Einstein and the Quantum (14 page)

It is important to understand that the theory of special relativity is completely independent of quantum theory and can be seen (along with the later general theory of relativity) to be the culmination of classical physics and its deterministic worldview. Special relativity makes sense of classical Maxwellian electromagnetic waves, and it does so without the somewhat embarrassing, unobservable ether. Two months after undermining Maxwell's equations with his heuristic theory of light quanta, he vindicates them from a host of
experimental challenges by banishing the ether. Talk about creative tension.

On the other hand, by getting rid of the ether, it was clear that Einstein was now prepared to accept waves that
do not travel in a medium
, “fundamental waves.” A few years later he made this thinking explicit: “
one can obtain a satisfactory theory
only if one drops the ether hypothesis. In that case the electromagnetic fields which constitute the light will no longer appear to be states of a hypothetical medium, but rather independent entities emitted by the source of light.” Since EM waves were, from this point of view, a completely new kind of entity in physics, perhaps they
could be
something in between a classical particle and a classical wave, as suggested by Einstein's notion of light quanta. En masse they exhibited interference like classical waves and hence could cancel one another out, but when exchanging energy with matter they acted like localized particles. Einstein was willing to entertain this contradiction. He, of all the physicists of his time, was the only one to really imagine that these two apparently conflicting concepts could be married. Over the next six years Einstein would devote the main part of his energies to consummating this difficult marriage.

CHAPTER 11

STALKING THE PLANCK

“
The three of us are fine
, as always. The little sprout has grown into quite an imposing impertinent fellow. As for my science, I am not all that successful at present. Soon I will reach the age of stagnation and sterility when one laments the revolutionary spirit of the young. My papers are much appreciated and are giving rise to further investigations. Professor Planck (Berlin) has recently written to me about that.”

Thus the twenty-seven-year-old Einstein wrote to his former Olympia Academy colleague Maurice Solovine in April 1906, in the after-math of his miracle year. The “little sprout” he spoke of was his first son,
¹
Hans Albert, who had been born on May 14, 1904, and was now coming up to his second birthday. At this point Einstein remained virtually unknown to the wider physics community, having personally encountered only a handful of physicists during his studies at the Poly and in subsequent research, none of whom were eminent theorists. He was working steadily eight hours a day, six days a week at the patent office, describing himself (with characteristic saltiness) to a friend as “
a respectable Federal ink pisser
with a decent salary.” His salary had recently grown a bit more respectable. In April of 1905 (in the midst of his creative epiphanies) he had again submitted a dissertation for a PhD to the University of Zurich, choosing his safest work of the moment as the topic, that on irregular molecular movement (“Brownian motion”) and the determination of Avogadro's number. This time Kleiner and the committee accepted the thesis, and as a consequence he was promoted to technical expert second class at the patent office, with a 15 percent increase in salary.

FIGURE 11.1.
Albert Einstein in 1904 with his wife, Mileva Maric, and his young son, Hans Albert. ETH-Bibliothek Zurich, Image Archive.

While Einstein personally was unknown to the great men of theoretical physics, his work had already, barely a year later, made a significant
impact. As the above quotation makes clear, Planck had already written to tell him his work was very much appreciated (although Planck's letter has not survived). One might have supposed that Einstein's work on light quanta, relating as it did to the central achievement of Planck's career, the blackbody radiation law, would have been the main object of Planck's attention and appreciation. However this was not the case. Nothing is known of Planck's reaction to Einstein's quantum hypothesis until his letter of July 1907, quoted above, definitively rejecting the idea of light quanta in vacuum. In contrast, Planck immediately embraced the special theory of relativity; he, not Einstein, gave the first public lecture on the subject (crediting Einstein of course) in the fall of 1905 shortly after the theory was published in September of that year. (As the theory editor of
Annalen der Physik
, Planck naturally would have seen the paper when it arrived at the end of June.)

Not only did Planck quickly publicize relativity theory, but he also paid it the highest compliment possible from a working scientist: he redirected his research to the study and extension of the theory. In 1906 he published the first major contribution to relativity theory not due to Einstein, a proof that relativistic mechanics was compatible with the “principle of least action,”
²
an alternative mathematical formulation of classical mechanics that was flexible enough to encompass the alterations of Newton's laws required by relativity theory. From 1906 to 1908 all of Planck's new research related to relativity theory. Because of Planck's stature in the field, his immediate “lively attention” to relativity theory endowed it with a credibility and importance that it otherwise might not have achieved for some time. Einstein acknowledged this in a tribute to Planck in 1913 when he stated, “
It is largely due to
the determined and cordial manner in which [Planck] supported this theory that it attracted notice so quickly among my colleagues in the field.”

That Planck reacted in this manner was completely consistent with his personality and philosophy of science; he recognized that relativity theory, as strange as it appeared to laypeople and to some physicists, in fact
completed
classical mechanics and made it compatible with Maxwell's electromagnetic theory. Einstein himself described it as “
simply a systematic
development of the electrodynamics of Maxwell and Lorentz.” He frequently emphasized the continuity of relativity theory with earlier physical principles: “There is a false opinion widely spread among the general public that the theory of relativity is to be taken as differing radically from the previous developments in physics…. The four men who laid the foundations of physics on which I was able to construct my theory are Galileo, Newton, Maxwell, and Lorentz.” Special relativity was a theory a purist like Planck could love. Wayward quanta of light, propagating in vacuum but still interfering like waves, impugning the integrity of Maxwell's equations; now that was a completely different matter. The best he could do was to forgive this impetuous genius a youthful indiscretion.

