Einstein and the Quantum (39 page)

This parallelism between atoms and light impressed Einstein greatly because, as he says in his second quantum gas paper, “
I believe that it is more
than just an analogy, since a material particle … can be represented by a … wave field, as de Broglie has stated in a remarkable thesis.” He continues: “
this oscillating field
—whose physical nature is still obscure—must, in principle permit itself to be demonstrated by phenomena corresponding to its motion. So a beam of gas molecules travelling through an opening must experience a bending, which is analogous to that of a beam of light.” That was the key to testing this extension of wave-particle duality. Whereas before, one was looking for
particulate
behavior of light waves, such as the localized collisions of photons and atoms in the photoelectric effect, now one should look for
wave
behavior of particles: a stream of atoms should interfere with itself as it passed near an edge, exhibiting the wave phenomenon known as diffraction. Einstein began suggesting such a search to physicists in September of 1924, shortly after reading de Broglie's thesis.
¹⁰

Einstein's endorsement and extension of de Broglie's matter wave concept was decisive in bringing this idea under serious study in the physics community. “
The scientific world of the time
hung on every one of Einstein's words, for he was then at the peak of his fame,” de Broglie noted. “By stressing the importance of wave mechanics, the illustrious scientist had done a great deal to hasten its development. Without his paper my thesis might not have been appreciated until very much later.” Even
with
his paper the thesis was viewed with suspicion; a student of Sommerfeld's recalled, “
the paper of de Broglie
was studied [in Munich] too; everyone had objections (they were not very difficult to find), and no one took the idea seriously.” Thus the search to observe matter waves, which began almost immediately after Einstein's second paper on the quantum gas, was mainly pursued in Einstein's name.

Walter Elsasser, the very first experimenter to find evidence in this direction, illustrates this with the introduction to his initial paper: “
By way of a detour
through statistical mechanics, Einstein has recently arrived at a physically very remarkable result. Namely he makes plausible the assumption that a wave field is to be associated with every translational motion of a material particle…. The hypothesis of such waves, already advanced by de Broglie before Einstein, is so strongly supported by Einstein's theory that it seems appropriate to look for experimental tests for it.” Within two years these tests rendered an unambiguous verdict: electrons beams do show wavelike interference. This discovery was so striking that de Broglie, with Einstein's strong support, received the Nobel Prize in Physics in 1929, a mere four years after his work became widely known.

But had this enormous success corroborated de Broglie's specific model, involving superluminal phase waves? Not at all. Almost no trace of this concept survives in modern quantum mechanics, and the vast majority of contemporary physicists are completely unaware of the basis on which de Broglie argued for matter waves. The only equation that survives from de Broglie's thesis work is so simple that Einstein could have written it down any time after 1905.
¹¹
This is the famous relation
λ
=
h
/
p
=
h
/
m
v, which relates the momentum,
p
=
m
v, of a massive particle to its “De Broglie wavelength.” It is obtained in a few lines by an extension, from light quanta to material particles, of Einstein's equation
E
=
hυ
.
¹²
This formula, correctly interpreted, is essential to modern quantum theory but arises there without any appeal to relativity theory or de Broglie's phase waves.

De Broglie himself, having completed his thesis by age thirty-one, never again made a fundamental contribution to physics, although he remained active in research, unlike Bose. He was an “
uninspiring
” classroom teacher, who started and ended his lectures precisely on time and permitted no questions during or after them. The research seminars he organized for many years were stilted affairs, with only brief open interchanges, described as “
dry and devoid of passion
.” The disciples that congregated around him were “
not of the highest intellectual caliber
” and created “an atmosphere of admiration, not to say adulation.” For example, it was considered bad form to refer to “quantum mechanics,” even after it became the standard term; one was supposed rather to say “wave mechanics,” as homage to de Broglie's seminal role. His research career spanned more than fifty years (he lived to age ninety-five), and it is now widely acknowledged that his influence was not very positive for the development of theoretical physics in France.

FIGURE 26.1.
Prince Louis De Broglie circa 1930. Academie des Sciences, Paris, courtesy AIP Emilio Segrè Visual Archives.

For much of his life de Broglie worked diligently within the standard quantum theory, which emerged from the work of Schrödinger, Heisenberg, and Bohr, although he initially opposed it in 1927. Then, in 1952, at age sixty, he again rejected this approach and joined Einstein in searching for a new and more aesthetically satisfying theory. In 1954, a year before his death, Einstein wrote touchingly to de Broglie, “
Yesterday I read
… your article on quanta and determinism, and your ideas, so clear, have given me great pleasure…. I must resemble the bird from the desert, the ostrich, hiding its head in the sands of Relativity rather than to face the malicious Quanta. Indeed, exactly like you, I am convinced that one must look for a substructure, a necessity that the present quantum theory hides.”

 

¹
Unlike Louis, Maurice was never awarded the prize, although his research was prominently cited in the 1922 prize, awarded to Niels Bohr.

²
Note that
c
here can no longer be thought of as
the
speed of light, but rather the limiting velocity of light as its frequency and hence mass goes to zero.

³
When measured appropriately, in terms of its energy density.

