had been so recognized.
3 “The specific heat decreases strongly at low temperatures … corresponding to the requirements of Einstein’s theory [that] it tends to zero” (Nernst, February 17, 1910).THE INDIAN COMET
I have ventured to send you the accompanying article for your perusal and opinion. I am anxious to know what you think of it. You will see that I have tried to deduce the coefficient 8πυ2/c3 in Planck’s Law independent of the classical electrodynamics, only assuming that the ultimate elementary region in the phase-space has the content h3.
This letter to Einstein from an unknown Indian scientist, received in early June, 1924, initiated one of the most extraordinary episodes in the modern history of science, culminating in Einstein’s final historic contribution to the structure of the new quantum theory. At the time of his writing, Satyendra Nath Bose was a thirty-year-old Reader (roughly equivalent to the rank of associate professor) at Dacca University in East Bengal. His previous five research papers had made no impact at all on contemporary research, and he had recently been informed that, due to a funding cutoff to the university, his appointment would not be extended more than a year. Moreover, the paper he was sending to Einstein had already been submitted for publication, and rejected, by the English journal Philosophical Magazine. He had admired Einstein for many years and had even produced a rather undistinguished translation of Einstein’s papers on general relativity into English, for distribution in India. Thus, through some combination of veneration and chutzpah, he hit upon the idea of sending this paper, which related closely to Einstein’s 1916 work on radiation theory, directly to the master, with an astonishing request:
I do not know sufficient German to translate the paper. If you think the paper worth publication, I shall be grateful if you arrange for its publication in Zeitschrift fur Physik. Though a complete stranger to you, I do not feel any hesitation in making such a request. Because we are all your pupils though profiting only by your teachings through your writings.
Yours faithfully, S. N. Bose.
Einstein by that time, as we have seen, was not just the most famous scientist of his time; he was one of the best-known individuals on the entire planet. He was deluged by letters from strangers, wanting his opinion on everything under the sun, while at the same time struggling to keep up his voluminous scientific correspondence with the large community of physicists with whom he had personal and professional relations. In addition, Einstein spoke very little English and had not been able to deliver his prestigious lectures in England during his visit in 1921 in the native tongue of his audience.1 The a priori probability that S. N. Bose’s paper would end up in the circular file, his work and his name lost to posterity, was extremely high.
But that is not what happened. Einstein read the paper shortly after its arrival, translated it, and sent it to the German journal Zeitschrift für Physik on July 2, 1924, with his strong endorsement. There it was subsequently published, with a note from Einstein appended. “In my opinion Bose’s derivation of the Planck formula signifies an important advance. The method used also yields the quantum theory of the ideal gas, as I will work out in detail elsewhere.” In fact, shortly thereafter Einstein translated a second paper, which he had received from Bose on the heels of the first one and which he sent on to the journal by July 7, 1924. This one did not elicit such a favorable opinion from the great man, and he published a critical comment along with it, while nonetheless supporting its publication.
With these magnanimous gestures from the sage, the die was cast. Bose would go on to become one of the most famous names in the history of modern physics. The term “boson” is used for one of the two fundamental categories of elementary particles in modern physics2 because such particles obey novel statistical laws first employed, although not announced, in Bose’s initial paper sent to Einstein. This category includes Einstein’s photons (light quanta) as well as roughly half of the atoms in the periodic table. But Bose’s discovery, like that of Planck twenty-four years earlier, was not as clear-cut as it has been portrayed, and again it would take Einstein to find the radical implications in it.
S. N. Bose was born into the rising middle class of English-educated Indians in 1894 in Calcutta. He father, an accountant, had a wide range of intellectual interests that he transmitted to his son, who, in addition to his great aptitude for mathematics, became deeply interested in poetry, music, and diverse languages. When he matriculated at the Presidency College in Calcutta in 1909, however, he chose to study science, at least partly because of its potential utility to the future Indian nation, as a wave of nationalism swept through his generation. His cohort was “a particularly brilliant lot—the famous 1909 batch of Presidency College … [which] in all its history has not seen the likes … since.” Bose completed his BSc in 1913 and MSc in 1915, taking first place in both examinations; but there was no obvious avenue for obtaining a doctorate, and the professorial ranks were still reserved for third-rate English academics at that time. Bose therefore went through a period as a striving outsider, not dissimilar to the career of Einstein at the same age.
He married early (prior to graduation) but, contrary to custom, refused to accept a dowry or other financial support from his wife’s family. Already responsible for a wife and son shortly after he received his MSc, he spent a year eking out a living through private tutoring, while trying to work toward a PhD in mathematics with a well-known professor, Ganesh Prasad. Prasad was noted for his aggressive criticism of prospective students and of their previous teachers, which typically cowed the candidates into silence. But Bose was “notorious for plain speaking.” In an echo of Einstein’s conflicts with authority figures such as Weber, Bose “dared to counter his adverse criticisms” and was summarily dismissed from consideration for PhD work with the comment, “you may have done well in the examination, but that does not mean you are cut [out] for research.” “Disappointed, I came away [and] decided to work on my own,” Bose recalled.
Like Einstein, he was turned down for low-level teaching jobs before being offered an entry-level lectureship in the new University College of Science in Calcutta, whose founder, Sir Asutosh Mookerjee, began hiring the cream of young Indian scientists, including the future Nobel laureate C. V. Raman. It was at the College of Science that Bose first began to learn about the exciting developments in physics in Europe associated with the names of Planck, Einstein, and Bohr. Here also he and his close friend, the physicist Meghnad Saha, obtained and translated important German works of physics, including Einstein’s papers on relativity theory.
In 1921 Bose accepted a faculty position at the new University of Dacca, in East Bengal, recently established by an ambitious vice-chancellor, P. J. Hartog. At Dacca he “spent many sleepless nights” trying to understand the Planck law, while at the same time teaching it to his students. He felt an obligation to present something to them that he himself found clear and consistent: “As a teacher who had to make these things clear to his students I was aware of the conflicts involved…. I wanted to know how to grapple with the difficulty in my own way…. I wanted to know.” In late 1923 he hit upon his new approach to deriving the law and sent off a manuscript to the Philosophical Magazine, where he had previously published papers on quantum theory, only to receive the reply, in the spring of 1924, that the referee’s decision had been negative. It was at this point that Bose took the bold step of sending the paper to Einstein, a strategy so speculative that its success appeared to violate the very principle of maximum entropy employed in the paper itself!
It was remarkable, but nonetheless true, that Planck’s blackbody formula remained somewhat mysterious a full twenty-four years after Planck’s initial derivation. It was not that anyone doubted any longer the validity of the formula, but the tortured reasoning Planck had used to derive it left physicists unsatisfied for decades. That is why Bose’s paper, titled “Planck’s Law and the Quantum Hypothesis.” was of interest to Einstein and others. As we saw earlier, Planck had been reluctant to treat radiation directly with statistical mechanics, and instead, using classical reasoning, he related the mean energy of radiation at a given frequency to the mean energy of idealized vibrating molecules (“resonators”). He then calculated the entropy of these resonators by introducing into the counting of states his quantized energy “trick.” A key factor in totaling up the number of allowed states in Planck’s method, which was not appreciated for quite some time, was that one could treat the units of energy belonging to each resonator as indistinguishable quantities (i.e., if resonator one had seven units of energy hυ and resonator two had nine units, one didn’t have to ask which units they were). In 1912 Peter Debye, an outstanding young theorist and future Nobel laureate, rederived the Planck law, not by counting resonator states, but by counting states of classical electromagnetic waves that could fit in the blackbody cavity and then ascribing to them the same average energy that Planck had assigned to each resonator. The counting of the number of allowable waves in the cavity led to the factor in the Planck radiation formula, 8πυ2/c3, to which Bose alludes in his letter to Einstein. This factor was very easy to find from classical wave physics but very hard to find from quantum principles, hence Bose’s emphasis on having found a quantum route to it.
It was clear that Einstein had been troubled by Planck’s derivation from his earliest works, but not until 1916 had he even tried to justify this law, succeeding marvelously with his “perfectly quantic” paper, which introduced the concepts of spontaneous and stimulated emission of photons, as well as providing strong arguments for the reality of photons. One major reason that Einstein was so happy with this work, and even called it “the derivation” of Planck’s formula, was that it did not at any point use the strange counting of the distribution of energy units that Planck had employed. Instead he managed to get the same answer by a different route, based on Bohr’s quantized atomic energy levels and his own plausible hypotheses about the balancing out of all emission and absorption processes. Bose, though, was not completely happy even with this method and claimed to have found an even more ideologically pure derivation.
Bose’s paper is concise in the extreme, running to less than two journal pages. He begins the work by laying out his motivation for presenting yet another approach to the Planck law. “Planck’s formula … forms the starting point for the quantum theory … [which] has yielded rich harvests in all fields of physics … since its publication in 1901 many types of derivations of this law have been suggested. It is acknowledged that the fundamental assumptions of the quantum theory are inconsistent with the laws of classical electrodynamics.” However, Bose continues, the factor 8πυ2/c3 “could be deduced only from the classical theory. This is the unsatisfactory point in all derivations.” Even the “remarkably elegant derivation … given by Einstein” ultimately relies on some concepts from the classical theory, which he identifies as “Wien’s displacement law” and “Bohr’s correspondence principle,” so that “in all cases it appears to me that the derivations have insufficient logical foundation.”
Einstein did not agree with this criticism and even took time out during his first, quite friendly letter to Bose to dispute it: “However I do not find your objection to my paper correct. Wien’s displacement law does not presuppose [classical] wave theory, and Bohr’s correspondence principle is not used. But this is unimportant. You have derived the first factor [8πυ2/c3] quantum-mechanically…. It is a beautiful step forward.” On both points Einstein was correct.3 Bose had come up with a more direct method of getting the result, the first to use only the photon concept itself, a tremendously appealing simplification.
Ever since Einstein’s 1905 paper on light quanta, there had been a glaring logical problem with taking quanta seriously as elementary particles. In dealing with the statistical mechanics of a gas of molecules, it is possible to derive all the important thermodynamics relations, such as the “ideal gas law” (PV = RT),4 without ever specifying any other system with which the gas molecules interact. It is enough to simply say that there exist other large systems (“reservoirs”) with which the gas can exchange energy. Then counting the states of the gas, using the classical method (no h!) pioneered by Boltzmann, leads to both the entropy and the energy distribution of the gas molecules, and eventually to all the known relations.5 The very same approach appeared to fail for quanta of light; it led to Wien’s incorrect radiation law, and not Planck’s. This was a major problem, which, along with the difficulty in explaining the interference properties of light, led to the consensus that light quanta weren’t “real” particles but some sort of heuristic construct. This consensus had survived even the awarding of the Nobel Prize to Einstein for the photoelectric effect. Bose’s work shows how to escape from the first of these dilemmas.
Bose sets out to count the possible states, W, into which many light quanta of energy hυ and momentum hυ/c can be distributed according to quantum principles. From this, by a variant of Planck’s method, he obtains the average entropy and energy of the photon gas.6 Step one is to consider a single light quantum, with energy hυand momentum hυ/c. If a photon were to be treated as a real, classical particle, one ought to be able to specify its state at each time by its position and its momentum. Physicists refer to such quantities as vectors, since they carry both a magnitude and a direction (e.g., the photon is 5 blocks northeast, with its momentum [always parallel to its direction of motion] due south). The momentum for a massive particle is just its mass times its velocity vector (when its speed is much less than c); but a photon’s speed (magnitude of velocity) is always equal to c, and Einstein has shown (e.g., in his 1916 work) that the magnitude of its momentum is hυ/c (not m times c, since the photon has zero mass).
Counting the position states of the photon is not the hard part of Bose’s argument; one assumes that the photon gas is enclosed in a box of volume V and it can be anywhere within V, with equal probability (this point was used by Einstein in his original arguments for the photon concept when analyzing the blackbody entropy back in 1905). Therefore Bose focuses on counting momentum states. Since the possible directions of photon motion are continuous and hence infinite, he has to employ an idea already proposed by Planck as early as 1906. Planck’s constant, h, defines a quantum limit on the smallest difference in momentum that can be resolved.7 Since all photons of frequency υ have the same magnitude of momentum, hυ/c, which is assumed equally likely to point in any direction, Bose can count their states by tiling the surface of a sphere of radius hυ/c with these “Planck cells.” From basic geometry and the assumption that the spherical shell is only one cell thick, he is then able to find the number of states,8 8πυ2/c3.
Up to this point, what Bose has done is logically appealing but not historic. It was his next step that caused his paper to become “the fourth and last of the revolutionary papers of the old quantum theory.” He still has to obtain the Planck form for the radiation law and not the Wien form. For this next, crucial step he has to calculate how many physically distinct ways there are to put many photons at the same time into these available states. But Bose does not appear to realize that the next step is the big one; instead he seems to think that the previous one was the most significant. He begins the relevant paragraph by saying, “It is now very simple to calculate the thermodynamic probability of a macroscopically defined state.” After a few definitions, he unveils his answer, a rather obscure combinatorial formula bristling with factorial symbols. This is the key intermediate step. From here on, finding the Planck formula is inevitable and just involves straightforward manipulations, within the competence of scores of his contemporaries.
This paltry written record leaves an enormous historical question. To what extent did Bose understand the key concept in his “revolutionary” derivation? For buried in Bose’s factorial formula is a very deep and bold assertion. This formula implies that interchange of two photons in the photon gas does not lead to different physical states, unlike the standard, classical, Boltzmann assumption for the atomic gas. Boltzmann, and everyone else after him, assumed that even though atoms were very small and presumably all “looked” the same, one could imagine labeling them and keeping track of them. And if photons were particles like atoms, one should be able to do the same thing. Photon 1 having momentum toward the north and photon 2 having momentum toward the south was a physically different condition from photon 2 north, photon 1 south. Bose, without saying a word about it in his paper, implicitly denied this was so!
Late in his life Bose was asked about this critical hypothesis concerning the microscopic world, which followed from his work. He replied with remarkable candor:
I had no idea that what I had done was really novel…. I was not a statistician to the extent of really knowing that I was doing something which was really different from what Boltzmann would have done, from Boltzmann statistics. Instead of thinking about the light quantum just as a particle, I talked about these states. Somehow this was the same question that Einstein asked when I met him: How had I arrived at this method of deriving Planck’s formula? Well, I recognized the contradictions in the attempts of Planck and Einstein, and applied statistics in my own way, but I did not think that it was different from Boltzmann statistics [italics added].
Recall that in his derivation Bose was guided by the knowledge of the end point, the precisely known formula for the Planck radiation law. So he did not have to convince himself in advance that his counting method was justified; it turned out to be the method that gave the “correct” answer, so it would seem to be justified a posteriori. He appears to have missed the fact that in asserting this new counting method, he had made a profound discovery about the atomic world, that elementary particles are indistinguishable in a new and fundamental sense.
Einstein, despite his initial enthusiasm for the prefactor derivation, very quickly grasped that the truly significant but puzzling thing about the Bose work was the postclassical counting method. Soon after receiving Bose’s paper, he expressed this in a letter to Ehrenfest: “[the Bose] derivation is elegant, but the essence remains obscure.” He would pursue and ultimately elucidate this obscure essence for the next seven months.
Bose, not having fully appreciated the novelty of his first paper, placed a great deal of emphasis on his second paper, which was dated June 14, 1924, and sent to Einstein immediately on the heels of the first one. This paper is not a rederivation of a known result, as is his previous paper, but an attempt to reformulate the quantum theory of radiation, in direct contradiction to Einstein’s classic 1916 work. The paper, titled “Thermal Equilibrium of the Radiation Field in the Presence of Matter,” proposes a bold hypothesis. While there still is a balance between quantum emission and absorption leading to equilibrium, the emission process is assumed to be completelyspontaneous and independent of the presence or absence of external radiation. Bose has eliminated Einstein’s hypothesis of stimulated emission, which he refers to as “negative irradiation,” saying it is “not necessary” in his theory. To make things balance out, he then has to assume that the probability of absorption also has a different and more complicated dependence on the energy density of radiation than Einstein assumed.
Einstein, who was a virtuoso at finding absurd implications of flawed theories since his days at the patent office, published a decisive critical note appended to his translation of this paper. First, he notes that Bose’s hypothesis “contradicts the generally and rightly accepted principle that the classical theory represents a limiting case of the quantum theory…. in the classical theory a radiation field may transfer to a resonator positive or negative energy with equal likelihood.” Second, Bose’s strange hypothesis about the nature of absorption implies that a “cold body should absorb [infrared radiation] less readily than the less intense radiation [at higher frequencies]…. It is quite certain that this effect would have been already discovered for the infrared radiation of hot sources if it were really true.” Because of these compelling arguments, Bose’s only published attempt to extend quantum theory had no influence on the field and is solely of historical interest.
Nonetheless, Einstein’s recognition of Bose’s first paper transformed his professional situation in an instant. Einstein’s supportive postcard to Bose congratulating him on his “beautiful step forward” was shown to the vice-chancellor at Dacca, and it “solved all problems.” Bose recalled, “that little thing [the postcard] gave me a sort of passport for [a two-year] study leave [in Europe] … on rather generous terms…. Then I also got a visa from the German consulate just by showing them Einstein’s card.”
By mid-October of 1924 Bose had arrived in Paris and was introduced to the noted physicist Paul Langevin, who was a personal friend of Einstein’s. Bose immediately wrote to Einstein, asking Einstein’s opinion on his second paper (which he was unaware had already been published with Einstein’s assistance) and expressing his desire to “work under you, for it will mean the realization of a long cherished hope.” Einstein quickly replied, “I am glad I shall have the opportunity soon of making your personal acquaintance,” then summarized his reasons for rejecting Bose’s conclusions in the second paper and concluded by saying, “we may discuss this together in detail when you come here.”
