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 will never 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.”
1 Even after he moved to the United States in 1933, Einstein never fully mastered the language, and one of his close collaborators, Leopold Infeld, said he functioned with “about 300 words, which he pronounced very weirdly.”
2 Bosons are the force-carrying particles of the fundamental fields. The most recent confirmed member of this group is the Higgs boson, related to the electroweak interaction. Atoms are not fundamental particles, but are composites of quarks and electrons that can still behave statistically as bosons.
3 The displacement law, which constrains the form of the Planck law but does not determine it, follows from general principles of thermodynamics and doesn’t require Maxwell’s equations. And Einstein had not used Bohr’s correspondence principle, which at that time related only to the mechanics of atoms, but had simply used the known coefficient of the Rayleigh-Jeans law. However, the latter did require some form of counting of waves, so at least the thrust of this objection by Bose had some merit.
4 Here P is pressure, V is volume, T is temperature, and R is the gas constant, related to Boltzmann’s constant, k, discussed earlier. A special case of this law is Boyle’s law, that gas pressure is inversely proportional to its volume at fixed temperature.
5 In the next chapter we will see that there actually were subtle flaws in this method, which Einstein would discover and then, through the application of Bose’s ideas, show that the real ideal gas will deviate from the classical behavior found by Boltzmann. But these deviations were not yet detectable, and the problem was not with the concept of a gas connected to unspecified reservoirs but with Boltzmann’s counting method.
6 Henceforth I will use “photon” (the modern term) and “light quanta” (Einstein and Bose’s term) interchangeably. The photon gas is the standard modern terminology.
7 Actually the relevant unit of “resolution” is a cell simultaneously in position and momentum, known as a “phase space volume” (mentioned in Bose’s first letter to Einstein); this cell has volume h3 (again as mentioned in Bose’s letter).
8 In this argument he includes at the end a final factor of 2 he needs to recover the correct coefficient by assuming that the concept of polarization of EM waves can be extended to photons. Since polarization is a property of waves and not particles, this step was not completely rigorous, as Einstein pointed out to Bose. Much later Bose would claim that he had proposed that the photon has a spin with two possible states, now the accepted theory, but that Einstein had rejected this view and “crossed it out” of the first paper.
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 k
now 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.
Einstein and the Quantum Page 26