Einstein and the Quantum (38 page)

De Broglie was determined to be a theorist, unlike his brother, but this was not nearly as felicitous a choice in France as in the German-speaking countries of Europe, which were leading the abstract developments of quantum theory. De Broglie's close friend and fellow theorist Leon Brillouin recalled, “
There was really no career
open for a theoretical physicist in the French organization. People who had curiosity for theory would go right away into pure mathematics…. Many of my colleagues told me ‘Are you crazy? To go into theoretical physics, there is no future.” Nonetheless Brillouin, and de Broglie, joined the small group around Paul Langevin who were working on quantum theory.

De Broglie's first thought was to focus on the theory of light quanta, which, unlike most other physicists, he had believed in since his first exposure to Einstein's work in 1912.

I never had any doubt
at this time about the existence of photons. I considered that Einstein had discovered them, that they raised many difficulties … but that in the end this was a problem to resolve, that one could not deny the existence of photons…. It must be noted that I was very young. I had not made any theoretical works and for that reason was not as attached to electromagnetic theory, as was Langevin…. Thus, I accepted this as something that must be required.

In addition, his brother Maurice had been very directly grappling with the particulate properties of light in his extensive research on the x-ray photoelectric effect. Louis and Maurice collaborated on the interpretation of these experiments in terms of Bohr's theory during 1920 and 1921; particularly striking was the fact that the x-ray radiation appeared capable of giving all its energy to a pointlike electron, an observation very difficult to reconcile with a picture of the x-rays as extended, spherically expanding waves. When Maurice spoke of his experiments at the 1921 Solvay Congress (which Einstein had skipped), he concluded that the radiation “
must be corpuscular
… or if undulatory, its energy must be concentrated in points on the surface of the wave.”

De Broglie's first significant contribution had a motivation remarkably similar to that of Bose: to derive “
a number of known results
of the theory of radiation … without the intervention of electromagnetic theory” (i.e., without using classical physics). “The hypothesis we adopt is that of light quanta.” However, unlike every other physicist working on the quantum problem, de Broglie was convinced that the key to unlocking it lay in the appropriate use of relativity theory, which he described as Einstein's “
incomparable insight
.” Thus, in the second paragraph of the paper, he introduces a very peculiar notion, that quanta are to be thought of as “atoms” of light, with a very small but nonzero mass. In almost two decades of quantum research neither Einstein nor anyone else in the field had made such an outré suggestion.
Photons (quanta) were conceived to have energy,
E
=
hυ
, and momentum,
p
=
hυ
/
c
, but their energy and momentum was not accompanied by a rest mass, precisely because photons, by definition, move at the speed of light and can never be at rest in
any
frame of reference. Nonetheless de Broglie asserts that photons
should
be conceived of as having a rest mass which satisfies Einstein's most famous equation,
E
=
mc
²
! Hence, he states, combining this with the Planck relation,
E
=
hυ
, this mass must equal
hυ
/
c
²
. Never mind that this mass would then vary continuously with the frequency of light, which seemed very odd—in the current work he simply assumes that this mass is “infinitely small,” while the speed of the photon is “infinitely close” to
c
, so that the usual relation
E
/
c
=
p
for photons still holds, to a very good approximation.
²