Despite the warmth of Einstein’s reply, Bose was reluctant to move on to Berlin immediately, in part because Einstein’s potent critique had made him unsure of his new proposal, which he wanted to refine further. In addition he seems to have found the change of cultures challenging; he decided to settle in Paris in the company of a local circle of Indian compatriot intellectuals. He justified this as follows: “because I was a teacher … and had to teach both theoretical and experimental physics … my motivation then became to learn all about the techniques I could in Paris … radioactivity from Madame Curie and also something of x-ray spectroscopy.” An interviewer of Bose in 1972 noted that “even more than forty years later one still has the impression that Bose was terribly intimidated by most Europeans.” This no doubt contributed to his disastrous interview with Madame Curie, concerning joining her lab. She had hosted a previous Indian visitor and had fixed it in her mind that the collaboration had failed because of his poor French. Thus she conducted her first interview with Bose entirely in English, and while welcoming him warmly, firmly insisted he would need four months of language preparation before starting work. Although “she was very nice,” Bose, who had studied French already for ten years, found no opportunity to interrupt her monologue. And so “I wasn’t able to tell her,” he later explained, “that I knew sufficient French and could manage to work in her laboratory.”
Despite this missed opportunity, Bose tarried in Paris nearly a full year learning x-ray techniques before working up the courage to move on to Berlin in October of 1925 and finally meeting Einstein a few weeks later. In the intervening year, Einstein had taken up Bose’s novel counting method and extended it to treat the quantum ideal gas, leading to truly remarkable discoveries, about which Bose was unaware. For Bose, “the meeting was most interesting…. he challenged me. He wanted to find out whether my hypothesis, this particular kind of statistics, did really mean something novel about the interaction of quanta, and whether I could work out the details of this business.” During Bose’s visit, Werner Heisenberg’s first paper came out on the new approach to quantum theory known as matrix mechanics (of which we will hear more later). Einstein specifically suggested that Bose try to understand “what the statistics of light quanta and the transition probabilities for radiation would look like in the new theory.”
However, Bose was not able to make progress. He seems to have had a difficult time assimilating these rapid new developments and wrote somewhat despairingly to a friend: “I have made an honest resolution of working hard during these months, but it is so hard to begin, when once you have given up the habit.” Bose received extensive access to the scientific elite of Berlin through Einstein’s patronage and experienced the whirlwind of excitement around the revolution in atomic theory. But no publication resulted from his stay in Europe, and in late summer of 1926 he returned to Dacca. By then the new quantum mechanics had passed him by.
FIGURE 24.1. S. N. Bose photographed in Paris in 1924. Courtesy of Falguni Sarkar, SN Bose Project, www.snbose.org.
Bose became a revered teacher and administrator in his subsequent career in India, but he published little, and nothing that has survived in the scientific canon. He continued to write to Einstein, periodically, and late in Einstein’s life tried to visit him in Princeton, but he was denied a visa because “your senator McCarthy objected to the fact that I had seen Russia first.” He eulogized Einstein eloquently upon his death: “His indomitable willnever bowed down to tyranny, and his love of man often induced him to speak unpalatable truths which were sometimes misunderstood. His name would remain indissolubly linked up with all the daring achievements of physical sciences of this era, and the story of his life a dazzling example of what can be achieved by pure thought.” For his own part, Bose seemed content with his role in scientific history, summing up his career aptly: “On my return to India I wrote some papers … they were not so important. I was not really in science any more. I was like a comet, a comet which came once and never returned again.”
CHAPTER 25
QUANTUM DICE
Just under two years before Einstein’s famous rejection of the new quantum mechanics with the memorable phrase “I … am convinced that [God] is not playing at dice,” Einstein himself, inspired by Bose, changed the laws governing the playing of dice. Bose had unwittingly introduced a new method of counting the states of a physical system in order to derive the Planck law from direct consideration of a gas of light quanta, treated as particles, not waves. It was Einstein who would now explain and extend this new representation of the microscopic world to resolve long-standing paradoxes in gas theory and to reveal dramatic and previously undreamed-of behavior of atomic gases at low temperature.
Einstein had become renowned as the young genius of statistical physics (“Boltzmann reborn”) through the sponsorship of Nernst fifteen years earlier, when Nernst realized that only Einstein’s radical quantum theory of the specific heat of solids would validate his own famous “heat theorem”: that the entropy of all systems should tend to zero as the temperature goes to zero. This fortunate confluence of Einstein’s quantum principles and the interests of the most powerful scientist in Germany had played a significant role in winning Einstein his comfortable Berlin existence, free of teaching and administrative responsibilities. Einstein was now to add a sequel to this story.
Nernst had been arguing since 1912 that something similar to Einstein’s freezing of particles into their lowest quantum states must occur for a gas of atoms or molecules at sufficiently low temperatures. However, how this would come about for a gas was a major puzzle. Gas particles are free to move over macroscopic distances, unlike electrons bound to atomic nuclei. In quantum theory, the larger the volume over which a particle is constrained to move, the smaller is its lowest allowed energy level, known as its “ground state.” When you worked out this amount of energy for a gas particle in a container of human scale, it was absurdly small compared with the thermal energy scale, kT, even when the temperature was reduced to a few degrees above absolute zero.1 So gas particles did not freeze out in the same way that vibrations of a solid did, according to the now-accepted form of Einstein’s 1907 theory, refined by Peter Debye. Planck, Sommerfeld, and others had analyzed gases from the point of view of quantum mechanics and had failed to find an entropy function that obeyed Nernst’s theorem. Of course, as we have learned, entropy is all about counting possibilities, and all the previous attempts had counted possibilities from the same point of view as Boltzmann. This point of view regarded atoms or molecules, even if identical in appearance, as distinct, distinguishable entities, in the same self-evident sense that a well-made pair of dice are identical in appearance but are distinct entities. It was this very obvious but very fundamental extrapolation from our macroscopic world that Bose had implicitly denied, and which Einstein would now explicitly deny. Einstein would yet again tell the world that our collective intuition about commonsense properties of the natural world is mistaken.
Einstein must have realized immediately, upon reading it, that Bose’s approach would allow him to resolve the decade-old problem of the quantum ideal gas. For on July 10, 1924, just a few weeks after receiving, translating, and submitting Bose’s first paper for publication, Einstein was reading his own paper, titled “Quantum Theory of the Monatomic Ideal Gas,” to the Prussian Academy. This was the work to which he had alluded already in his famous “Comment of the Translator” published at the end of Bose’s first paper: “The method used here also yields the quantum theory of the ideal gas, as I will show in another place.” He minces no words in his opening to the gas theory paper: “A quantum theory of the … ideal gas free of arbitrary assumptions did not exist before now. This defect will be filled here on the basis of a new analysis developed by Bose…. What follows can be characterized as a striking impact of Bose’s method.”
In the next section of the paper Einstein directly follows the same computational method that Bose applied to the gas of light quanta, now applied to a gas of atoms. The analysis differs in only two significant ways. First, as Bose correctly assumed, a gas of photons loses energy as it is cooled simply by the disappearance of photons. As we already know, according to quantum theory, a photon is absorbed and disappears when it excites an electron in an atom to a higher energy level (and similarly can appear out of nothing when that atoms reemits energy and the electron quantum jumps back down to the lower level). This is the process that Einstein analyzed in detail in his famous 1916 paper, which set Bose on his quest for the perfect derivation of Planck’s law. The total number of photons inside a box decreases as the box is cooled. The situation for an atomic gas is quite different. Atoms cannot just disappear,2 so in analyzing the atomic gas, unlike the photon gas, Einstein has to add the constraint that the number of gas particles is fixed. Second, unlike photons, which always move at the speed of light, gas particles can lose energy simply by slowing down. For an ideal gas, which is the case Einstein is considering, in fact all the atomic energy is in the kinetic energy of motion of the atoms.3
With the constraint of a fixed number of atoms, Einstein correctly derives all the fundamental equations of the quantum ideal gas, which turn out to be substantially more complicated than those for the photon gas and do not lead to a relatively simple formula (“equation of state”) analogous to PV = RT, which describes the classical gas. Thus Einstein has to employ a subtler mode of analysis of these equations. He identifies a “degeneracy parameter,” a ratio of variables that, if much larger than one, will lead back to the classical equation, PV = RT, but, if it approaches one, will lead to a new and different gas law. Thus this parameter measures the “quantumness” of the gas, and since it decreases with decreasing temperature, the theory implies that quantum effects will become more and more important the colder the gas becomes. To see if these deviations from the usual law will be observable, he plugs in numbers and finds that for a typical gas at room temperature this degeneracy parameter is very large, about 60,000, consistent with observations that all gases at room temperature obey the classical law (PV = RT) extremely well, and that the gas molecules obey the equipartition theorem, Emol= 3kT/2, with no hint of quantum effects.
Next he analyzes what form the quantum corrections to the usual behavior will take if the temperature, and hence the degeneracy parameter, can be decreased to the point where deviations from classical behavior are no longer too small to observe. Sure enough, he finds that the energy per particle begins to drop below the equipartition value; so some precursor of quantum freezing is beginning to take place in the gas despite its macroscopic scale. Thus his results hint that Bose’s statistical method will restore Nernst’s theorem even for the ideal gas.
It seems unlikely that Einstein realized the full implications of Bose’s approach when he wrote this first paper, since when he introduces Bose’s new counting method, just like Bose, he does not explain or defend it with even a single sentence. Evidently the realization of just how strange the implications of this new statistical theory are had not yet fully dawned on Einstein. Hence his comment to Ehrenfest, in a letter sent two days after presenting his paper to the academy, admitting that the essence of the new approach is “still obscure.” By September, two months later, he hints in a letter to Ehrenfest that things are becoming clear but that the implications are so strange as to raise doubts: “the theory is pretty, but is there also any truth in it?” By early December he was ready to commit: “The thing with the quantum gas turns out to be very interesting,” he wrote again to Ehrenfest. “I am increasingly convinced that very much of what is true and deep is lurking behind it. I am happily looking forward to the moment when I can quarrel with you about it.” So what did Einstein realize about Bose’s method that makes its implications so interesting and deep? How can something as mundane as a statistical counting method lead to a revolution in our physical worldview?
Any serious gambler knows that the laws of statistics are laws of nature, just as surely as is gravitational attraction. Games of chance are based on systems that are chaotic and unpredictable, such as a ball bouncing around on a rotating roulette wheel, or a pair of dice flung forcefully onto a surface. Since each toss is slightly different, and the final resting position of the dice depends sensitively on the small details of each throw, these events are effectively random processes, in which the probability that each face will turn up is the same, and equal to one-sixth. Moreover, what face turns up on one of the dice is completely independent of what face turns up on the other die. From these simple principles it is possible to work out the consequences of rolling a pair of dice many times, to the point where a casino can make an extremely reliable income from dice-based games.
Games of chance, such as dice or cards, are all based on the same underlying statistical principle: each specific configuration of the basic units (cards, dice, coins) is equally likely; this is exactly the same assumption as underlies the entropy concept in statistical physics. The atomic world behaves like a huge number of many-faced dice, constantly being rolled and rerolled; in fact Bose’s combinatorial formula is essentially a statement about the number of states available when a huge aggregate of many-sided dice are thrown. To understand the strangeness of his answer, consider the simple case of throwing two dice. The available configurations are naturally specified by a pair of numbers, the number facing upward on die one and the number facing upwards on die two; for example (1, 4) is a specific configuration in which die one shows a 1 and die two shows a 4. Each of the thirty-six possible pairs [(1, 1), (1, 2), (2, 1), … (5, 6), (6, 6)] is then equally likely to occur. However, the statistics gets somewhat more interesting when one looks, not at a specific configuration, but at the total score in a throw, the sum of the two numbers defining a configuration. Now one quickly realizes that there are six configurations adding up to seven (i.e., six ways to roll a seven) and there is only one way to roll a two. Thus the chance of rolling a seven is 6/36 = 1/6, and of rolling a two is 1/36. These calculations, and all other statistical properties of dice, follow directly from the fact that there are two distinct, independent dice, each of which randomly shows one of its faces when thrown, and that each throw is independent.
Now, if you have a pair of different-color dice (e.g., die one red, die two blue) and you keep track as you roll many times, you will surely find that (red = 3, blue = 4) and (red = 4, blue = 3) occur roughly an equal number of times, and you can tell that some of your sevens come from (3, 4) and some from (4, 3). However, suppose someone makes for you a pair of dice so perfectly matched that they are completely identical to your eye, and you put the dice in a closed box and shake them before making the throw. In this case every time you get a four and a three you will not be able to tell whether it is (3, 4) or (4, 3). Do you expect this to make any difference in the probability of getting a seven? Absolutely not. This probability is a law of physics: there are two distinct, independent physical possibilities, which the laws of dynamics may or may not lead to in a given roll, and we must add the probabilities for each to occur to get the right answer. It matters not at all if we can tell which possibility actually occurred.
What, then, about the behavior of two atoms (or electrons) being distributed by some complex microscopic dynamics into, say, six different quantum energy levels? The two atoms are then like two “quantum dice,” and the energy level each atom occupies is analogous to the face of the die that comes up. If atoms are independent, distinct objects, no matter how much they look identical, one would have to conclude that having atom one in level three, and atom two in level four, is a different possibility from atom one in four and atom two in three. And therefore that these two possibilities must both contribute to the number of possible states (i.e., both contribute to the entropy of the system). One would be wrong.
This is the mind-bending, if unappreciated, assumption behind Bose’s method of counting light quanta, which Einstein adopted for atoms and which he must have fully grasped only sometime after his first paper on the atomic ideal gas. The new principle is that, in the atomic realm, the interchanging of the role of two identical particles does not lead to a distinct physical state. This has nothing to do with whether a physicist chooses to regard these states as the same, or doesn’t know how to distinguish them: they are not distinct. This is an ontological and not an epistemological assertion.
How do we know this? Consider again our quantum dice. According to Bose-Einstein statistics, there are now only twenty-one possible configurations, not thirty-six. The six doubles are still there as before [(1, 1), (2, 2) …]. The number of these states didn’t change when we switched over to quantum dice; even with classical statistics there is only one way to get snake eyes, or double deuces, etcetera. But now, for the thirty other configurations, where the two numbers are different, we identify them pairwise, leaving only fifteen. Configurations (3, 4) and (4, 3) are merged into a single entity of “three-four-and-four-three-ness,” and similarly for all the other unlike pairs. Now, suddenly, our dice behave differently. Instead of seven being the most likely score, six, seven, and eight are all equally likely and have probability 1/7 of occurring. (With the new rules one might be tempted to sneak a pair of quantum dice into a classical casino and make a killing.)
But there is a further change in the probabilities, which has a profound significance in physics. With the Bose-Einstein approach, the probability of rolling doubles has greatly increased. Classically the chance of rolling doubles is 6/36 = 1/6 = 16.6 percent; switching to the quantum dice makes it 6/21 = 28.5 percent, increasing the odds of doubles by more than 70 percent. With Bose-Einstein statistics there are fewer configurations available in which the particles do different things, and as a result the particles have a tendency to bunch together in the same states! And the more particles there are, the more there is the tendency to bunch. For three quantum “dice” the probability of rolling triples is more than twice as large as it would be if the classical statistics of distinguishable dice held sway. With a trillion trillion quantum particles, as in a mole of gas, this effect is enormous; it literally changes the behavior of matter.
Fine, but do we really care that much about what happens when you swap atoms? Well, we should. Because it is very hard to think of atoms as particles in our usual everyday sense when they lack this individuality. After all, just as we could imagine painting one die red and the other blue (i.e., labeling them), can’t we somehow label atom one and atom two, and distinguish them? No, we can’t (according to Einstein). Atoms are fundamentally indistinguishable and impossible to label. Nature is such that they are not separate entities, with their own independent trajectories through space and time. They exist in an eerie, fuzzy state of oneness when aggregated. So the Bose-Einstein statistical worldview, coming from a different direction, reinforces the concept of wave-particle duality, in this case applied to both light and matter, and heralds the emerging discovery that the microscopic world exists in a bizarre mixture of potentiality and actuality.
Einstein lays out this revolutionary idea in his second paper, read to the Prussian Academy on January 8, 1925, where he also predicts a totally unexpected condensation phenomenon that would have a profound influence on quantum physics up to the present. He introduces the new paper as follows: “When the Bose derivation of Planck’s radiation formula is taken seriously, then one is not permitted to ignore it as a theory of the ideal gas; when it is correctly applied, the radiation is recognized as a gas of quanta, so the analogy between the gas of quanta and the gas of molecules must be a complete one. In the following, the earlier development will be supplemented by something new, which seems to me to increase the interest of the subject.”
The interesting “something new” is first presented as a mathematical paradox. In his first paper he derived an equation relating the density of the quantum ideal gas to the temperature of the gas. Upon close inspection one notices an odd feature of this equation. On the left-hand side of the equation sits the density of the gas in a container, a quantity that can be increased indefinitely simply by compressing the volume of the container, which is kept at a fixed temperature.4 But on the right-hand side is a mathematical expression that varies with temperature but cannot get larger than a certain maximum value if the temperature is fixed. This leads to an apparent contradiction, as Einstein points out: this equation violates the “self-evident requirement that the volume and temperature of an amount of gas can be given arbitrarily.” What happens, he asks, when at fixed temperature one lets the density increase by compressing it into a smaller volume until the density becomes greater than the maximum allowed?
Having posed the question, he brilliantly resolves it with a bold hypothesis: “I maintain that in this case … an increasing number of the molecules go into the quantum state numbered 1 [the ground state], the state without kinetic energy…, a separation occurs; a part [of the gas] ‘condenses,’ the rest remains a ‘saturated ideal gas.’ ” Here he is making an analogy to an ordinary gas, like water vapor, which when cooled reaches a temperature at which it begins to condense partly into a liquid while still retaining a particular ratio of liquid to vapor.5 The reason that his hypothesis resolves the paradox is that in deriving the relation of density to pressure in his original paper he made an innocent mathematical transformation, which amounted to neglecting the single quantum state of the gas where each molecule has zero energy.6 Never before in the history of statistical physics had the neglect of a single state made any difference to the value of a thermodynamic property of a gas, such as its density. On the contrary, the number of states involved, as we saw earlier, is normally unimaginably large, and physicists routinely make approximations that neglect billions of states without giving it a second thought. But Einstein unerringly recognized that in this new world of Bose-Einstein statistics, this single zero-energy state would gobble up a macroscopic fraction of all the molecules, creating a novel quantum “liquid,” now known as a Bose-Einstein condensate.
