Einstein and the Quantum (35 page)

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 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
h
³
(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,
E
_mol
= 3
kT
/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.