It must be noted that in the modern quantum theory the mass of the photon
is
precisely zero, and a finite photon mass played no role in the first full formulations of quantum mechanics, which emerged a mere four years later. Thus this forced marriage of the two most famous equations of modern physics (
E
=
mc
²
and
E
=
hυ
) ended in a speedy divorce. However, in the first paper de Broglie uses this hypothesis only to rederive a known result of electromagnetic theory, that light exerts pressure and that this pressure is half the value one finds for ordinary nonrelativistic particles.
³
From this point on in the paper he uses standard statistical and thermodynamic relations, very much as in Einstein's work of 1902–1906, to derive the law for blackbody radiation, arriving not at Planck's formula but rather at Wien's law.
⁴
The reason for this is that he has, like so many before, assumed that the quanta are statistically independent, and, as was discussed in connection with the work of Bose, this assumption inevitably leads to Wien's approximation to the Planck law. He misses completely the strange statistical attraction
that Bose stumbled upon, and Einstein would elucidate, two years later. The paper is significant for two reasons. First, de Broglie introduces the method of counting single photon states using units defined by Planck's constant, which Bose would rediscover a couple of years later.
⁵
Second, de Broglie expressly assumes that the formulas of relativity theory are used for “
atoms of light
” and thus asserts that relativistic mechanics is of central importance in quantum theory, whereas previously it played a very minor role. This novel point of view would shortly lead him to a historic breakthrough.

After his initial foray into the theory of light quanta in 1922, de Broglie became convinced that there must be some symmetry between the behavior of “atoms of light” and other massive particles, such as electrons or atoms. According to his friend Brillouin, he was fascinated by the experimental images of radioactive decay processes, in which massive particles (e.g., electrons and positrons) are emitted from the atomic nucleus and follow curved tracks due to the force exerted by a magnetic field, while the simultaneously emitted photon makes a straight track, since it lacks electric charge. Apparently de Broglie intuitively felt, “
Well, all this must be very similar
. Either they are all waves or they are all particles … [so he tried] to see if he couldn't make everything waves.” De Broglie recalled that the key ideas “
developed rapidly in the summer
of 1923,” perhaps in July. “
I got the idea that one had to extend
[wave-particle] duality to the material particles, especially to electrons.” In September and October of that year he submitted three short notes for publication, which contained “the essential things” that later entered into his thesis.

In pondering how to associate a wave with material particles, de Broglie had been struggling with an apparent paradox. Just as had he had for photons, he persisted in combining the two great equations of Einstein and Planck and hence associating a frequency with the particle's rest mass:
m
=
hυ
/
c
²
, but now he took the bold step of applying
this formula to electrons, for which the mass was not unmeasurably small. De Broglie insisted that this was “
a meta law of Nature
, [that] to each … proper mass
m
, one may associate a periodic phenomenon of frequency
υ
=
mc
²
/
h
.” In other words de Broglie postulated a sort of internal “vibration” of every particle, which acted like a ticking clock, even when it was at rest. Moreover, if this is so, reasoned de Broglie, then when the particle moves at velocity v, its “ticking” must slow down, because Einstein's theory of relativity predicts the universal effect of
time dilation
: clocks are measured to run more slowly when in motion relative to an observer. So the moving particle will appear to vibrate at a lower frequency,
⁶
υ
₁
, than it does when at rest.

However, de Broglie at the same time considered the most basic postulate of relativity theory, that the laws of physics are the same in all frames of reference, and hence that the energy of the moving particle must still be related to
some
frequency via Planck's constant (i.e.,
E
=
hυ
must still hold when the particle is moving). But since the energy of the particle is
larger
in the frame in which it is moving (it has kinetic energy, ½
m
v
²
, in addition to its rest mass energy), then to satisfy the Planck relation it must have a
higher
frequency,
⁷
υ
₂
, than its “rest frequency,”
υ
. So there were two frequencies one should associate with the particle motion, one
larger
than its rest frequency, one
smaller
; which one was physically relevant?

Both, was de Broglie's answer. While the particle is moving, “it
glides on its wave
, so that the internal vibration of the particle (
υ
₁
) is in phase with the vibration of the wave (
υ
₂
), at the point where it finds itself.” The “fictive wave” that guides the particle moves at just the right speed so that the wave peak coincides with the peak of the particle oscillations; the particle is like a lucky surfer, permanently attached to the crest of the perfect wave. De Broglie showed that, for this to be so, the velocity of the “phase wave” (as he termed his fictive waves) must have a specific value,
V
phase
=
c
²
/v, which is faster than the speed
of light.
⁸
Because the phase waves moved faster than light, they could carry no energy, according to relativity theory, but served only to guide the particle motion.