The generosity of the “Bose-Einstein” designation is not widely appreciated, as few physicists realize that Bose said not a word about the quantum ideal gas in his seminal paper. The paper that does predict quantum condensation belongs to Einstein alone, and it is a masterwork. The boldness of the young rebel combines with the technical virtuosity of the mature creator of general relativity to reach breathtaking conclusions with complete self-assurance. A lesser physicist would either not have noticed the subtle mathematical error introduced by the neglect of a single state or, even if noticed, would likely have dismissed its logical implications as so bizarre as to indicate some fundamental error. The reason that this condensation phenomenon seems so strange, even today, is that condensation of an ordinary gas is caused by the weak attraction between the gas molecules, which becomes important only when the gas is relatively dense. But Einstein is considering the theory of the ideal gas, in which such molecular interactions are assumed to be completely absent. His condensation phenomenon is driven purely by the newly discovered quantum “oneness” of identical particles, not by a force like electromagnetism, but by this strange statistical “pseudoforce” that Einstein was the first to recognize. Proposing it as a real physical phenomenon was an act of great courage.
Bose-Einstein condensation is now one of the fundamental pillars of condensed-matter physics; it underlies the phenomena of superconductivity of solids and superfluidity of liquids such as helium at low temperatures,7which have been the subject of five Nobel prizes. These substances have substantial interaction forces between atoms and electrons, unlike the ideal gas of Einstein’s theory, although it is clear theoretically that the “statistical attraction” of bosonic particles plays the key role in generating their unique properties. Nonetheless, it was big news and Nobel-worthy yet again when, in 1995, atomic physicists finally realized a holy grail of the field. They created an atomic gas with negligible interactions, cold enough8 to observe pure Bose-Einstein condensation—Einstein’s last great experimental prediction, coming to fruition a lifetime after its first statement.
And again, as before, Einstein’s progress was a step too far for the physics world; it was ignored by the leading atomic physicists, such as Bohr, Sommerfeld, and Max Born, and by the soon-to-be-famous Werner Heisenberg and Wolfgang Pauli. Even Einstein’s admired colleague Planck and his great friend Ehrenfest, the former student of Boltzmann, thought it was unacceptable; they believed that the Bose-Einstein statistical method was just plain wrong. In the condensate paper Einstein responds explicitly to their criticism: “Ehrenfest and others have reported that in the Bose theory of radiation and in my [ideal gas theory], the quanta or molecules do not act in a manner statistically independent of each other … this is entirely correct…. The formula [for counting states according to Bose and Einstein] expresses indirectly an implicit hypothesis about the mutual influence of the molecules of a totally new and mysterious kind.” That’s it. Statistical independence (which here means the classical method of counting configurations) is out; statistical attraction is in. Quantum particles bunch; get used to it.
Einstein then goes on to show that only Bose-Einstein statistics can save two things statistical physicists hold dear: Nernst’s Third Law of thermodynamics and the additivity of entropy (i.e., when one combines two systems their total entropy is the sum of their individual entropies), a less obvious but fundamental requirement for thermodynamics to make sense. Having carried physics to the very brink of the new quantum theory, he rests his case.
While Einstein’s breakthrough did not have much impact on the mainstream of quantum research, a lesser-known but already well-respected physicist had been following carefully Einstein’s papers on the quantum ideal gas, the Austrian Erwin Schrödinger, who had recently been appointed professor of theoretical physics at the University of Zurich (Einstein’s first academic position). Schrödinger had previously worked on both general relativity theory and on the quantum ideal gas, and he had met Einstein recently in Innsbruck, shortly after his first paper on the ideal gas had been published. He wrote to Einstein in February of 1925, when he did not yet know of Einstein’s landmark second paper on condensation, expressing skepticism about the validity of his application of Bose’s method to atoms in his first paper. Einstein replied a few weeks later with characteristic good humor: “your reproach is not unjustified, although I have not made a mistake in my paper…. In the Bose statistics, which I use, the quanta or molecules are not considered as being mutually independent objects.” He then drew a small diagram illustrating the case of just two particles in two states, pointing out that there is only one configuration in which the particles are in different states instead of two, as there would be in classical statistics. To Schrödinger, Einstein’s reply was a revelation: “only through your letter did the uniqueness and originality of your statistical method of calculation become clear to me,” he wrote in November of 1925. “I had not grasped it before at all despite the fact that Bose’s paper had already come out…. [Bose’s work] did not seem particularly interesting to me. Only your theory of gas degeneracy is really something fundamentally new.”
FIGURE 25.1. First page of Einstein’s letter to Schrödinger of February 28, 1925 in which he explains how Bose statistics differs from classical statistics, using the diagram at the bottom of the page. The example is equivalent to two quantum coins: on the right Einstein lists the four states of two ordinary coins, whereas on the left he lists the three states of quantum coins, since heads-tails and tails-head are indistinguishable quantum mechanically. Austrian Central Library for Physics.
Schrödinger wanted to follow up on the suggestion, which Einstein had made at the end of the paper, that the strange statistics could be understood using the concept of matter waves, recently introduced by the young French physicist Louis de Broglie. Within months this newly inspired research direction would fundamentally change the emerging modern form of quantum theory, and enshrine Herr Schrödinger in the eternal pantheon of physics.
1 Such extremely low temperatures were now available to physicists since 1908, following the Nobel Prize–winning research of the Dutch physicist H. Kammerling Onnes on liquefying and cooling helium. Einstein and Onnes were well acquainted, having met at the First Solvay Congress, and interacted frequently during Einstein’s regular visits to Leiden.
2 At very high energies there are processes in which massive particles, such as an electron and a positron, annihilate one another and disappear, but such processes are not relevant here.
3 It is a good approximation to neglect the interaction energy between atoms, based on the assumption that the gas is dilute enough that most of the time atoms are well separated, so that their mutual interaction is weak.
4 The gas can be kept at a fixed temperature as its density (and hence its pressure) increases simply by making the container out of a heat-conducting material and putting it in contact with a large “bath” that will add or subtract the necessary heat.
5 This state of an ordinary gas-liquid is referred to as “phase-separated,” and the gas is termed “saturated,” hence Einstein’s use of the term in this new context.
6 Quantum mavens might worry that having a particle at rest in a finite volume violates the uncertainty principle. They would be right; “zero energy” here means an essentially infinitesimal energy, much less than the natural thermal energy scale kT, determined by the macroscopic size of the container for the gas.
7 Superconductivity is a phenomenon whereby, at low temperatures, a solid such as aluminum conducts electricity with no resistance and hence no dissipation of energy; superfluidity is a similar effect in liquids, whereby they lose their viscosity and flow without dissipating energy.
8 The required temperature in these experiments was 170 billionths of a degree above absolute zero, by far the coldest temperature ever created by man at that time.
THE ROYAL MARRIAGE: E = mc2 = hυ
I said to myself that classical physics wasn’t sufficient, that all of the ancient edifice … was shaken and that it was necessary to reconstruct the edifice; but I didn’t think it was necessary to change fundamental notions, I thought that it was by the introduction of new things, completely unknown, that one would take into account quanta…. For example by a synthesis between waves and particles … it was more or less the point of view of Einstein.
—LOUIS DE BROGLIE
“A younger brother of the de Broglie known to us has made a very interesting attempt to interpret the Bohr-Sommerfeld quantization rules in his dissertation. I believe that it is the first feeble ray of light to illuminate this, the worst of our physical riddles. I have also discovered something that supports his construction.” So Einstein wrote to Lorentz in December of 1924, just as he was completing his masterpiece on the quantum ideal gas. In the midst of his cogitations on the meaning of Bose statistics, during the summer of 1924, a second bolt from the blue singed his mail slot. This revelation came in the form of a doctoral thesis, sent to him by his old friend Paul Langevin, the French physicist, with the somewhat skeptical assessment, “[the thesis] is a bit strange, but after all, Bohr also was a bit strange, so see if it is worth something all the same.” Just as with Bose, Einstein was willing to look at this young man’s ideas with an open mind and recognize the crucial insight they represented. He wrote back to Langevin with a warm and eloquent endorsement: “Louis de Broglie’s work has greatly impressed me. He has lifted a corner of the great veil. In my work I [have recently] obtained results that seem to confirm his. If you see him please tell him how much esteem and sympathy I have for him.”
Louis de Broglie, like Bose, had venerated Einstein from his earliest exposure to modern physics, but, unlike Bose, he had good reason to expect that Einstein might someday take his ideas seriously. Louis’ older brother, Maurice, was one of the most distinguished experimental physicists in France and had been one of two scientific secretaries at the First Solvay Congress of 1911 (hence Einstein’s allusion to the “de Broglie known to us” in his letter to Lorentz). Moreover, the de Broglies were an eminent noble family of France; Maurice and Louis counted among their predecessors ministers, generals, and famous literary figures. Maurice himself held the title of Duke de Broglie, whereas Louis had the inherited rank of Prince of the Holy Roman Empire, awarded to all the direct descendants of his ancestor Duke Victor-François in reward for his martial feats, by order of the emperor Francis I.
Louis Victor Pierre Raymond de Broglie was born on August 15, 1892, the youngest of the five offspring of his father Victor de Broglie. His brother, Maurice, was the second child, seventeen years older than Louis and first son (hence duke). He served as a father figure and mentor to Louis, particularly after the death of Victor in 1906; “at every stage of my life and career,” Louis wrote to his brother, “I found you near me as guide and support.” Maurice had begun a career as a naval officer in 1895 but, against the wishes of his family, abandoned this path in 1904 and devoted himself to science, taking the rather unusual step of installing a private laboratory in the family town house on the tony Rue Chateaubriand, adjacent to the Arc de Triomphe. He would become an enormously influential figure in French physics, successor to Langevin at the College de France, and nominated for the Nobel Prize multiple times, including in the same year, 1925, when his younger brother was first nominated.1
this little brother had become a charming child, slender, svelte, with a small laughing face, eyes shining with mischief, curled like a poodle…. His gaiety filled the house. He talked all the time, even at the dinner table, where the most severe injunctions to silence could not make him hold his tongue…. He had a prodigious memory … and seemed to have a particular taste for … political history…. He improvised speeches inspired by accounts of the newspapers and could recite unerringly complete lists of the Ministers of the Third Republic…. A great future as a statesman was predicted for Louis.
After finishing his secondary education in 1909, graduating from the elite Lycée Janson de Sailly, and obtaining his licence in historical studies, he was encouraged to “continue and prepare the diploma in history.” However, this early passion was dying out, and confusion about his future set in. “I could see that to do that it was necessary to go often to the library, make a large bibliography and such things. That didn’t appeal to me too much, and that was the beginning of the year of my ‘moral crisis.’ ” He studied law for a year while hesitating “between several intellectual directions … and [I] wasn’t very much in agreement with my brother, who would have preferred that I do either the Ecole Polytechnique or study diplomacy…. I wanted to do neither, and yet I didn’t want to follow him [into physics], which increased the crisis a bit.” Finally he committed himself to a course in advanced mathematics, and by the end of the year, according to his brother, “The hesitations are over, he has crossed the Rubicon, and the course of his thoughts has turned towards physics and more particularly theoretical physics.”
The precise timing of this crisis and conversion is not clear, but it overlapped closely with Maurice’s service as secretary at the First Solvay Congress, which afforded Louis a unique window into the ferment of the new atomic theory. De Broglie recalled: “I began to think about quanta from the moment that my brother gave me the notes of the Solvay Congress of 1911, probably at the beginning of 1912,” and “with the ardor of my age I became enthusiastically interested in the problems that had been treated and I promised myself to devote all my efforts to achieve an understanding of the mysterious quanta, which Max Planck had introduced ten years earlier into theoretical physics, but whose deep significance had not yet been grasped.” Two years later, in 1913, he graduated with his licence de science, having performed brilliantly on his exams. “His enthusiasm was returning,” Maurice recalled, “with the certainty of being at last on the right track.”
However, just as the young man was on the verge of entering the intellectual fray of quantum research, all of Europe was engulfed in a more primitive conflict, which led to an indefinite extension of the brief military service to which de Broglie had been called after graduation. He was eventually assigned to a wireless telegraphy unit and found himself working underneath the Eiffel Tower, where a transmitter had been installed. And while his brother recalled that Louis “regretted the interruption of his meditations” on quantum theory and complained of “inspiration broken to pieces,” Louis himself spoke of the experience somewhat fondly, “because that put mein relation with … real things. That made a certain impression, and led me to examine quite closely all the theories of the propagation of waves, all the theories of electronics … that certainly contributed greatly to clarifying my ideas on all this.” Decommissioned in 1919, de Broglie was ready to begin his quest in earnest.
De Broglie was determined to be a theorist, unlike his brother, but this was not nearly as felicitous a choice in France as in the German-speaking countries of Europe, which were leading the abstract developments of quantum theory. De Broglie’s close friend and fellow theorist Leon Brillouin recalled, “There was really no career open for a theoretical physicist in the French organization. People who had curiosity for theory would go right away into pure mathematics…. Many of my colleagues told me ‘Are you crazy? To go into theoretical physics, there is no future.” Nonetheless Brillouin, and de Broglie, joined the small group around Paul Langevin who were working on quantum theory.
De Broglie’s first thought was to focus on the theory of light quanta, which, unlike most other physicists, he had believed in since his first exposure to Einstein’s work in 1912.
I never had any doubt at this time about the existence of photons. I considered that Einstein had discovered them, that they raised many difficulties … but that in the end this was a problem to resolve, that one could not deny the existence of photons…. It must be noted that I was very young. I had not made any theoretical works and for that reason was not as attached to electromagnetic theory, as was Langevin…. Thus, I accepted this as something that must be required.
In addition, his brother Maurice had been very directly grappling with the particulate properties of light in his extensive research on the x-ray photoelectric effect. Louis and Maurice collaborated on the interpretation of these experiments in terms of Bohr’s theory during 1920 and 1921; particularly striking was the fact that the x-ray radiation appeared capable of giving all its energy to a pointlike electron, an observation very difficult to reconcile with a picture of the x-rays as extended, spherically expanding waves. When Maurice spoke of his experiments at the 1921 Solvay Congress (which Einstein had skipped), he concluded that the radiation “must be corpuscular … or if undulatory, its energy must be concentrated in points on the surface of the wave.”
De Broglie’s first significant contribution had a motivation remarkably similar to that of Bose: to derive “a number of known results of the theory of radiation … without the intervention of electromagnetic theory” (i.e., without using classical physics). “The hypothesis we adopt is that of light quanta.” However, unlike every other physicist working on the quantum problem, de Broglie was convinced that the key to unlocking it lay in the appropriate use of relativity theory, which he described as Einstein’s “incomparable insight.” Thus, in the second paragraph of the paper, he introduces a very peculiar notion, that quanta are to be thought of as “atoms” of light, with a very small but nonzero mass. In almost two decades of quantum research neither Einstein nor anyone else in the field had made such an outré suggestion. Photons (quanta) were conceived to have energy, E = hυ, and momentum, p = hυ/c, but their energy and momentum was not accompanied by a rest mass, precisely because photons, by definition, move at the speed of light and can never be at rest in any frame of reference. Nonetheless de Broglie asserts that photons should be conceived of as having a rest mass which satisfies Einstein’s most famous equation, E = mc2! Hence, he states, combining this with the Planck relation, E = hυ, this mass must equal hυ/c2. Never mind that this mass would then vary continuously with the frequency of light, which seemed very odd—in the current work he simply assumes that this mass is “infinitely small,” while the speed of the photon is “infinitely close” to c, so that the usual relation E/c = p for photons still holds, to a very good approximation.2
It must be noted that in the modern quantum theory the mass of the photon is precisely zero, and a finite photon mass played no role in the first full formulations of quantum mechanics, which emerged a mere four years later. Thus this forced marriage of the two most famous equations of modern physics (E = mc2 and E = hυ) ended in a speedy divorce. However, in the first paper de Broglie uses this hypothesis only to rederive a known result of electromagnetic theory, that light exerts pressure and that this pressure is half the value one finds for ordinary nonrelativistic particles.3 From this point on in the paper he uses standard statistical and thermodynamic relations, very much as in Einstein’s work of 1902–1906, to derive the law for blackbody radiation, arriving not at Planck’s formula but rather at Wien’s law.4 The reason for this is that he has, like so many before, assumed that the quanta are statistically independent, and, as was discussed in connection with the work of Bose, this assumption inevitably leads to Wien’s approximation to the Planck law. He misses completely the strange statistical attraction that Bose stumbled upon, and Einstein would elucidate, two years later. The paper is significant for two reasons. First, de Broglie introduces the method of counting single photon states using units defined by Planck’s constant, which Bose would rediscover a couple of years later.5 Second, de Broglie expressly assumes that the formulas of relativity theory are used for “atoms of light” and thus asserts that relativistic mechanics is of central importance in quantum theory, whereas previously it played a very minor role. This novel point of view would shortly lead him to a historic breakthrough.
After his initial foray into the theory of light quanta in 1922, de Broglie became convinced that there must be some symmetry between the behavior of “atoms of light” and other massive particles, such as electrons or atoms. According to his friend Brillouin, he was fascinated by the experimental images of radioactive decay processes, in which massive particles (e.g., electrons and positrons) are emitted from the atomic nucleus and follow curved tracks due to the force exerted by a magnetic field, while the simultaneously emitted photon makes a straight track, since it lacks electric charge. Apparently de Broglie intuitively felt, “Well, all this must be very similar. Either they are all waves or they are all particles … [so he tried] to see if he couldn’t make everything waves.” De Broglie recalled that the key ideas “developed rapidly in the summer of 1923,” perhaps in July. “I got the idea that one had to extend [wave-particle] duality to the material particles, especially to electrons.” In September and October of that year he submitted three short notes for publication, which contained “the essential things” that later entered into his thesis.