Planck finally did address the light-quantum hypothesis in the letter of July 1907 quoted above (“[I] assume that processes in vacuum are described exactly by Maxwell's equations”), but apparently only in response to repeated prodding by Einstein, whose own preceding letters to Planck have been lost. After stating his belief in the validity of Maxwell's equations, and that the “quantum of action” (
h
) pertained only to the exchange of electromagnetic energy with matter, he continued, “
but more urgent than
this
surely rather old question
is at the moment the question of the admissibility of your relativity principle … as long as the proponents of the principle of relativity constitute such a modest little band as is now the case, it is doubly important that they agree among themselves” (italics added). Seven years after slipping discontinuity into physics through the back door, Planck still did not see this for the epoch-making event it was. In contrast, by the end of December 1906, Einstein had already realized that Planck's quantum of action was not going to remain trapped in Pandora's radiation cavity, and that the challenge it presented to the worldview of physicists was more fundamental than that of relativity theory.

As mentioned above, in his 1905 paper on light quanta Einstein had sidestepped a direct confrontation with Planck by crediting the Planck radiation law with being “
sufficient to account for all
observations made so far” but basing all his conclusions on the Wien approximation to it, which is valid for short-wavelength radiation. His resolute insistence that the statistical mechanics of the time could only give an impossible answer—
kT
of energy for each allowed wavelength in the cavity, leading to the ultraviolet catastrophe—indicates that at that time Einstein regarded Planck's “derivation” of his radiation law, employing the artifice of the energy element,
hν
, as highly suspect if not downright incorrect. By March of 1906, almost exactly a year after publishing his revolutionary light-quantum hypothesis, he had apparently reconsidered this view, and submitted a paper arguing that the Planck formula
requires
the concept of light quanta.

In a study published last year
I showed that the Maxwell Theory of electricity in conjunction with the theory of electrons leads to results that contradict the evidence on black-body radiation. By a route described in that study I was led to the view that light of frequency
ν
can only be absorbed or emitted in quanta of energy
³
hν
…. This relationship was developed for a range that corresponds to the range of validity of Wien's radiation formula.

At the time it seemed to me that in a certain respect Planck's theory of radiation constituted a counterpart [alternative] to my work. New considerations, which are being reported [here], showed me, however, that the theoretical foundation on which Mr. Planck's radiation theory is based differs from the one that would emerge from Maxwell's theory and the theory of electrons, precisely because Planck's theory makes implicit use of the aforementioned hypothesis of light quanta.

These are the opening words of the 1906 paper. Note here the consistency with his 1905 paper, beginning first with the statement that
conventional theory leads to a blackbody radiation law in contradiction to Planck's law (and for good measure he restates this incorrect law, which led to the ultraviolet catastrophe in the second section of the 1905 paper). Then he reiterates that his quantum hypothesis was based only on the Wien limit and not on the full Planck law. Finally he explicitly states that when he wrote his 1905 paper he believed that there was a tension, if not an outright contradiction, between the Planck law and the heuristic theory of light quanta. What had changed his mind on this?

Planck had introduced the quantization of energy for the “molecules” (he called them resonators) in his blackbody as a counting device, leaving it quite unclear as to whether this was a physical hypothesis or a mathematical convenience. From this hypothesis he derived the entropy of the resonators and then by further manipulations determined the distribution of energy among frequencies of radiation in the body. Now, in the 1906 paper, Einstein starts from exactly the same equation as Planck, relating the energy of radiation at frequency
ν
to the average energy of each molecular resonator in the black body. In 1905 he had calculated the average oscillator energy by conventional statistical mechanics and got the answer that each one had the same energy,
kT
, which when transferred to radiation and added up over an infinite number of possible wavelengths gave infinity. Now he decides to tinker with conventional statistical mechanics. He writes a mathematical expression for the entropy of one of Planck's resonators, one that he (and independently Gibbs) had found several years earlier, which looks different from Boltzmann's famous
S
=
k
log
W
but which he shows is mathematically equivalent. In his new expression, instead of counting states, one finds the entropy by adding up contributions from all the possible energies of each resonator.
⁴
He finds that when he allows the energies to take continuous values, as they do in Newtonian physics, he gets an expression for the entropy that leads to the ultraviolet catastrophe. But if he simply uses Planck's restriction, that
energy can increase only in steps of
hν
, the Planck law follows in a few steps of algebra. He then puts his cards on the table:

Hence we must view the following proposition
as the basis underlying Planck's theory of radiation: The energy of an elementary resonator can only assume values that are integral multiples of
hν
; by emission and absorption, the energy of a resonator changes by jumps of integer multiples of
hν
.

This in itself is not much of a mathematical advance over Planck; Einstein has just rearranged the mathematical route to Planck's formula in a way he finds more congenial and intuitive. To Einstein this approach makes it clear that the quantization of energy is not just a mathematical convenience but a hypothesis about nature, and one very closely allied to his hypothesis of quanta of light. Anyone who had read Planck's derivation carefully might have realized the same thing. In fact, however, we know of no other physicist of the time who
did
realize this, except for the omniscient Lorentz, and as we shall see, Lorentz ultimately drew the wrong conclusion from this realization—that the Planck formula must be wrong. Einstein does not even entertain the possibility that the Planck formula might be wrong—from his earliest work on the subject he seems to have accepted this law as an established experimental fact that must be dealt with (a view held by a knowledgeable few in Germany, but not more generally). Instead he attempts, as he did earlier in 1905, to find the law's
meaning
by linking it to the quantum hypothesis, but he is unsparingly honest about the flaws in his and Planck's reasoning. The starting point for Planck, and for him in the current paper, is a mathematical relationship between the energy of thermal radiation and the average energy of matter (resonators) in contact with that radiation,
a relationship
that was found on the basis of assuming the validity of Maxwell's equations
. To then deduce the Planck law, one inserts a quantum hypothesis that is foreign to, and apparently contradicts, Maxwell's theory—hardly a compelling chain of logic.