So that is de Broglie's picture: every particle has some unspecified internal oscillation, which must remain in phase with a mysterious steering wave that directs its motion but which moves ahead faster than light so as to remain always “in resonance” with the particle's oscillation. Even by the standards of the new quantum theory, this was a rather wild invention, and, if anything, Langevin greatly understated the case when he told Einstein it was “a bit strange.”

But de Broglie had at least one further result that supported his extreme conjectures. He took this picture and applied it to an electron circulating around a hydrogen atom. The electron would move around in a circular orbit, continually emitting these phase waves, which would zoom ahead of the particle and lap the particle almost 19,000 times for each electronic circuit. Again he asked the question: what was required so that each time a wave crest and the particle coincided, the particle oscillation and the wave oscillation were in phase? Almost miraculously, he was able to show that this requirement was equivalent to the Bohr-Sommerfeld rule for the allowed electron orbits in hydrogen.
⁹
It was this result that apparently impressed Einstein, who referred to it, in his paper on Bose-Einstein condensation, as “
a very remarkable geometric interpretation
of the Bohr-Sommerfeld quantization rule.”

While de Broglie's key ideas were developed in the fall of 1923, he prepared a longer and more comprehensive document as his thesis in the spring of 1924, and gave it to his adviser, Langevin, who clearly was concerned about whether by accepting the work he would be endorsing nonsense. “
It looks far-fetched
to me,” was his initial reaction, shared with a colleague. At some point in the spring Langevin spoke to Einstein, who was intrigued and agreed to look at a copy. Einstein “
read my thesis during the summer
of 1924” de Broglie recalled, and
wrote the very favorable report quoted above. “As M. Langevin had great regard for Einstein, he counted this opinion greatly, and this changed a bit his opinion with regard to my thesis.”

De Broglie then defended his thesis in November of 1924 before a distinguished but bewildered “jury” (as it is termed in France), chaired by the future Nobel laureate Jean Perrin and including the famous mathematician Elie Cartan, the eminent crystallographer Charles Mauguin, and his adviser, Langevin. While their verdict was positive and de Broglie was congratulated for his “
remarkable mastery
,” a student who attended the thesis defense later remarked, “never has so much gone over the heads of so many.” Maurice de Broglie sought a candid opinion from Perrin and was told, “all I can tell you is that your brother is very intelligent.”

De Broglie's thesis had come to Einstein's attention at the perfect time. Einstein was now deep into his second paper analyzing the statistical properties of the quantum atomic gas. And in addition to his realization that the principle of indistinguishability of particles is implicit in Bose statistics, and the possibility of quantum condensation this implied, he had made one more major mathematical discovery. Just as he had done for the gas of light quanta, which he had analyzed in the seminal work of 1909, he now looked at the
fluctuations
of the energy in a particular volume of the quantum gas of particles. For both the photon gas and for a gas of atoms in a box, the energy in a small region of the box can vary randomly in time, while maintaining the same energy on average. This is simply a reflection of the ceaseless give-and-take that corresponds to thermal equilibrium. Einstein's earliest insight into the wave-particle duality of light had come in 1909, when he derived a formula for the typical magnitude of these energy fluctuations. He found that it consisted of two contributions, one of which could be explained by the interference of light waves, but the other of which looked exactly like the fluctuations expected from a gas of particles with energy
E
=
hυ
. It was this latter, particulate term that was a revelation in 1909, supporting Einstein's hypothesis of light quanta and prompting him to declare that the future quantum theory would involve a “fusion” of the wave and particle concepts. Now, after adopting Bose statistics, he finally had the correct theory to evaluate the same quantity for the
atomic gas. He found that
exactly the same structure occurs
: the fluctuations are the sum of two terms, a “particle term” and a “wave term.” But in this case it is the “wave term” that is the surprise.