In pondering how to associate a wave with material particles, de Broglie had been struggling with an apparent paradox. Just as had he had for photons, he persisted in combining the two great equations of Einstein and Planck and hence associating a frequency with the particle’s rest mass: m = hυ/c2, but now he took the bold step of applying this formula to electrons, for which the mass was not unmeasurably small. De Broglie insisted that this was “a meta law of Nature, [that] to each … proper mass m, one may associate a periodic phenomenon of frequency υ = mc2/h.” In other words de Broglie postulated a sort of internal “vibration” of every particle, which acted like a ticking clock, even when it was at rest. Moreover, if this is so, reasoned de Broglie, then when the particle moves at velocity v, its “ticking” must slow down, because Einstein’s theory of relativity predicts the universal effect of time dilation: clocks are measured to run more slowly when in motion relative to an observer. So the moving particle will appear to vibrate at a lower frequency,6 υ1, than it does when at rest.
However, de Broglie at the same time considered the most basic postulate of relativity theory, that the laws of physics are the same in all frames of reference, and hence that the energy of the moving particle must still be related to some frequency via Planck’s constant (i.e., E = hυ must still hold when the particle is moving). But since the energy of the particle is larger in the frame in which it is moving (it has kinetic energy, ½ mv2, in addition to its rest mass energy), then to satisfy the Planck relation it must have a higher frequency,7 υ2, than its “rest frequency,” υ. So there were two frequencies one should associate with the particle motion, one larger than its rest frequency, one smaller; which one was physically relevant?
Both, was de Broglie’s answer. While the particle is moving, “it glides on its wave, so that the internal vibration of the particle (υ1) is in phase with the vibration of the wave (υ2), at the point where it finds itself.” The “fictive wave” that guides the particle moves at just the right speed so that the wave peak coincides with the peak of the particle oscillations; the particle is like a lucky surfer, permanently attached to the crest of the perfect wave. De Broglie showed that, for this to be so, the velocity of the “phase wave” (as he termed his fictive waves) must have a specific value, Vphase = c2/v, which is faster than the speed of light.8 Because the phase waves moved faster than light, they could carry no energy, according to relativity theory, but served only to guide the particle motion.
So that is de Broglie’s picture: every particle has some unspecified internal oscillation, which must remain in phase with a mysterious steering wave that directs its motion but which moves ahead faster than light so as to remain always “in resonance” with the particle’s oscillation. Even by the standards of the new quantum theory, this was a rather wild invention, and, if anything, Langevin greatly understated the case when he told Einstein it was “a bit strange.”
But de Broglie had at least one further result that supported his extreme conjectures. He took this picture and applied it to an electron circulating around a hydrogen atom. The electron would move around in a circular orbit, continually emitting these phase waves, which would zoom ahead of the particle and lap the particle almost 19,000 times for each electronic circuit. Again he asked the question: what was required so that each time a wave crest and the particle coincided, the particle oscillation and the wave oscillation were in phase? Almost miraculously, he was able to show that this requirement was equivalent to the Bohr-Sommerfeld rule for the allowed electron orbits in hydrogen.9 It was this result that apparently impressed Einstein, who referred to it, in his paper on Bose-Einstein condensation, as “a very remarkable geometric interpretation of the Bohr-Sommerfeld quantization rule.”
While de Broglie’s key ideas were developed in the fall of 1923, he prepared a longer and more comprehensive document as his thesis in the spring of 1924, and gave it to his adviser, Langevin, who clearly was concerned about whether by accepting the work he would be endorsing nonsense. “It looks far-fetched to me,” was his initial reaction, shared with a colleague. At some point in the spring Langevin spoke to Einstein, who was intrigued and agreed to look at a copy. Einstein “read my thesis during the summer of 1924” de Broglie recalled, and wrote the very favorable report quoted above. “As M. Langevin had great regard for Einstein, he counted this opinion greatly, and this changed a bit his opinion with regard to my thesis.”
De Broglie then defended his thesis in November of 1924 before a distinguished but bewildered “jury” (as it is termed in France), chaired by the future Nobel laureate Jean Perrin and including the famous mathematician Elie Cartan, the eminent crystallographer Charles Mauguin, and his adviser, Langevin. While their verdict was positive and de Broglie was congratulated for his “remarkable mastery,” a student who attended the thesis defense later remarked, “never has so much gone over the heads of so many.” Maurice de Broglie sought a candid opinion from Perrin and was told, “all I can tell you is that your brother is very intelligent.”
De Broglie’s thesis had come to Einstein’s attention at the perfect time. Einstein was now deep into his second paper analyzing the statistical properties of the quantum atomic gas. And in addition to his realization that the principle of indistinguishability of particles is implicit in Bose statistics, and the possibility of quantum condensation this implied, he had made one more major mathematical discovery. Just as he had done for the gas of light quanta, which he had analyzed in the seminal work of 1909, he now looked at the fluctuations of the energy in a particular volume of the quantum gas of particles. For both the photon gas and for a gas of atoms in a box, the energy in a small region of the box can vary randomly in time, while maintaining the same energy on average. This is simply a reflection of the ceaseless give-and-take that corresponds to thermal equilibrium. Einstein’s earliest insight into the wave-particle duality of light had come in 1909, when he derived a formula for the typical magnitude of these energy fluctuations. He found that it consisted of two contributions, one of which could be explained by the interference of light waves, but the other of which looked exactly like the fluctuations expected from a gas of particles with energy E = hυ. It was this latter, particulate term that was a revelation in 1909, supporting Einstein’s hypothesis of light quanta and prompting him to declare that the future quantum theory would involve a “fusion” of the wave and particle concepts. Now, after adopting Bose statistics, he finally had the correct theory to evaluate the same quantity for the atomic gas. He found that exactly the same structure occurs: the fluctuations are the sum of two terms, a “particle term” and a “wave term.” But in this case it is the “wave term” that is the surprise.
This parallelism between atoms and light impressed Einstein greatly because, as he says in his second quantum gas paper, “I believe that it is more than just an analogy, since a material particle … can be represented by a … wave field, as de Broglie has stated in a remarkable thesis.” He continues: “this oscillating field—whose physical nature is still obscure—must, in principle permit itself to be demonstrated by phenomena corresponding to its motion. So a beam of gas molecules travelling through an opening must experience a bending, which is analogous to that of a beam of light.” That was the key to testing this extension of wave-particle duality. Whereas before, one was looking for particulate behavior of light waves, such as the localized collisions of photons and atoms in the photoelectric effect, now one should look for wave behavior of particles: a stream of atoms should interfere with itself as it passed near an edge, exhibiting the wave phenomenon known as diffraction. Einstein began suggesting such a search to physicists in September of 1924, shortly after reading de Broglie’s thesis.10
Einstein’s endorsement and extension of de Broglie’s matter wave concept was decisive in bringing this idea under serious study in the physics community. “The scientific world of the time hung on every one of Einstein’s words, for he was then at the peak of his fame,” de Broglie noted. “By stressing the importance of wave mechanics, the illustrious scientist had done a great deal to hasten its development. Without his paper my thesis might not have been appreciated until very much later.” Even with his paper the thesis was viewed with suspicion; a student of Sommerfeld’s recalled, “the paper of de Broglie was studied [in Munich] too; everyone had objections (they were not very difficult to find), and no one took the idea seriously.” Thus the search to observe matter waves, which began almost immediately after Einstein’s second paper on the quantum gas, was mainly pursued in Einstein’s name.
Walter Elsasser, the very first experimenter to find evidence in this direction, illustrates this with the introduction to his initial paper: “By way of a detour through statistical mechanics, Einstein has recently arrived at a physically very remarkable result. Namely he makes plausible the assumption that a wave field is to be associated with every translational motion of a material particle…. The hypothesis of such waves, already advanced by de Broglie before Einstein, is so strongly supported by Einstein’s theory that it seems appropriate to look for experimental tests for it.” Within two years these tests rendered an unambiguous verdict: electrons beams do show wavelike interference. This discovery was so striking that de Broglie, with Einstein’s strong support, received the Nobel Prize in Physics in 1929, a mere four years after his work became widely known.
But had this enormous success corroborated de Broglie’s specific model, involving superluminal phase waves? Not at all. Almost no trace of this concept survives in modern quantum mechanics, and the vast majority of contemporary physicists are completely unaware of the basis on which de Broglie argued for matter waves. The only equation that survives from de Broglie’s thesis work is so simple that Einstein could have written it down any time after 1905.11 This is the famous relation λ = h/p = h/mv, which relates the momentum, p = mv, of a massive particle to its “De Broglie wavelength.” It is obtained in a few lines by an extension, from light quanta to material particles, of Einstein’s equation E= hυ.12 This formula, correctly interpreted, is essential to modern quantum theory but arises there without any appeal to relativity theory or de Broglie’s phase waves.
De Broglie himself, having completed his thesis by age thirty-one, never again made a fundamental contribution to physics, although he remained active in research, unlike Bose. He was an “uninspiring” classroom teacher, who started and ended his lectures precisely on time and permitted no questions during or after them. The research seminars he organized for many years were stilted affairs, with only brief open interchanges, described as “dry and devoid of passion.” The disciples that congregated around him were “not of the highest intellectual caliber” and created “an atmosphere of admiration, not to say adulation.” For example, it was considered bad form to refer to “quantum mechanics,” even after it became the standard term; one was supposed rather to say “wave mechanics,” as homage to de Broglie’s seminal role. His research career spanned more than fifty years (he lived to age ninety-five), and it is now widely acknowledged that his influence was not very positive for the development of theoretical physics in France.
FIGURE 26.1. Prince Louis De Broglie circa 1930. Academie des Sciences, Paris, courtesy AIP Emilio Segrè Visual Archives.
For much of his life de Broglie worked diligently within the standard quantum theory, which emerged from the work of Schrödinger, Heisenberg, and Bohr, although he initially opposed it in 1927. Then, in 1952, at age sixty, he again rejected this approach and joined Einstein in searching for a new and more aesthetically satisfying theory. In 1954, a year before his death, Einstein wrote touchingly to de Broglie, “Yesterday I read … your article on quanta and determinism, and your ideas, so clear, have given me great pleasure…. I must resemble the bird from the desert, the ostrich, hiding its head in the sands of Relativity rather than to face the malicious Quanta. Indeed, exactly like you, I am convinced that one must look for a substructure, a necessity that the present quantum theory hides.”
1 Unlike Louis, Maurice was never awarded the prize, although his research was prominently cited in the 1922 prize, awarded to Niels Bohr.
2 Note that c here can no longer be thought of as the speed of light, but rather the limiting velocity of light as its frequency and hence mass goes to zero.
3 When measured appropriately, in terms of its energy density.
4 De Broglie mentions in passing that if one were to consider not just isolated atoms of light but “a mixture of monatomic, diatomic, triatomic” molecules of light, Planck’s law could be obtained, but then dismisses this as requiring “some arbitrary hypotheses.” He and others followed up this idea, but it was superseded by the concepts of Bose statistics.
5 Like Bose, de Broglie then finds his answer is off by a factor of 2, and he needs to insert this factor “by hand” to account for the two possible polarizations of light (which is a concept of classical electromagnetism, not present in the theory of light quanta at that time).
7 Again from relativity theory, υ2 = υ0(1 –v2/c2)−1/2, so it is higher than υ0 by just the same factor that υ1 is lower.
8 Since any massive particle’s velocity must be less than the speed of light, the wave velocity, Vphase = (c/v)c, is necessarily greater than the speed of light.
9 Suitably generalized to include relativistic effects.
10 De Broglie also had suggested such a search, for interference of electrons, roughly a year earlier.
11 Much later, under direct questioning from the physicist I. I. Rabi, Einstein allowed that he did indeed think of the famous equation λ = h/p for matter waves before de Broglie but didn’t publish because “there was no experimental evidence” for it.
12 The logic is as follows: for a photon, E = hυ, and for light waves E = pc. If we assume both relations hold and use the relationship of wave frequency to wavelength, υ = c/λ, we get λ = h/p. If we assume the same relation holds for massive particles moving slowly compared with the speed of light, so that p = mv, we find λ = h/mv. The full quantum derivation of this is based on Schrödinger’s equation and doesn’t rely on the assumptions that are used in this simple argument.CHAPTER 27
THE VIENNESE POLYMATH
Physics does not consist only of atomic research, science does not consist only of physics, and life does not consist only of science.
—ERWIN SCHRÖDINGER
“When you began this work you had no idea that anything so clever would come out of it, had you?” This question was addressed to the Austrian theorist Erwin Schrödinger sometime in the fall of 1926. The questioner was a young female admirer of the thirty-nine-year-old physicist, whose unusual marriage allowed for many such “friendships.” The work in question was that leading to the most famous equation of quantum mechanics, the “wave equation,” named after its inventor. Schrödinger’s scientific colleagues were less restrained in their praise. The reserved Planck effused, “I have read your article the way an inquisitive child listens in suspense to the solution of a puzzle which he has been bothered about for a long time.” Einstein, who learned of the work from Planck, wrote simply, “the idea of your article shows real genius.”
At the time of this seminal work, Schrödinger was a professor at the University of Zurich, occupying the very same chair that Einstein had once held as his first academic position.1 Schrödinger was in the midst of what he called his “First Period of Roaming,” during which he moved between various positions, as had Einstein fifteen years earlier, ascending the academic hierarchy. Indeed, in 1927, after the great triumph of his wave equation, Schrödinger would end up as Einstein’s colleague in Berlin, after receiving the signal honor of succession to the chair of the recently retired Planck. Even before that, Einstein and Schrödinger had become allies in the struggle and competition to create the new atomic theory, and they shared certain intellectual habits. Schrödinger, like Einstein, did almost all his research alone, unlike the other school of quantum theory involving Bohr, Sommerfeld, Max Born, Werner Heisenberg, Pascual Jordan, and Wolfgang Pauli, who primarily worked collaboratively. Also, Schrödinger and Einstein had a sincere respect for and interest in philosophy,2 and they shared a similar philosophy of science, influenced by the positivism of Ernst Mach but with a strong note of idealism.
However, unlike Einstein, Schrödinger had been appointed at Zurich primarily for his breadth of knowledge, outstanding mathematical abilities, and brilliant intellect—not because of any breakthrough attached to his name. In 1926, when he finally wrote his name into the history of science, he was already thirty-nine years old, well past the age when radical breakthroughs are expected from a theoretical physicist. And in fact his style of research had never before involved a daring leap into the unknown; instead his modus operandi was to criticize and improve the work of others.
In my scientific work … I have never followed one main line, … my work … is not entirely independent, since if I am to have an interest in a question, others must also have one. My word is seldom the first, but often the second, and may be inspired by a desire to contradict or to correct, but the consequent extension may turn out to be more important than the correction.
In a sense, his work culminating in the wave equation was in that vein, building strongly on the insights of Einstein and de Broglie, but in this instance the extension was of historic consequence. In fact the state of quantum theory in 1925 called for just such an outsider, a critic who understood the two main lines of research, the Bohr-Sommerfeld atomic theory and the Einstein–Bose–De Broglie statistical theory of quanta, but who had a sentimental attachment to neither.
Erwin Schrödinger himself, while a man of great personal magnetism, was not known for his sentimental attachments. In his autobiographical sketch, written in his seventies, he reflected that he’d had only one close friend in his entire life and that he had “often been accused of flirtatiousness, instead of true friendship.” Flirtatiousness understates his behavior with respect to the opposite sex. He ends his sketch with the most titillating of disclaimers. “I must refrain from drawing a complete picture of my life, as I am not good at telling stories; besides, I would have to leave out a very substantial part of the portrait, i.e. that dealing with my relationships with women.” Thus we do not learn, for example, the name of the mystery woman (not his wife) who accompanied him on the Christmas ski vacation of 1925 during which the wave equation was discovered.3
Born in 1887 and raised in an imperial Vienna that represented the flowering of art and culture at the turn of the century, Erwin Schrödinger was closer to Einstein’s generation than he was to the rising cohort of brilliant young theorists (Heisenberg, Pauli, Dirac)4 who would join him in driving the quantum revolution to completion. An only child, raised by a doting mother and aunts, he showed great intellectual talent from an early age. His father had studied chemistry at university, and pursued serious interests in art and botany, but contented himself with running the family linoleum business, while investing his son with his unrealized professional aspirations. Homeschooled until the age of eleven, Schrödinger then attended the elite Akademisches Gymnasium, Vienna’s oldest secondary school, where he was the top student in his class for eight straight years. “I was a good studentin all subjects, loved Mathematics and Physics, but also the strict logic of the ancient grammars (Latin and Greek),” he recalled. Unlike Einstein, the independent-minded Schrödinger managed to get along with his teachers and, in looking back, could “only find words of praise for my old school.” His intellectual facility astonished his classmates, one of whom recounted: “I can’t recall a single instance in which our Primus5 ever could not answer a question.”
When he matriculated at the University of Vienna in 1906, his brilliance was already widely known; a friend, Hans Thirring, recalls encountering a striking blond young man in the mathematics library and being told by a fellow student, sotto voce, “das ist der Schrödinger.”6 Their first meeting instilled in Thirring the conviction that “this man is really somebody special … a fiery spirit at work.” By the time Schrödinger reached adulthood his erudition was legendary; he lectured comfortably in German, English, French, and Spanish, recited and wrote poetry (even publishing a volume late in life), and became a true expert in the philosophy of Schopenhauer and the Hindu spiritual texts, the Upanishads. Schrödinger “would translate Homer into English from the original Greek, or old Provencal poems into German,” and insisted throughout his life that study of the ancient Greek thinkers was not something for his “hours of leisure” but was “justified by the hope of some gain in the understanding of modern science.” It was said of Schrödinger’s physics articles that “if it were not for the mathematics, they could be read with pleasure as literary essays.”
After settling on physics as his main focus at the end of his undergraduate years, Schrödinger went on to graduate work, primarily in experimental physics or in theoretical topics relating to the experimental work going on at the university. “I learnt to appreciate the significance of measuring. I wish there were more theoretical physicists who did.” However, by the end of this period, around 1914, when he obtained his habilitation, he had decided that he was personally unsuited to be an experimenter and that Austrian experimental physics was second rate. Nonetheless he continued to do some laboratory work, and his reputation as a broadly trained physicist, conversant with both experiment and theory, would be of great value when he began searching for academic positions.
