Einstein and the Quantum (36 page)

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,
⁷
which 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 enough
⁸
to observe pure Bose-Einstein condensation—Einstein's last great experimental prediction, coming to fruition a lifetime after its first statement.