Einstein and the Quantum (10 page)

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.

CHAPTER 8

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
=
mc
²
. 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.