Schrödinger was poised to dive into the rushing currents of change in theoretical physics in 1914, with Bohr’s atomic theory newly hatched and Einstein’s general relativity on the near horizon. But, as it did for de Broglie, the Great War intervened. Schrödinger was called into service as an artillery officer, and he served in that capacity for three years before being transferred to the meteorology service. In general Schrödinger’s military assignments were not among the most challenging or dangerous, and he mainly suffered from boredom, and a certain degree of depression, during this period. However, early on in his tour of duty, in October of 1915, he was caught up in one of the major battles around the Isonzo River on the Italian front and received a citation for “his fearlessness and calmness in the face of recurrent heavy enemy artillery fire.”
During his war service he wrote to his many women friends, but only one visited him at the front, a young woman from Salzburg named Annemarie Bertel, whom he had met through friends in 1913. She admired and adored Schrödinger from their first meeting: “I was impressed by him because, first of all, he was very good-looking.” They would marry in 1920, and within a few years the marriage evolved into a close, but nonmonogamous, relationship, with both fairly openly engaging in affairs, although Erwin was certainly the more active in this regard. For Annie (as she was known), this was the price of involvement with a great man. “I know it would be easier to live with a canary bird than with a race horse. But I prefer the race horse.”
When Schrödinger returned full time to physics research in 1918, he was not particularly focused on the problems of quantum theory. He had learned theoretical physics at university from Fritz Hasenohrl, a leading disciple of the great Boltzmann, who along with Maxwell and Gibbs founded statistical mechanics. Boltzmann had died by suicide in 1906, the same year that Schrödinger began his studies; but his atomic worldview now prevailed; it had become a pillar of modern physics. “No perception in physics has ever seemed more important to me than that of Boltzmann,” Schrödinger recounted, “despite Planck and Einstein.”
During the war he had filled several notebooks with statistical calculations very much in the spirit of Einstein’s early work on Brownian motion and diffusion. Upon returning to civilian life he published two papers based on these notes, the second of which, dealing with fluctuations in the rate of radioactive decay, is the longest article he ever produced, stretching to sixty journal pages. It was a tour de force of applied mathematics, and it announced to the world that he was to be taken seriously as a statistical physicist. In the same period he also published his first paper on quantum theory, focusing on further developments in Einstein’s quantum theory of specific heat, as well as two short papers analyzing the equations of general relativity. In yet another nod to Einstein’s work, in 1919, he performed an experiment trying to distinguish between the wave and particle theories of light, using a very small source. The experiment was similar in a general sense to the failed experiment that Einstein proposed in 1921 (his “monumental blunder”) and gave similarly equivocal results.
Schrödinger was establishing his research style as a critic and polymath, one able to work expertly in many subfields at once, who took the ideas of others and either demolished them or clarified and extended them. Although his radiation experiment had not had been a major success, it resulted in an invitation from Sommerfeld to visit Munich, where he became enamored of the (old) quantum theory of atomic spectra, due to Bohr and elaborated in great detail by the “beautiful work” of the Sommerfeld school.7 By 1920 he had been appointed full professor at Breslau, and he threw himself into research on atomic spectra, something Einstein had never been willing to do. By January of 1921 he had produced a step forward in the theory of alkali atoms, leading to a correspondence with Bohr, who wrote: “[your paper] interested me very much … some time ago I made exactly the same consideration.”8 He would continue to make respectable, but not decisive, contributions to the Bohr-Sommerfeld theory regularly, into the fateful year of 1925, when the old theory would be overthrown by two revolutions, one of his own making.
FIGURE 27.1. Erwin Schrödinger circa 1925. AIP Emilio Segrè Visual Archives, Physics Today Collection.
By 1922 he had been recruited to Zurich and was a certified expert in both modern quantum theory and modern statistical physics, but still a virtuoso without a masterpiece of his own. Almost all his work for the next four years would be on either atomic spectra or the statistical mechanics of gases; surprisingly, it was the latter that led him to his great discovery, with more than a nudge from Albert Einstein. As we have seen, in February of 1925, shortly after the publication of Einstein’s key paper on the quantum theory of the ideal gas and Bose-Einstein condensation, Schrödinger wrote to Einstein respectfully but firmly suggesting that his paper contained an error. When, in his reply, Einstein explained to him how the new statistics worked, the scales dropped from Schrödinger’s eyes, and he was entranced by the “originality of [Einstein’s] statistical method.” He immediately set out to deepen his understanding of this new form of statistical physics, which he would soon describe as “a radical departure from the Boltzmann-Gibbs type of statistics.”
By July of 1925 he had produced a typically insightful but incremental response, a paper titled “Remarks on the Statistical Definition of Entropy for the Ideal Gas,” which contrasted Planck’s definition of entropy for the gas with that of Einstein. Planck for some time had been suggesting a weaker form of indistinguishability of gas particles than that of Bose and Einstein,9 which was sufficient to save Nernst’s law but didn’t lead to the weird statistical attraction that is implied by Bose-Einstein statistics. Schrödinger realized that Planck’s method was illogical because it got rid of too many states. Recall that Bose-Einstein’s new counting method, when applied to dice, would insist that the two dice “states” (4, 3) and (3, 4) are just one state, so that for each such unequal pair one should count only one state, not two, reducing the number of states and hence the entropy of the system. However, there is no such reduction for doubles (there is only one way to roll snake eyes); so there is no reduction in the number of double “states” for quantum versus classical dice. Yet Planck’s method, once you understood it deeply, boiled down to counting each double as only half a state, which was clearly wrong. Schrödinger says exactly this: “in order that two molecules are able to exchange their roles, they must really have different roles … one is [then] almost automatically led to that definition of the entropy of the ideal gas which has recently been introduced by A. Einstein [Bose-Einstein statistics].” In a quaint custom of the time, this rather significant criticism of Planck was read to the Prussian Academy by Planck himself, on behalf of Schrödinger.
Einstein was impressed by this exegesis, which he himself apparently had not appreciated; in September of 1925 he wrote to Schrödinger again: “I have read with great interest your enlightening considerations on the entropy of ideal gases.” He then sketched for Schrödinger another approach to the ideal gas problem, which he had worked through crudely, leading to results that he found puzzling. When Schrödinger wrote back to Einstein on November 3, in addition to applauding Einstein’s development of Bose statistics he proposed to carry through Einstein’s alternative approach in detail, which he was able to do in a scant few days. He was less troubled than Einstein by the answer he found, which confirmed Einstein’s original argument, and proposed a joint publication: “the basic idea is yours … and you must decide about the further fate of your child…. I need not emphasize the fact that it would be a great honor for me to be allowed to publish a joint paper with you.”
A touching exchange ensued, in which Einstein insists he should not be a coauthor, “since you have performedthe whole work; I feel like an ‘exploiter,’ as the socialists call it.” Schrödinger immediately demurs: “not even jokingly would I have … thought of you as an “exploiter” … one might say: ‘when kings go building, wagoners have work.’ ” On December 4 he sent Einstein a complete draft of the paper with the second author slot blank, but Einstein presented the paper to the Prussian Academy without his own signature. Einstein’s intuition was again right; there were subtle errors in the reasoning, and the approach itself turned out to be unwieldy, in contrast to Einstein’s first approach, which is still found in modern texts.
However, only a couple of weeks later, Schrödinger had yet another ideal gas paper in the works, which led directly to the wave equation. Just prior to his letter to Einstein in early November, Schrödinger had finally managed to get hold of de Broglie’s thesis, which he had sought because of Einstein’s strong endorsement of it in his work on the quantum gas. Having now got hold of the thesis, Schrödinger tells Einstein that “because of it, section 8 [the wave-particle section] of your second [quantum gas] … paper has also become completely clear to me for the first time.” Two weeks later, writing to another physicist, he remarks that he is very much inclined toward a “return to wave theory.”
His final ideal gas paper, submitted December 15, just before he left for his Christmas holiday, was a last effort to digest Einstein’s new quantum statistics; it is titled simply “On Einstein’s Gas Theory.” He sets out to derive the same answers as Einstein without accepting the strange state counting of Bose, which he thinks requires too great a “sacrificium intellectus.” The way out, he says, is “nothing else but taking seriously the De Broglie–Einstein wave theory of the moving particles, according to which the particles are nothing more than a kind of ‘white crest’ on a background of wave radiation.” He then formulates the mathematical problem differently from Einstein but shows how to reach exactly the same final equations. From the perspective of a modern physicist10the two calculations are equivalent and have the same meaning, but Schrödinger felt that by interpreting the fundamental objects as waves instead of as particles the weirdness of indistinguishability was somehow made more palatable. He concludes by speaking of the particles as “signals” or “singularities” embedded in the wave, highly reminiscent of Einstein’s failed ideas of 1909–1910 and his later idea of “ghost fields” guiding the particles developed in the early 1920s. Except that Schrödinger clearly now thinks that the particles are the “ghosts,” the ephemera, since by viewing waves as fundamental he has “explained” Bose-Einstein statistics in a natural way. Later he would say, “wave mechanics was born in statistics.”
A few days before Christmas 1925 Schrödinger set off to a familiar mountain lodge in the Swiss village of Arosa, determined to find a new equation to describe these matter waves. Although he took his skis (he was an expert alpinist) and was accompanied by an unnamed “old girlfriend,”11 it seems that this trip was really focused on wave equations. De Broglie, despite Einstein’s praise, had not produced a new governing equation, similar to Maxwell’s electromagnetic wave equation, that could predict or explain the remaining mysteries of the atom. He had produced some suggestive mathematical relations, which made contact with the old quantum theory of Bohr and Sommerfeld, but only at the most elementary level. The central mystery of quantum theory, quantization, was not really resolved by de Broglie’s work. Why wasn’t nature continuous? Why are only certain energies allowed for electrons bound to atomic nuclei?
Schrödinger saw an answer. Classical waves, or vibrations, in a confined medium have certain natural constraints on their properties. Consider a violin string of a certain length, L, clamped down at each end. The notes it can play arise from vibrations of the strings; these vibrations are waves of sideways displacement in the string, but they cannot have a continuously varying wavelength as is possible for waves in an open (essentially infinite) medium. The longest wavelength, λ, they can have is twice the string length. Why is this the longest? First, consider that, for any wave on a string, the string’s displacement must be zero at the points where it is clamped down. The simplest form of displacement the string can have away from these fixed points is for it to be displaced everywhere in the same sideways direction (left or right) at a given instant, so that the maximum displacement is in the middle and it decreases back to zero at both clamped ends. This displacement then oscillates back and forth, causing sound waves of a certain pitch. Since we measure wavelength by the distance between points in the medium that take us through a peak and a trough, this shape corresponds to half the full wavelength, so the wavelength is λ = 2L. This will determine the lowest note the violin can play (for a given string tension). The next-lowest note will have λ = L, implying that there will be no displacement at the center of the string (even though that point isn’t clamped down). In general the only allowed wavelengths are λ = 2L/n, where n is a whole number (n = 1, 2, 3 …). That’s the point: for confined waves, nature produces whole numbers automatically. De Broglie had hinted at this, but now Schrödinger realized that this was the key to getting the quantum into quantum theory.
The mathematics of the old quantum theory had not done this in a natural way. Bohr and his followers had taken mathematical expressions that are continuous (i.e., don’t involve whole numbers exclusively) and had simply restricted them to whole numbers by fiat. Schrödinger, by contrast, was looking for an equation that simply did not have continuous solutions, one in which each solution would be connected to a whole number organically.
As noted above, wave equations, through the quantization of the wavelength, have this property. They are differential equations, describing continuous change in space and time; but when the waves are confined, only certain wavelengths are physically possible. So Schrödinger was looking for a matter-wave differential equation to describe electrons: it would have to describe a matter field and an extended “disturbance” of the field that varied in space and time but was confined to the vicinity of the atomic nucleus by the electron’s attraction to the nucleus.
Maxwell’s electromagnetic wave equation was the only fundamental wave equation of physics at that time, but because photons are massless, it had no safe place for the introduction of Planck’s constant (as Einstein had learned to his dismay fifteen years earlier). Schrödinger’s challenge was to fashion a wave equation, modeled on Maxwell’s, that included Planck’s constant as well as the physical constants e and m, representing the charge and mass of the electron. Maxwell’s wave equation does, however, contain the wavelength, λ, of the EM waves which then lead to quantized values and whole numbers, when it is written for “standing waves” confined to a specific region. So the key idea was to write an equation similar to the electromagnetic wave equation but for a new wave field, a “matter wave” described by a mathematical expression now known as the “wavefunction,” and to replace λ using de Broglie’s relation λ = h/p = h/mv.
With this approach Schrödinger had an equation containing h, Planck’s constant, and m, the electron mass. The electron velocity, v, can be eliminated from the equation in favor of the difference between its total energy and potential energy. The potential energy of the electron orbiting the nucleus of course depends on its charge, e. The resulting “time-independent Schrödinger equation” for hydrogen contains the “holy trinity,” h, m, and e. And now the moment of triumph: since only certain wavelengths are allowed, that implies that the only unknown in the equation, the total energy of the electron, can only take on certain allowed values. The energy of electrons in an atom is quantized, not by fiat, but due to the fundamental properties of waves confined to a fixed region in space.12
After returning from his “ski” trip, Schrödinger immediately subjected his new equation to the acid test: could it reproduce the energy levels, grouped into different “shells,” for the case of atomic hydrogen, which were known from spectral measurements for decades and “explained” in an ad hoc manner by the old quantum theory? The answer was a resounding yes. The details of the calculations took only a few weeks, and on January 27, 1926, the first of Schrödinger’s seminal papers was received at the Annalen der Physik. It states the breakthrough thus: “in this paper, I wish to consider … the simple case of the hydrogen atom … and show that the customary quantum conditions can be replaced by another postulate, in which the notion of ‘whole numbers’ … is not introduced…. The new conception is capable of generalization and strikes, I believe, very deeply at the true nature of the quantum rules.”
After presenting the detailed solution of his hydrogen equation, he briefly touches on its interpretation and its origins. “It is, of course, strongly suggested that we try to connect the [wavefunction] with some vibration process in the atom, which would more nearly approach reality than the electron orbits [of Bohr-Sommerfeld theory],” but he feels that this is premature, since the theory needs further development. However, “Above all, I wish to mention that I was led to these deliberations by the suggestive papers of M. Louis de Broglie … I have lately shown that the Einstein gas theory can be based on [such] considerations … the above reflections on the atom could have been represented as a generalization from those on the gas model.” Later Schrödinger would say, “My theory was inspired by L. de Broglie … and by brief, yet infinitely far-seeing remarks of A. Einstein.”
This first paper was followed in rapid succession by five more in just six months, in which Schrödinger, the consummate craftsman, working alone, determined essentially all the known properties of atomic spectra from the solutions of his wave equation. It was a breathtaking display, about which even his competitor Born would later remark, “what is more magnificent in theoretical physics than Schrödinger’s six papers on wave mechanics?” The normally reserved Sommerfeld called Schrödinger’s equation “the most astonishing among all the astonishing discoveries of the twentieth century.” Hence, by June of 1926, physicists had uncovered most of the new laws and mathematical methods necessary to describe physics on the atomic scale; they just didn’t yet know what they meant. However, an interpretation was soon to emerge, one that would challenge the philosophical principles that both Schrödinger and Einstein held dear.
1 When Einstein held the theoretical physics position is was only at the level of an associate professorship (extraordinarius); it was subsequently upgraded to a full professorship (ordinarius).
2 Schrödinger’s interest in philosophy was so great that in 1918, before the war ended, he had been planning to “devote himself to philosophy” more than physics, only to find that the chair he expected to receive, in Czernowitz, Ukraine, had disappeared along with Austrian control of the region.
3 While Schrödinger was discreet in this final public document, in 1933, to his diary, he confided that he never slept with a woman “who did not wish, in consequence, to live with me for all her life.” There is some evidence to back this up.
4 An English theoretical physicist, Paul Adrien Maurice Dirac, was the last of the trio of wunderkind to play a founding role in quantum mechanics, along with Pauli and Heisenberg. Born in 1902, he was even younger than Heisenberg, and upon hearing Heisenberg speak at Cambridge in July of 1925, he shortly afterward invented his own, mathematically elegant version of the quantum equations. A few years later he discovered the “Dirac Equation,” the quantum wave equation that takes into account the effects of relativity. However, he interacted little with Einstein during the period 1925–26 and so is not very relevant to our historical narrative; Einstein remarked of him in August 1926, “I have trouble with Dirac. This balancing on the dizzying path between genius and madness is awful.”
5 First in class.
6 “That is the Schrödinger.”
7 It is likely that it is at this time he studied and understood Einstein’s 1917 reformulation of the Bohr-Sommerfeld theory, which he later praised so highly.
8 This is always a great compliment from one physicist to the other, unless the former is angling for priority.
9 For expert readers, this is the “division by N!” in the partition function (state counting), which is approximately correct at high temperature and is still used in modern texts.
10 Einstein immediately saw this equivalence, writing to Schrödinger after his paper appeared, “I see no basic difference between your work on the theory of the ideal gas and my own.”
11 Hermann Weyl, a distinguished physicist and a friend of Schrödinger’s, later famously commented that Schrödinger “did his great work during a late erotic outburst in his life.”
12 This simple argument, while certainly part of Schrödinger’s reasoning, was not how he first presented the equation and omits his failed initial attempt to come up with an equation consistent with Einstein’s relativity theory.CHAPTER 28
CONFUSION AND THEN UNCERTAINTY
If we are still going to have to put up with these damn quantum jumps, I am sorry that I ever had anything to do with quantum theory.
—SCHRÖDINGER TO BOHR, OCTOBER 1926
“I am convinced that you have made a decisive advance with your formulation of the quantum condition, just as I am equally convinced that the Heisenberg-Born route is off the track.” Thus Einstein wrote to Schrödinger in late April of 1926. The Heisenberg-Born route, a different approach to the “quantum conditions,” introduced the term “quantum mechanics” as a more rigorous replacement for the nebulous conceptual structure of “quantum theory.” This method had begun to bear fruit six months earlier than Schrödinger’s, and unlike his work it arose independently of Einstein’s recent successes with the quantum gas.
It represented the radical point of view that since atoms were, practically speaking, impossible to observe in space and time, one should stop attempting to describe them by space-time orbits as in classical mechanics. Instead one should develop a description in terms of observable atomic variables, which might not themselves be easily visualized, such as the absorption frequencies for light incident on the atom, and how strongly each frequency was absorbed. The first breakthrough using this approach had come from the twenty-three-year-old prodigy Werner Heisenberg, who formulated his method in July of 1925. By the time of Schrödinger’s work, Einstein had been ambivalently struggling with this new framework for quite some time already, since Heisenberg was working in the research group of his close friend Max Born in Göttingen. Born had immediately informed him of Heisenberg’s initial sighting of a New World of the atom, writing in a letter dated July 15, 1925, that Heisenberg’s paper “appears rather mystifying, but is certainly true and profound.”
Heisenberg was not the only young genius to find his way to Born’s research team in Göttingen. Six of Born’s research assistants and one of his PhD students would go on to win the Nobel Prize,1 and three of them—Enrico Fermi, Wolfgang Pauli, and Heisenberg—would contribute cornerstones to the rising quantum edifice. Born, only three years younger than Einstein, was from Prussian Silesia, and was of Jewish descent (like many of Einstein’s closest friends). He had been appointed associate professor at Berlin from 1915 to 1919, arriving just in time to observe Einstein’s awe-inspiring success with general relativity theory. He, and his wife Hedwig, formed a lifelong friendship with Einstein, although Born maintained as well a certain reverence for his friend, whom he would refer to, after his death, as “my beloved master.” Born made seminal contributions to physics and eventually won the Nobel Prize himself, but he was not an imposing intellect, and he sometimes had trouble keeping up with his brilliant wards. Of Pauli, who was renowned for his critical brilliance, he said, “I was from the beginning quite crushed by him … he would never do what I told him to do.” Heisenberg, he recalled, was quite different: “he looked like a simple peasant boy, with short, fair hair, clear bright eyes and a charming expression” when he arrived, “very quiet and friendly and shy…. Very soon I discovered he was just as good in the brains as the other one.”
After a few months spent visiting Niels Bohr in the fall of 1924, Heisenberg returned to Göttingen with the germ of an idea for a completely new quantum theory of the atom, distinct from the old Bohr-Sommerfeld approach. This approach, while it worked for hydrogen and a few other atoms, appeared to be breaking down for more complicated atoms and molecules. In fact, by 1924 more than a decade had passed since Bohr’s pathbreaking work, and a full quarter century since that of Planck; many physicists were beginning to wonder if the fundamental laws of the atom were simply beyond human ken. In May of 1925 the enfant terrible, Pauli, wrote despairingly to a friend, “right now physics is very confused once again—at any rate it’s much too difficult for me and I wish I were a movie comedian or some such.” However, Heisenberg was just about to shake the field out of its malaise.
FIGURE 28.1. Werner Heisenberg circa 1927. AIP Emilio Segrè Visual Archives, Segrè Collection.
Heisenberg’s idea was to take the continuous trajectory of a particle, which in classical physics is represented by the three Cartesian coordinates x, y, z that vary continuously with time, and replace each coordinate with a list of numbers arrayed in rows and columns, rather like a Sudoku puzzle. Each number in the list is not fixed, but oscillates in time sinusoidally, with a characteristic frequency. When applied to electrons in an atom, the frequencies corresponded to the observable “transition frequencies” at which the atom would absorb and emit light. First, however, Heisenberg considered the most basic “toy problem” of mechanics, the familiar linear harmonic oscillator (mass on a spring). He was able to show that using his new definition for position, and a similar one for momentum, the energy of the oscillator was conserved; that is, it didn’t change in time as long as the energy took the special values found by Planck so long ago, quantized in steps of hυ (where υ is the frequency of the oscillator). So here, in Heisenberg’s new arithmetic, the whole numbers of quantum theory also arose naturally from the math and were not imposed externally, just as they would later appear naturally in Schrödinger’s wave approach. Heisenberg first discovered this while recovering from an allergy attack on the North Sea island of Helgoland, and he was so excited that he stayed up all night working, and then, lying on a rock watching the sun rise, he thought to himself, “well something has happened.”
Indeed something had. This mode of thinking was simply orthogonal to everything physicists had been trying to do in atomic theory, and it broke the impasse. Heisenberg informed his friend Pauli, who was elated, saying that the idea gave him renewed “joie de vivre and hope … it’s possible to move forward again.” Heisenberg wrote up his initial ideas with the boldly stated goal of establishing a new quantum mechanics, “based exclusively on relationships between quantities which are in principle observable.” Born, with another talented student, Pascual Jordan, quickly realized that Heisenberg’s “lists of numbers” were objects that mathematicians refer to as matrices, and that the rule for combining them that Heisenberg had invented was the known rule for multiplying matrices. An odd thing about representing physical magnitudes by matrices is that when multiplying matrices, in general x times y is not equal to y times x. This curiosity would end up having a deep significance in the final theory. Within a few months Born, Heisenberg, and Jordan were able to put together a definitive paper announcing the rules for calculating observable quantities in the new quantum mechanics, which, in their version, would become known also as “matrix mechanics.”
Einstein, despite Born’s endorsement, reacted suspiciously to the breakthrough from the beginning, writing to Ehrenfest in September 1925 with a typically earthy judgment: “Heisenberg has laid a big quantum egg. In Göttingen they believe in it (I don’t).” Despite his skepticism, he realized that a substantial advance had been made, telling Besso in December 1925 that matrix mechanics was “the most interesting thing that theory had produced in recent times”; but he could not resist a dig at its odd structure, “a veritable witches’ multiplication table … exceedingly clever and because of its great complexity safe against refutation.” Sarcasm notwithstanding, he studied the theory closely and discovered several technical objections, which he communicated to Jordan.2
Bose recalled that upon his arrival in Berlin in the fall of 1925, “Heisenberg’s paper came out. Einstein was very excited about the new quantum mechanics. He wanted me to try to see what the statistics of light-quanta … would look like in the new theory.” But Einstein’s reservations were beginning to win out; early in 1926 he wrote to Ehrenfest, “more and more I tend to the opinion that the idea, in spite of all the admiration [I have] for [matrix mechanics], is probably wrong.” Just as he was hardening his negative view, in January the newly reenergized Pauli showed how to derive the basic hydrogen spectrum using matrix mechanics, an apparently decisive proof that the theory was on the right track. Of course Schrödinger was just at that time deriving the same result by the quite different method of his wave equation.
Schrödinger’s approach was superficially much more congenial to the classical physics worldview, based as it was on a continuum wave equation in space and time, similar to that of Maxwell, and seeming to arrive at quantized energies via the familiar properties of vibrating waves. Einstein, Planck, Nernst, and Wien, the reigning royalty of German physics, all jumped on the Schrödinger bandwagon immediately. Born, now a bit under siege, later recalled that Schrödinger’s paper “made much more of an impression than ours. It was as though ours didn’t exist at all. All the people said now we have the real quantum mechanics.” However, Born would soon have a key ally; Niels Bohr had been moving toward a view that the conventional space-time picture of the atom was fatally flawed, and his force of personality would eventually prevail, although not without some further twists and turns.
Initially the two sides believed that they were faced with a choice between two fundamentally different theories, so that Einstein, in the same letter to Ehrenfest in which he called matrix mechanics “probably wrong,” described Schrödinger’s innovation as “not such an infernal machine [as matrix mechanics], but a clear idea—and logical in its application.” And a few weeks later, in early May, he told Besso, “Schrödinger has come out with two excellent papers on the quantum rules, which present some profound truths.” But the period of either/or decision making was brief. A dramatic change in the debate occurred at one of the famed Berlin colloquia, where Einstein often presided. A young student, Hartmut Kallmann, recorded the events. “People were packed into the room as lectures on Heisenberg’s and Schrödinger’s theories were given. At the end of these reports Einstein stood up and said, ‘Now just listen! Up until now we have had no exact quantum theory, and now suddenly we have two. You will agree with me that these two exclude each other. Which theory is correct? Perhaps neither is correct.’ At that moment—I shall never forget it—Walter Gordon stood up and said: ‘I have just returned from Zurich. Pauli had proved that the theories are identical.’ ”3 Actually by mid-March Schrödinger, prior to Pauli (who never even bothered to publish his proof), was able to show that the equations of matrix mechanics followed from his wave equation and vice versa; matrix mechanics could be used to derive the Schrödinger equation. The two theories were indeed mathematically equivalent.
At this point the debate shifted to the question of the meaning of the new theory, and the aesthetic and conceptual merits of the two different formulations. Already, in his paper proving their equivalence, Schrödinger had slipped in a jibe against the matrix approach, saying that he was “discouraged, if not repelled” by the difficulty of its methods and its lack of transparency. And he repeatedly stated that his approach was the more “visualizable,” prompting a fed-up Heisenberg to declare in a letter to Pauli, “what [he] writes about Anschaulichkeit[visualizability] makes scarcely any sense…. I think it is crap.”
Matters came to a head in July, when Schrödinger made a “victory tour” of the conservative physics centers of Berlin, where they had begun recruiting him to replace Planck, and Munich, where Wien and Sommerfeld were in charge. By coincidence Heisenberg was in Munich when Schrödinger spoke, and he raised some unresolved issues for wave mechanics in the question period at the end of the lecture. Before Schrödinger could respond, Heisenberg was almost “thrown out of the room” by Wien, who thundered, “young man, Professor Schrödinger will certainly take care of all these questions in due time. You must understand that we are now finished with all that nonsense about quantum jumps.” A shaken Heisenberg wrote immediately to Bohr, who responded by inviting Schrödinger to Copenhagen. A marathon session of conceptual arm wrestling ensued, ending with Schrödinger in bed exhausted and sick, but unconverted. The key point that Bohr insisted upon is that while Schrödinger’s wave equation appeared to restore a continuous description of nature, when applied to atoms it would inevitably lead back to the fundamental discontinuity of natural processes implied by quantum phenomena. At about the same time Einstein and his close friend Max Born were wrestling with exactly this issue.
For Einstein, the mathematical equivalence of the two theories simply extended his doubts about matrix mechanics to wave mechanics. He was not immune to the exhilaration felt by his colleagues as the historic puzzles of atomic structure were being unraveled almost on a weekly basis. After another colloquium, at which the evidence for the newly discovered spin of the electron was presented, Bose ran into him on a streetcar: “we suddenly found him jumping [into] the same compartment where we were, and forthwith he began talking excitedly about the things we have just heard. He has to admit that it seems a tremendous thing, considering the lot of things which these new theories correlate and explain, but he is very much troubled by the unreasonableness of it all. We were silent, but he talked almost all the time; unconscious of the interest and wonder that he was exciting in the minds of the other passengers.”
The unreasonableness that Einstein felt now focused mainly on the meaning of Schrödinger’s wavefunction, which somehow represented the behavior of electrons bound to atomic nuclei. Schrödinger originally tried to argue that his matter waves could accumulate in a localized region of atomic dimensions, carrying along a bump or “crest” that behaved like a particle. But further study soon showed that such a “wave packet” could not cohere over long times; the math was actually very similar to that of light waves, and the failure of this idea reprised Einstein’s own failure to find particulate behavior in Maxwell’s wave equation back in 1910. It is likely that Einstein spotted this problem very quickly. A fallback position, taken by Schrödinger subsequently, was to assert that there simply are no electron particles; the “real electron” is a wave of electric charge density, spread out in space on dimensions somewhat larger than the atom. But there was a further basic problem with this picture. Einstein expressed this in June of 1926 in a letter to a colleague, Paul Epstein: “We are all here fascinated by Schrödinger’s new theory of quantum levels … strange as it is to introduce a field in q-space, the usefulness of the idea is quite astonishing.” What was this “q-space” that Einstein found so strange?
All the waves that were known to physicists at that time were represented as an oscillating field or disturbance in our normal three-dimensional space, even electromagnetic waves, which, according to Einstein, didn’t require a medium (the ether) to exist. The number of particles in a wave did not enter the equation except through the density of the medium, as for sound waves, or through the intensity of the wave, for electromagnetic waves. But electron waves could not be represented that way. Isolated free electrons could be studied, and their charge could be measured; it was –e, the same in magnitude (and opposite in sign) from that of the proton. The periodic table of elements requires that hydrogen have a single electron, helium two electrons, lithium three, and so on. So to describe electrons in helium, for example, one needed to have a wave equation for two electrons in different quantum states, with total electric charge adding up to 2 times –e. There was only one way to do this, mathematically speaking, using Schrödinger’s equation: the “two-electron” wavefunction had to “live” in a six-dimensional space, three dimensions for the first electron and three more for the second. Moreover, the wavefunction for electrons in a large atom such as uranium-235 had to live in a 705-dimensional space! This was “q-space,” an abstract space that “copies” our three-dimensional space N times in order to represent N electrons. Einstein recognized this strange feature in his very first letter to Schrödinger, and to him it was an enormous clue that a classical wave picture might not be restored through Schrödinger’s equation.
Einstein was not the only person puzzling over how to interpret Schrödinger’s waves. Max Born also had grave reservations about the idea that the electron was a conventional wave. He worked closely with the noted experimentalist James Franck, who did measurements of electron beams colliding with atoms. “Every day,” he recalled, “I saw Franck counting particles and not measuring continuous wave distributions.” The moment he learned about the Schrödinger equation, he had an intuition about what its matter waves represented. He recalled Einstein’s idea that the electromagnetic field is a “ghost field” guiding the photons. “I discussed this with himvery often. He said that as long as there was nothing better, one can [use this approach].” But now, for matter waves, Born felt this was the true picture: the Schrödinger wavefunction represented a guiding wave of probability. Mathematicians and physicists were already used to the idea of assigning probabilities to a continuous space, essentially by dividing the space into infinitesimal regions. Born argued that Schrödinger’s wavefunction represented such a probability density,4 which actually moved deterministically in space as a wave but simply described how likely it would be to find an electron particle in each particular region of space.
By the end of June 1925 Born went public with his idea, submitting a paper titled “Quantum Mechanics of Collision Phenomena.” In it he formulates the problem of a directed matter wave (representing a stream of electrons in a beam directed at an atom) that interacts with the electric field of the atom and then “scatters” in all directions, just as water waves hitting a post send out circular waves in all directions. Could this really mean that each electron “breaks up” and goes in all directions like a smeared-out electrical “oil slick”? That is exactly the view that Schrödinger wants to take, but Born is having none of it. He insists that the expanding circular wave just determines the probability of finding a whole, pointlike electron emerging in particular direction. To test this idea you need to do the same experiment over and over again and count the number of electrons that go in each direction. “Here the whole problem of determinism arises. From the point of view of our quantum mechanics, there exists no quantity which in an individual case causally determines the effect of the collision…. I myself tend to give up determinism in the atomic world.” In his view we need to adopt a weaker form of determinism: “the motion of particles follows probabilistic laws, but the probability itself propagates according to the law of causality.”
When Schrödinger learned of Born’s interpretation, he was incensed and engaged him in an “acrimonious debate.” As Born recalled, “he believed that [matter waves] meant some continuous distribution of matter and I was very much opposed to it [because of Franck’s experiments]…. he was very offensive, as he always was when somebody objected to [his ideas].” Schrödinger’s opposition notwithstanding, the Born probabilistic interpretation of the wave function was widely adopted almost immediately, and was the basis of Born’s eventual Nobel Prize. However, the person who had inspired Born’s critical step, Einstein, was among the few holdouts. In November of 1926 Born wrote to his dear friend: “I am entirely satisfied, since my idea to look upon the Schrödinger wave field as a ‘[ghost field]’ in your sense proves better all the time…. Schrödinger’s achievement reduces itself to something purely mathematical; his physics is quite wretched.” But by this time Einstein’s reservations had solidified into an unshakable conviction. Just a few days later he sent Born his famous and crushing response: “Quantum mechanics calls for a great deal of respect. But some inner voice tells me that this is not the true Jacob. The theory offers a lot, but it hardly brings us closer to the Old Man’s secret. For my part, at least, I am convinced he doesn’t throw dice.”
Some months earlier Einstein had met privately with Heisenberg to discuss quantum mechanics. Heisenberg had presented his view that the new theory should restrict itself to describing observable quantities, and not unobservable electron orbits. Einstein rejected this view, leading Heisenberg to rejoin, “isn’t that precisely what you have done with relativity theory.” Einstein responded, “possibly I did use this form of reasoning … but it is nonsense all the same.5 … It is the theory which decides what can be observed.” This conversation stuck with Heisenberg, and a year later, while pondering the meaning of quantum mechanics, it came back to him. “It must have been one evening after midnight when I suddenly remembered my conversation with Einstein, and particularly his statement, ‘it is the theory which decides what we can observe.’ I was immediately convinced that the key to the gate that had been closed so long must be sought right here.” Within days he had used the new quantum mechanics to prove his uncertainty principle. One could observe the position of an electron very accurately, or the momentum of an electron very accurately, but not both at the same time. That’s what the theory had decided. Even this realization, so fiercely opposed by Einstein, had been stimulated by his own insight.
1 In addition to Heisenberg, Pauli, and Fermi, the others were Max Delbruck (PhD), Eugene Wigner, Gerhard Herzberg, and Maria Goeppert-Mayer, the second woman to win the prize in physics.
2 Einstein wrote a number of letters to Heisenberg in this period, all of which have been lost. At least one, Heisenberg recalled, was signed “in genuine admiration.”
3 Kallmann most likely erred in his memory of the city, since Gordon was working with Pauli in Hamburg at that time. Schrödinger was in Zurich.
4 More precisely, it is the absolute square of the wavefunction that represents a probability density.
5 Later, in replying to the same reproach from his friend Philipp Frank, Einstein responded with the pithy retort, “A good joke should not be repeated too often.”CHAPTER 29
NICHT DIESE TÖNE
All the fifty years of conscious brooding have brought me no closer to the answer to the question, “what are light quanta?” Of course today every rascal thinks he knows the answer, but he is deluding himself.
—EINSTEIN TO BESSO, 1951
“Here I sit in order to write, at the age of 67, something like my own obituary … [this] does … not come easy—today’s person of 67 is by no means the same as was the one of 50, of 30 or of 20. Every reminiscence is colored by today’s being what it is, and therefore by a deceptive point of view.” Einstein, in the autobiographical sketch he thus begins, confirms his initial disclaimer. Readers hoping to learn from the man himself amusing anecdotes or details of his personal life were disappointed; the article of forty-six pages is a rather dense treatment of his philosophy of science, the evolution of physical theory, and then his actual contributions to science, ending with a technical statement of his latest attempt at a unified field theory. However, his revolutionary work on light quanta, and his groundbreaking quantum theory of specific heat of 1905–1907, merit only one long sentence. His early discovery of wave-particle duality gets a bit less than one page, ending in a remark that the current quantum theoretical explanation for it is “only a temporary way out.” His foundational work on the quantum theory of radiation and the spectacular discovery of Bose-Einstein condensation get no mention at all. He devotes much more space to his critique of quantum mechanics than to his contributions thereto. In contrast, relativity theory, special and general, is laid out in beautiful and exacting detail.
After the decisive year of 1926, in which he rejected the new quantum theory as the ultimate description of reality, he briefly sought to show, via his classic method of gedankenexperiments, that the theory contained internal contradictions. However, fairly soon he accepted the consistency of its logical structure with the comment “I know this business is free of contradictions, but in my view it contains a certain unreasonableness.” By September of 1931 he would graciously nominate both Heisenberg and Schrödinger for the Nobel Prize, with the comment “I am convinced that this theory undoubtedly contains a part of the ultimate truth.”
But despite this grudging endorsement, Einstein himself never applied the quantum formalism to a specific physics problem for the rest of his career, except in the context of a famous critical paper written in 1935 with younger collaborators, Podolsky and Rosen. The article drew attention through a thought experiment to the “spooky action at a distance” implied by quantum theory, which the authors claimed made the theory an incomplete description of reality. Modern realizations of this “EPR” experiment have fully confirmed the existence of this effect, a counter-intuitive correlation between distant particles. Such effects are referred to as “entanglement”; they form the basis of much of the new field of quantum information science. Many now consider Einstein’s recognition and prediction of the EPR effect as his last major contribution to physics.
Not only Einstein but also de Broglie and Schrödinger, the two quantum pioneers whom he had championed, made little contribution to the further application of quantum theory, and both ended up joining Einstein in rejecting it on philosophical grounds. As a consequence, the history of the discovery/invention of quantum theory was told from the perspective of Bohr, Heisenberg, Born, and their legions of students and collaborators. (Einstein, de Broglie, and Schrödinger had no students or collaborators in their works on quantum theory.) The matrix mechanicians, whose approach was instantly devalued following Schrödinger’s discovery of the “real quantum mechanics,” simply appropriated that work and gave it the interpretation that fit their understanding (and, it must be admitted, the experimental evidence). Ironically, Schrödinger was correct; his method was much more intuitive and visualizable than that of Heisenberg and Born, and it has become the overwhelmingly preferred method for presenting the subject. But with Born’s probabilistic interpretation of the wave-function, Heisenberg’s uncertainty principle, and Bohr’s mysterious complementarity principle,1 the “Copenhagen interpretation” reigned supreme, and the term “wave mechanics” disappeared; it was all quantum mechanics. The limitations on human knowledge of the physical world implied by these concepts were accepted by all practicing physicists. To this new generation Einstein became known primarily for relativity theory, admired by all, and secondarily for his stubborn refusal to accept the elegant new atomic theory of everything.
However, if one takes stock of the conceptual pillars of the new theory, in light of the historical record, a rather different picture emerges. Einstein surely shares with Planck the discovery of quantization of energy, as Planck never accepted that the quantum of action implied quantization of mechanical energy until many years after Einstein had become the first to proclaim it. It was Einstein who first realized that quantized energy levels explained the specific heat of solids, which justified the Third Law of thermodynamics and brought chemists such as Nernst into the quantum arena. Einstein, in his paper on light quanta, discovered the first force-carrying particles, photons, now the paradigm for all the fundamental forces. Following up on this, he discovered the wave-particle duality of light and, in 1909, based on his rigorously correct fluctuation argument, predicted that a “fusion theory” must emerge to reconcile the two views. In 1916 his quantum theory of radiation combined the ideas of Bohr, Planck, and his own light quanta to put Planck’s blackbody law on a firm basis. Here he introduced, for the first time, the core concept of intrinsic randomness in atomic processes, which the mature theory would accept as fundamental. He also introduced the notion of the probability to make a quantum jump, and he distinguished between spontaneous and stimulated transitions, ideas fundamental to, for example, the invention of the laser. And during 1924–1925 he elevated Bose statistics from obscurity, explained what it meant and why it had to be correct, and derived the mind-boggling condensation phenomenon it implied, something undreamt of by Bose himself. Finally, without ever publishing it, he developed the rule of thumb that electromagnetic wave intensity could be thought of as determining a probability to find photons in a certain region of space, the idea that stimulated Born’s crucial interpretation of matter waves.
In summary: quantization of energy, force-carrying particles (photons), wave-particle duality, intrinsic randomness in physical processes, indistinguishability of quantum particles, wave fields as probability densities—these are most of the key concepts of quantum mechanics. As Born would later say, “Einstein is therefore clearly involved in the foundation of wave mechanics and no alibi can disprove it.” The magnitude of these achievements? Four Nobel Prizes would be about right, instead of the one he received, grudgingly, in 1922. Not that Einstein cared much for such accolades.
Why did Einstein, who clearly understood the structure of the new theory and the necessity of introducing radical concepts to explain the atom, refuse to accept that theory and hold out for a very different resolution of the quantum dilemma? In my opinion this was the result of both his life experiences in doing science and his fundamental motivation for choosing that life.
Twice in his scientific career Einstein had wandered so far from the mainstream that even the many colleagues who already regarded him as an historic genius simply dismissed his views as wildly speculative and not to be taken seriously. Special relativity was not such a case, building as it did on the work of Lorentz and others, although certainly it unveiled a spectacular physical and epistemological insight that was uniquely Einsteinian. The first time was, of course, the light quantum “nonsense,” for which Planck felt compelled to apologize when nominating him to the Prussian Academy, and which Bohr still ridiculed a full two decades after it was proposed. The second time was with the theory of general relativity. In the latter case the idea did not elicit ridicule but simply incomprehension. There was no crisis in gravitation theory that required a radical resolution; what was this eccentric mastermind doing anyway?
The development of general relativity had proceeded unevenly, with dead ends and backtracking, technical errors ultimately corrected, and then an epiphany as the beautiful final equations emerged and predicted correctly the precession of Mercury and the bending of starlight. Einstein recalled the struggle thus: “the years of searching in the dark for a truth that one feels but cannot express, the intense desire and the alternation of confidence and misgiving until one breaks through to clarity and understanding, are known only to him who has experienced them himself.” Much later, in rejecting the modern quantum theory, he remarked, “it is my experiences with the theory of gravitation which determines my expectations.”
Moreover, just before the promulgation of the new quantum theory, in the summer of 1925, Einstein had experienced a similar vindication of his faith in the existence of light quanta. In 1924 Bohr and collaborators had put forth a new approach to the interaction of light and matter that sacrificed the principle of conservation of energy and momentum and introduced statistical considerations into the theory, although not in a manner that would turn out to agree with the final quantum theory. In print Bohr flatly stated that the theory of light quanta was “obviously not [a] satisfactory solution of the problem of light propagation.” Einstein staunchly opposed Bohr’s new theory, since he believed that the conservation laws must be exact or his beloved thermodynamics would be undermined. He also pointed to dramatic recent experiments by the American physicist Arthur Holly Compton that seemed to confirm the conservation laws for the collisions of an x-ray photon with an electron (treated as particles). However, it still could be argued that Compton’s experiments left open the possibility that momentum and energy were only conserved on average and not in each individual collision. This possibility was ruled out late in 1924 by landmark experiments of Bothe and Geiger in which the individual collisions were measured and shown conclusively to obey the conservation laws for two particles.
In January of 1925 Born wrote to Bohr, “the other day I was in Berlin. There everybody is talking about the result of the Bothe-Geiger experiment, which decided in favor of light-quanta. Einstein was exultant.” In April of 1925, two months before Heisenberg’s coup de destin, Bohr conceded that it was time “to give our revolutionary efforts[to banish light quanta] as honorable a funeral as possible.” Einstein’s apparently infallible intuition had triumphed one last time. Thus when the newest statistical theory of the atom, the Heisenberg-Born-Schrödinger synthesis, commandeered the stage, Einstein must have felt a sense of déjà vu. Just hold out long enough, and again he would be proved right.
Einstein’s most famous objection to the theory was the “dice complaint”: its insistence on the intrinsic randomness of individual events and the abandonment of rigid causality. But Schrödinger’s q-space picture had actually undermined Einstein’s objection to a probabilistic theory. Eugene Wigner, a leading figure in the second generation of quantum pioneers, was studying in Berlin in the early 1920s, and recalled that Einstein was quite “fond” of his guiding field concept, “[which] has a great similarity with the present picture of quantum mechanics,” but “he never published it … [because] it is in conflict with the conservation principles.” However, in the N-dimensional space of Schrödinger’s waves, the conservation laws survived, even if the outcomes were indeterminate. In quantum mechanics, if two particles collide, even if one has full knowledge of the particle properties before the collision, it is impossible to predict, in each individual case, in which directions the two particles will be traveling after the collision. One can only state the probability that they emerge from the collision in a certain pair of directions. Nonetheless, there is zero probability that the two particles emerge with a different total momentum and energy than they went in with. They are like a magic pair of coins, which when flipped individually give you heads or tails randomly and with equal probability, but when flipped as a pair always come up with opposite faces showing. So quantum indeterminacy still respects the conservation laws.
Perhaps for this reason Einstein’s later critiques of quantum theory focused less on its indeterminacy and more on its strange epistemological status. In quantum mechanics the actual act of measurement is part of the theory; those magic coins just mentioned exist in a state of (heads, tails)-(tails, heads) uncertainty until they are measured, and then they are forced to “decide” which state they are in. This is true even if the coins are flipped very far apart, implying that obtaining knowledge of one coin, through measurement, “changes” the state of the other coin an arbitrary distance away. This is the “spooky action at a distance” that Einstein detested, now known as “quantum entanglement.” But beyond its apparent tension with relativity theory, the entire conceptual structure seems to break down the barrier between the “real world” of objective nature and the subjective world of human perception. “Do you really believe that the moon exists only if I look at it?” he used to say. Such a notion fundamentally challenged Einstein’s credo.
In his autobiography he states, “Physics is an attempt to conceptually grasp reality as it is…, independently of its being observed.” In a letter to Born, late in his life, he amplified on this theme: “We have become Antipodean in our scientific expectations. You believe in the God who plays dice and I in complete law and order in a world which objectively exists, and which I, in a wildly speculative way, am trying to capture.” The importance of this dichotomy, the transitory, subjective, and ultimately insignificant individual versus the eternal order of the Cosmos, was central to his personal philosophy. As a very young man he rejected the “nothingness of the hopesand strivings which chases most men restlessly through life.” In this way one could “satisfy the stomach” but not the “thinking and feeling being.” But he soon realized that there was another way to live: “out yonder there was this huge world, which stands independently of us human beings and which stands before us like a great eternal riddle, at least partially accessible to our thinking and inspection. The contemplation of this world beckoned us like a liberation.” When he had risen to the apex of success in this pursuit, he spoke these words in tribute to Max Planck: “I believe … that one of the strongest motives that leads men to art and science is the escape from everyday life with its painful crudity and hopeless dreariness, from the fetters of one’s own everyday desires … a finely tempered nature longs to escape from personal life into the world of objective perception and thought.”
In Beethoven’s Ninth Symphony, after three movements of breathtaking beauty, the composer interrupts the final movement with the baritone’s thunderous introduction to the chorale section, “Oh friends, not these notes” (nicht diese töne). As spectacular as his previous creations had been, the composer was searching for something different, something better. Similarly, Einstein could not hear the musicality of his quantum creations, and would spend the rest of his life in search of the final movement that would bring his atomic symphony to a harmonious resolution.
1 A philosophical principle about the impossibility of a unified picture of the atomic world, the utility of which is controversial.APPENDIX 1: THE PHYSICISTS
In order of appearance:
Max Planck (1858–1947): German theorist, expert on thermodynamics, and Nobel laureate (1918) who introduced the first quantum ideas and his famous constant, h, in order to explain the blackbody radiation law.
Wilhelm Wien (1864–1928): German Nobel laureate (1911) who did the first important theoretical work on the blackbody radiation law, leading in 1896 to Wien’s law, which is now known to be an approximation to the correct Planck law.
Heinrich Weber (1843–1912): German experimentalist and researcher in thermodynamics. He was head of the Physics Department at the Zurich Polytechnic when Einstein was a student and clashed with him. His measurements of the temperature variation of specific heats influenced Einstein’s 1907 quantum theory of specific heat.
Marcel Grossmann (1878–1936): Swiss mathematician and classmate of Einstein’s at the Zurich Poly. His family connections played the key role in Einstein’s receiving the patent office job in Bern. Later, in 1913, he became a professor at ETH and collaborated with Einstein on a fundamental paper in General Relativity Theory.
Mileva Maric (1875–1948): Promising physics student who became Einstein’s first wife and, after failing to obtain her diploma, did not pursue a career in physics.
Sir Isaac Newton (1642–1726): Founder of classical mechanics through Newton’s three laws and the invention of calculus. If you are reading this book you know who he is.
Michael Faraday (1791–1867): English scientist whose experiments led to the concept of electric and magnetic fields; he was greatly admired by Einstein.
James Clerk Maxwell (1831–1879): Scottish theoretical physicist who first found the complete equations of classical electromagnetism, which are named for him. He was also a pioneer of statistical mechanics.
Ludwig Boltzmann (1844–1906): Austrian theoretical physicist who, along with Maxwell and Gibbs, founded the discipline of statistical mechanics. He discovered the fundamental microscopic law of entropy, S = k log W, where k is a fundamental constant of nature known as Boltzmann’s constant.
Josiah Willard Gibbs (1839–1903): American physicist and mathematician who, along with Boltzmann and Maxwell, founded statistical mechanics.
Hendrick Antoon Lorentz (1853–1928): Dutch theorist and Nobel laureate (1902) who initially doubted the validity of the Planck law. He became a close friend of, and father figure to Einstein, who regarded him as the greatest thinker he had ever met.
Lord Rayleigh (1842–1919): English mathematical physicist and Nobel laureate (1904); he was an expert on wave theory, particularly acoustics. He proposed the Rayleigh-Jeans law based on classical statistical mechanics, which leads to the incorrect prediction of the ultraviolet catastrophe.
James Jeans (1877–1946): English theoretical physicist and astronomer who contributed to, and also championed, the Rayleigh-Jeans law.
Svante Arrhenius (1859–1927): Swedish physicist and physical chemist, Nobel laureate (1903) for his work on electrolysis, who influenced the establishment and awarding of the Nobel Prizes in Physics and in Chemistry.
Arnold Sommerfeld (1868–1951): German theoretical physicist and leader in the development of the Bohr-Sommerfeld approach to the quantum theory of atoms.
Johannes Stark (1874–1957): German experimental physicist and Nobel laureate (1919), expert on the photoelectric effect, who led the anti-Semitic physics movement under the Nazis.
Walther Nernst (1864–1941): German physical chemist, inventor, and Nobel laureate (1920); he proposed the Third Law of thermodynamics and recruited Einstein to Berlin.
Niels Bohr (1885–1962): Danish physicist and Nobel laureate (1922), who proposed the first successful quantum theory of the atom and played a leading role in interpreting the final form of quantum mechanics.
Ernest Rutherford (1871–1937): New Zealand–born British physicist, Nobel laureate in chemistry (1908). His experiments revealed the nuclear structure of the atom.
Paul Ehrenfest (1880–1933): Austrian Jewish physicist who became professor in Leiden and a significant contributor to quantum theory. He was one of Einstein’s closest friends.
Arthur Eddington (1882–1944): English astrophysicist who led the eclipse expedition that confirmed Einstein’s theory of general relativity.
Satyendra Nath Bose (1894–1974): Indian theoretical physicist who first proposed the correct statistical method of treating quantum particles as indistinguishable in his paper on photons, sent to Einstein in 1924.
Erwin Schrödinger (1887–1961): Austrian theoretical physicist and Nobel laureate (1933). He invented the wave equation approach to quantum mechanics, which is the main approach used in modern physics.
Duke Louis de Broglie (1892–1987): French theoretical physicist and Nobel laureate (1929). He proposed the idea of matter waves, the complement to Einstein’s notion of quanta of light, and influenced Einstein’s work on the quantum gas of atoms.
Max Born (1882–1970): German Jewish theoretical physicist and Nobel laureate (1954). He played a major role in formalizing the Heisenberg approach to quantum mechanics, known as matrix mechanics, as well as in providing the probabilistic interpretation of Schrödinger waves. He was one of Einstein’s closest friends.
Werner Heisenberg (1901–1976): German theoretical physicist and Nobel laureate (1932). He invented the first correct formulation of modern quantum mechanics, matrix mechanics, in 1925. Two years later he proposed his uncertainty principle.
Wolfgang Pauli (1900–1958): Austrian theoretical physicist and Nobel laureate (1945). A brilliant but caustic personality, he discovered that electrons do not obey Bose statistics, because only one electron can occupy a given quantum state, the Pauli exclusion principle.APPENDIX 2: THE THREE THERMAL RADIATION LAWS
OVERVIEW
All objects emit electromagnetic radiation because they contain some amount of thermal energy that excites some of their atoms and molecules, at any one time, to higher energy states. These excited atoms and molecules then emit radiation (photons) and fall back down to their lowest (ground) state, while others are continually being excited, which maintains energy balance (thermal equilibrium). The amount of thermal energy an object has increases with its temperature, and so it emits more energetic, higher-frequency radiation as its temperature increases. But thermal radiation, unlike radio waves or laser light, for example, is not emitted at a single frequency; it is emitted over a broad band of frequencies, with the most radiation coming out at a specific frequency (the “peak” of the radiation curve), which depends on the temperature.
The key question challenging physicists circa 1900 was how much energy in the form of radiation is emitted in each band of frequencies for a perfect emitter at a given temperature. Because of a principle of thermodynamics known as Kirchoff’s law, a perfect emitter must also be a perfect absorber of radiation (i.e., a perfectly black object), called a blackbody, and the radiation it emits is called blackbody radiation. That doesn’t necessarily mean it will appear black to the eye; if it is heated to a sufficiently high temperature, it will glow at optical frequencies, which are visible to the eye. Thus the holy grail for the physics of heat at the time was the determination of the universal mathematical formula describing the energy of heat radiation (per unit volume) in a given frequency band for a given temperature; we will call this formula “the thermal radiation law.” There were three radiations laws of historical importance.
1. The Planck law: This law was proposed and then derived by Max Planck in the fall of 1900, when it became clear that the Wien law failed. In order to justify this new law, Planck had to introduce the concept of quantization of energy (although he did not put it that way), which set the quantum revolution in motion. It required the introduction of Planck’s constant, h, which appears in the radiation law and in the fundamental relation ε = hυ, relating the allowed increments of the energy, E, of a vibrating molecule to its frequency of vibration, υ. His law fit the experimental data very well and still does to this day. It is the correct radiation law according to modern physics.
2. The Wien law: This law was proposed by Wilhelm Wien in the early 1890s and was believed by many, including Planck, to be correct until 1900. It turns out to be an approximation to the correct Planck law that works well when one looks at frequencies that are higher than the peak frequency of the Planck law for a given temperature. For the achievable temperatures for blackbody experiments at the time of these first measurements, the higher frequencies were at or near the visible portion of the electromagnetic spectrum, and hence more easily measured, than those below the peak frequency, which were at infrared wavelengths.
3. The Rayleigh-Jeans law: A version of this law was proposed provisionally by Lord Rayleigh in 1900, and it turns out to be an approximation to the correct Planck law at low frequencies, that is, well below the peak frequency predicted by the Planck law. It was then proposed in its correct form by Rayleigh, with input from James Jean, in 1905; at roughly the same time it was derived but then rejected by Albert Einstein. Einstein rejected because it led to the “ultraviolet catastrophe.” This ominous descriptor was invented by the physicist Paul Ehrenfest because the Rayleigh-Jeans law predicts that the total energy in thermal radiation should be infinite. Einstein believed this property ruled out the law, whereas Jeans argued for a loophole that kept the theory in play until roughly 1911, when the Planck law became universally accepted.
This section assumes that one knows the properties of the exponential function, ex. Here the letter e stands for the irrational number that is the base of the natural logarithm. The object to be calculated to describe thermal radiation is the so-called spectral energy density of radiation, ρ(υ,T), which describes the energy of radiation emitted by a blackbody in a small interval centered at frequency υ for an object at absolute temperature T. According to Planck the correct form of this function is
where h is Planck’s constant, k is Boltzmann’s constant, and c is the speed of light. Note that the factor in parentheses in the numerator is the one that Bose mentioned in his letter to Einstein in 1924, which contained his new derivation of the radiation law.
The exponential function gets very large compared to 1 when its argument, in this case the ratio hυ/kT, is larger than 1 (i.e., when hυ > kT). When this is the case, one can neglect the term –1 in the denominator of the Planck law, and one finds the approximate form of the radiation law
(2) ρ(υ,T) ≈ (8πhυ3/c3) e−hυ/kT (Wien law).
This is the form of the radiation law proposed by Wilhem Wien, which we now know is a good approximation for frequencies υ >> kT/h. This defines what is meant by “high frequency”; it is a relative term, and it depends on the temperature of the blackbody. Visible light is considered high frequency at the temperature of the earth’s surface but not at the temperature of the sun’s surface.
In the other limit, of low frequencies, when the υ << kT/h, the exponential function becomes close to the value 1. In fact ex ≈ 1 + x (here x stands for hυ/kT), and if we put this into equation (1) for the Planck law, a factor of hυcancels in between the numerator and denominator, and with it all trace of the quantization of energy inherent in the Planck law vanishes. The resulting formula is
Note that this approximation to the thermal radiation law shows that for a given frequency, the amount of energy is proportional to the temperature. This is the experimental clue that led Planck to the correct radiation law. However, for a given temperature the energy density is also proportional to the square of the frequency. If this were the correct radiation law for all frequencies, one would reach the absurd conclusion that the energy density for high frequencies tends to infinity. This is the ultraviolet catastrophe that Einstein objected to in 1905.
The graph in figure A2.1 compares the three radiation laws.
FIGURE A2.1. Graph of the three proposed Radiation Laws, with energy density on the vertical axis and frequency on the horizontal axis, for a blackbody at room temperature. The lightest gray curve, which simply increases to infinity, is the Rayleigh-Jeans Law; note that it agrees with the Planck Law (black curve) for very low frequencies. The black curve is the Planck Law with a peak and then a decay, but with a more rapid rise at low frequencies than the Wien Law (gray). The Wien Law disagrees substantially with the Planck Law below the peak, but agrees very well with it above the peak. Inset is a blow up of the little dashed box in the graph at low frequencies, to show the excellent agreement between the Planck and Rayleigh-Jeans Law, and large disagreement with Wien’s Law that was first found experimentally circa 1900. Courtsey of Alex Cerjan.
REFERENCES
EINSTEIN’S WRITINGS AND CORRESPONDENCE
Einstein, Albert. “Autobiographical Notes.” In Albert Einstein: Philosopher-Scientist, pp. 1–94. Edited by P. A. Schilpp. La Salle: Open Court, 1970.
Einstein, Albert. The Born-Einstein Letters, 1916–1955: Friendship, Politics and Physics in Uncertain Times. Translated by Irene Born. New York: MacMillan, 1971.
Einstein, Albert. The Collected Papers of Albert Einstein. Translated by Anna Beck and consultation by Don Howard. 12 vols. Princeton: Princeton University Press, 1987–2009. References are to the English translations unless otherwise noted.
Einstein, Albert. Einstein Besso Correspondance, 1903–1955. Translated into French by Pierre Speziali. Paris: Hermann, 1972. English translations in the text by the author.
Einstein, Albert. Ideas and Opinions. Translated by Sonja Bargmann. New York: Random House, 1954.
BIOGRAPHICAL WORKS ON EINSTEIN
Bernstein, Jeremy. Albert Einstein. Edited by Frank Kermode. New York: Penguin Books, 1973.
Calaprice, Alice. The Quotable Einstein. Princeton: Princeton University Press, 1996.
D’Amour, Thibault. Once Upon Einstein. Translated by Eric Novak. Wellesley: A. K. Peters, 2006.
Dukas, Helen, and Banesh Hoffman, eds. Albert Einstein: The Human Side. Princeton: Princeton University Press, 1979.
Folsing, Albrecht. Albert Einstein: A Biography. Translated and abridged by Ewald Osers. New York: Penguin Press, 1998.
Frank, Phillip. Einstein: His Life and Times. Translated by George Rosen and edited by Schuichi Kusaka. New York: Da Capo Press, 1947.
French, A. P., ed. Einstein: A Centenary Volume. Cambridge, MA: Harvard University Press, 1979.
Hentschel, Ann M., and Gerd Grasshoff. Albert Einstein: Those Happy Bernese Years. Bern: Staempfli, 2005.
Highfield, Roger, and Paul Carter. The Private Lives of Albert Einstein. London: Faber & Faber, 1993.
Hoffmann, Banesh, with the collaboration of Helen Dukas. Albert Einstein: Creator and Rebel. New York: Viking Press, 1972.
Isaacson, Walter. Einstein: His Life and Universe. New York: Simon & Schuster, 2001.
Levenson, Thomas. Einstein in Berlin. New York: Bantam Books, 2003.
Moszkowski, Alexander. Conversations with Einstein. Translated by Henry L. Brose. New York: Horizon Press, 1970.
Neffe, Jurgen. Einstein: A Biography. Translated by Shelley Frisch. New York: Farrar Strauss Giroux, 2007.
Pais, Abraham. Einstein Lived Here. New York: Oxford University Press, 1994.
Pais, Abraham. Subtle Is the Lord. Oxford: Oxford University Press, 2005.
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Seelig, Carl. Albert Einstein: A Documentary Biography. Translated by Mervyn Savill. London: Staples Press, 1956.
Woolf, Harry, ed. Some Strangeness in Proportion: Einstein Centennial. Reading: Addison-Wesley, 1980.
EINSTEIN AND QUANTUM THEORY
Bolles, Edmund Blair. Einstein Defiant: Genius versus Genius in the Quantum Revolution. Washington, DC: John Henry Press, 2005.
Klein, Martin J. “Einstein and Wave-Particle Duality.” The Natural Philosopher, vol. 3, 1964, pp. 1–49.
Stachel, John. “Einstein and the Quantum” and “Bose and Einstein.” In Einstein from B to Z, vol. 9, pp. 367–444. Edited by Don Howard. Boston: Birkhauser, 2002.
QUANTUM THEORY AND QUANTUM MECHANICS
Haar, D. Ter. The Old Quantum Theory. Oxford: Pergamon Press, 1967.
Hermann, Armin. The Genesis of Quantum Theory (1899–1913). Cambridge, MA: MIT Press, 1971.
Kuhn, Thomas S. Black-Body Theory and the Quantum Discontinuity, 1894–1912. Chicago: University of Chicago Press, 1978.
Lindley, David. Uncertainty: Einstein, Bohr, and the Struggle for the Soul of Science. New York: Doubleday, 2007.
Mehra, Jagdish, and Helmut Rechenberg. The Historical Development of Quantum Theory, vols. 1–5. New York: Springer-Verlag, 1982–1987.
Pais, Abraham. Inward Bound: Of Matter and Forces in the Physical World. New York: Clarendon Press, 1986.
Van der Waerden, Bartel Leendert, ed. Sources of Quantum Mechanics. Amsterdam: North-Holland, 1967.
Wheaton, B. R. The Tiger and the Shark: Empirical Roots of Wave-Particle Dualism. Cambridge: Cambridge University Press, 1983.
Abragam, A. “Louis De Broglie.” Biographical Memoirs of Fellows of the Royal Society, vol. 34, 1988, pp. 22–41.
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Coffey, Patrick. Cathedrals of Science. Oxford: Oxford University Press, 2008.
Crawford, Elisabeth. “Arrhenius, the Atomic Hypothesis, and the 1908 Nobel Prizes in Physics and Chemistry.” Isis, vol. 75, 1984, pp. 503–22.
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ORIGINAL SCIENTIFIC RESEARCH ARTICLES (CHRONOLOGICAL)
1. Max Planck, “On an Improvement of Wien’s Equation for the Spectrum,” Proceedings of the German Physical Society, vol. 2, p. 202 (1900); reprinted in translation in Haar, The Old Quantum Theory, 79–81.
2. Max Planck, “On the Theory of the Energy Distribution Law of the Normal Spectrum,” Proceedings of the German Physical Society, vol. 2, p. 237 (1900); reprinted in translation in Haar, The Old Quantum Theory, 82–90.
3. Lord Rayleigh, “Remarks upon the Law of Complete Radiation,” Philosophical Magazine, vol. 49, pp. 539–40 (1900); reprinted in Scientific Papers by Lord Rayleigh, vol. 6, doc. 260, pp. 483–485, Dover, New York (1964).
4. Lord Rayleigh, “The Law of Partition of Kinetic Energy,” Philosophical Magazine, vol. 49, pp. 98–118 (1900); reprinted in Scientific Papers by Lord Rayleigh, vol. 6, doc. 253, pp. 433–451, Dover, New York (1964).
5. Albert Einstein, “On the General Molecular Theory of Heat,” Annalen der Physik, vol. 14, pp. 354–362 (1904); reprinted in CPAE, vol. 2, doc. 5, pp. 68–77.
6. Albert Einstein, “On a Heuristic Point of View concerning the Production and Transformation of Light,” Annalen der Physik, vol. 17, pp. 132–148 (1905); reprinted in CPAE, vol. 2, doc. 14, pp. 86–103.
7. Albert Einstein, “On the Electrodynamics of Moving Bodies,” Annalen der Physik, vol. 17, pp. 891–921 (1905); reprinted in CPAE, vol. 2, doc. 23, pp. 140–171.
8. Albert Einstein, “On the Theory of Light Production and Light Absorption,” Annalen der Physik, vol. 20, p. 199 (1906); reprinted in CPAE, vol. 2, doc. 34, pp. 192–199.
9. Albert Einstein, “Planck’s Theory of Radiation and the Theory of Specific Heat,” Annalen der Physik, vol. 22, pp. 180–190 (1907); reprinted in CPAE, vol. 2, doc. 38, pp. 214–224.
10. Albert Einstein, “On the Present Status of the Radiation Problem,” Physikalische Zeitschrift, vol. 10, pp. 185–193 (1909); reprinted in CPAE, vol. 2, doc. 56, pp. 357–375.
11. Albert Einstein, “On the Development of Our Views concerning the Nature and Constitution of Radiation,” Physikalische Zeitschrift, vol. 10, pp. 817–826 (1909), presented at the 81st Meeting of the German Scientists and Physicians, Salzburg, September 21, 1909; reprinted in CPAE, vol. 2, doc. 60, pp. 379–394.
12. “Discussion Following the Lecture: On the Development of Our Views concerning the Nature and Constitution of Radiation,” Physikalische Zeitschrift, vol. 10, pp. 825–826 (1909), presented at the 81st Meeting of the German Scientists and Physicians, September 21, 1909; reprinted in CPAE, vol. 2, doc. 61, pp. 395–398.
13. Albert Einstein, “On the Present State of the Problem of Specific Heats,” Proceedings of the Solvay Conference, October 30–November 3, 1911; reprinted in CPAE, vol. 2, doc. 26, pp. 419–420.
14. Niels Bohr, “On the Constitution of Atoms and Molecules,” Philosophical Magazine, vol. 26, p. 1 (1913); reprinted in The Old Quantum Theory, by D. Ter Haar, pp. 132–159.
15. Albert Einstein, “Emission and Absorption of Radiation in Quantum Theory,” Proceedings of the German Physical Society, vol. 18, pp. 318–323 (1916); reprinted in CPAE, vol. 6, doc. 34, pp. 212–216.
16. Albert Einstein, “On the Quantum Theory of Radiation,” Physikalische Gesellschaft Zurich, Mitteilungen, vol. 18 (1916); reprinted in CPAE, vol. 6, doc. 38, pp. 220–233.
17. Albert Einstein, “On the Quantum Theorem of Sommerfeld and Epstein,” Proceedings of the German Physical Society, vol. 19 (1917); reprinted in CPAE, vol. 6, doc. 45, pp. 434–443.
18. S. N. Bose, “Planck’s Law and the Light Quantum Hypothesis,” Zeitschrift für Physik, vol. 26, p. 178 (1924); reprinted in O. Theimer and B. Ram, “The Beginning of Quantum Statistics,” American. Journal of Physics, vol. 44, pp. 1056–1057 (1976).
19. S. N. Bose, “Thermal Equilibrium in the Radiation Field in the Presence of Matter,” Zeitschrift für Physik, vol. 27, p. 384 (1924); reprinted in O. Theimer and B. Ram, “Bose’s Second Paper: A Conflict with Einstein,” American Journal of Physics, vol. 45, pp. 242–246 (1976).
20. Albert Einstein, “Quantum Theory of the Monatomic Ideal Gas,” Proceedings of the Prussian Academy of Sciences, vol. 22, p. 261 (1924); reprinted in translation in I. Duck and E.C.G. Sudarshan, eds., Pauli and the Spin-Statistics Theorem, World Scientific, Singapore (1997), 82–87.
21. Albert Einstein, “Quantum Theory of the Monatomic Ideal Gas, Part Two,” Proceedings of the Prussian Academy of Sciences, vol. 1, p. 3 (1925); reprinted in translation in I. Duck and E.C.G. Sudarshan, eds., Pauli and the Spin-Statistics Theorem, World Scientific, Singapore (1997), 89–99.
22. Albert Einstein, “On the Quantum Theory of the Ideal Gas,” Proceedings of the Prussian Academy of Sciences, vol. 3, p. 18 (1925); reprinted in translation in I. Duck and E.C.G. Sudarshan, eds., Pauli and the Spin-Statistics Theorem, World Scientific, Singapore (1997), 100–107.
23. Louis de Broglie, “Black Radiation and Light Quanta,” Journal de Physique et le Radium, vol. 3, p. 422 (1922); reprinted in Selected Papers on Wave Mechanics by Louis de Broglie and Leon Brillouin, vols. 1–7, Blackie and Sons, London (1928).
24. Louis de Broglie, “A Tentative Theory of Light Quanta,” excerpt from Philosophical Magazine, vol. 47, p. 446 (1924); reprinted in I. Duck and E.C.G. Sudarshan, eds., 100 Years of Planck’s Quanta, chapter 4, World Scientific, Singapore (2000), 128–141.
25. Louis de Broglie, “Studies on the Theory of Quanta,” PhD thesis, originally published in Annales de Physique, vol. 3, p. 22 (1925).
26. Erwin Schrödinger collected nine of his seminal papers on the wave equation into a volume titled Abhandlungen der Wellenmechanik (Treatise on Wave Mechanics), which was originally published in 1927. These papers are available in English translation in E. Schrödinger, Collected Papers on Wave Mechanics, Chelsea Publishing, New York (1978). The nine papers are titled “Quantisation as a Problem of Proper Values, Parts I, II, III, IV,” “The Continuous Transition from Micro-to Macro-Mechanics,” “On the Relation between the Quantum Mechanics of Heisenberg, Born and Jordan, and That of Schrödinger,” “The Compton Effect,” “The Energy-Momentum Theorem for Material Waves,” and “The Exchange of Energy According to Wave Mechanics.” Note that the term “proper value” was the chosen translation for the German term Eigenvalue, which has become standard mathematical terminology in English as well.
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