⁴
De Broglie mentions in passing that if one were to consider not just isolated atoms of light but “
a mixture of monatomic, diatomic, triatomic
” molecules of light, Planck's law could be obtained, but then dismisses this as requiring “some arbitrary hypotheses.” He and others followed up this idea, but it was superseded by the concepts of Bose statistics.

⁵
Like Bose, de Broglie then finds his answer is off by a factor of 2, and he needs to insert this factor “by hand” to account for the two possible polarizations of light (which is a concept of classical electromagnetism, not present in the theory of light quanta at that time).

⁶
According to relativity theory,
υ
₁
=
υ
₀
(1 –v
²
/
c
²
)
^1/2
.

⁷
Again from relativity theory,
υ
₂
=
υ
₀
(1 –v
²
/
c
²
)
^−1/2
, so it is higher than
υ
₀
by just the same factor that
υ
₁
is lower.

⁸
Since any massive particle's velocity must be less than the speed of light, the wave velocity,
V
_phase
= (
c
/v)
c
, is necessarily greater than the speed of light.

⁹
Suitably generalized to include relativistic effects.

¹⁰
De Broglie also had suggested such a search, for interference of electrons, roughly a year earlier.

¹¹
Much later, under direct questioning from the physicist I. I. Rabi, Einstein allowed that he did indeed think of the famous equation
λ
=
h
/
p
for matter waves before de Broglie but didn't publish because “
there was no experimental evidence
” for it.

¹²
The logic is as follows: for a photon,
E
=
hυ
, and for light waves
E
=
pc
. If we assume both relations hold and use the relationship of wave frequency to wavelength,
υ
=
c
/
λ
, we get
λ
=
h
/
p
. If we assume the same relation holds for massive particles moving slowly compared with the speed of light, so that
p
=
m
v, we find
λ
=
h
/
m
v. The full quantum derivation of this is based on Schrödinger's equation and doesn't rely on the assumptions that are used in this simple argument.

CHAPTER 27

THE VIENNESE POLYMATH

Physics does not consist
only of atomic research, science does not consist only of physics, and life does not consist only of science.

—ERWIN SCHRÖDINGER

“
When you began this work
you had no idea that anything so clever would come out of it, had you?” This question was addressed to the Austrian theorist Erwin Schrödinger sometime in the fall of 1926. The questioner was a young female admirer of the thirty-nine-year-old physicist, whose unusual marriage allowed for many such “friendships.” The work in question was that leading to the most famous equation of quantum mechanics, the “wave equation,” named after its inventor. Schrödinger's scientific colleagues were less restrained in their praise. The reserved Planck effused, “
I have read your article
the way an inquisitive child listens in suspense to the solution of a puzzle which he has been bothered about for a long time.” Einstein, who learned of the work from Planck, wrote simply, “
the idea of your article
shows real genius.”

At the time of this seminal work, Schrödinger was a professor at the University of Zurich, occupying the very same chair that Einstein had once held as his first academic position.
¹
Schrödinger was in the midst of what he called his “First Period of Roaming,” during which
he moved between various positions, as had Einstein fifteen years earlier, ascending the academic hierarchy. Indeed, in 1927, after the great triumph of his wave equation, Schrödinger would end up as Einstein's colleague in Berlin, after receiving the signal honor of succession to the chair of the recently retired Planck. Even before that, Einstein and Schrödinger had become allies in the struggle and competition to create the new atomic theory, and they shared certain intellectual habits. Schrödinger, like Einstein, did almost all his research alone, unlike the other school of quantum theory involving Bohr, Sommerfeld, Max Born, Werner Heisenberg, Pascual Jordan, and Wolfgang Pauli, who primarily worked collaboratively. Also, Schrödinger and Einstein had a sincere respect for and interest in philosophy,
²
and they shared a similar philosophy of science, influenced by the positivism of Ernst Mach but with a strong note of idealism.

However, unlike Einstein, Schrödinger had been appointed at Zurich primarily for his breadth of knowledge, outstanding mathematical abilities, and brilliant intellect—not because of any breakthrough attached to his name. In 1926, when he finally wrote his name into the history of science, he was already thirty-nine years old, well past the age when radical breakthroughs are expected from a theoretical physicist. And in fact his style of research had never before involved a daring leap into the unknown; instead his modus operandi was to criticize and improve the work of others.

In my scientific work
… I have never followed one main line, … my work … is not entirely independent, since if I am to have an interest in a question, others must also have one. My word is seldom the first, but often the second, and may be inspired by a desire to contradict or to correct, but the consequent extension may turn out to be more important than the correction.

In a sense, his work culminating in the wave equation
was
in that vein, building strongly on the insights of Einstein and de Broglie, but in this instance the extension was of historic consequence. In fact the state of quantum theory in 1925 called for just such an outsider, a critic who understood the two main lines of research, the Bohr-Sommerfeld atomic theory and the Einstein–Bose–De Broglie statistical theory of quanta, but who had a sentimental attachment to neither.

Erwin Schrödinger himself, while a man of great personal magnetism, was not known for his sentimental attachments. In his autobiographical sketch, written in his seventies, he reflected that he'd had only one close friend in his entire life and that he had “
often been accused of flirtatiousness
, instead of true friendship.” Flirtatiousness understates his behavior with respect to the opposite sex. He ends his sketch with the most titillating of disclaimers. “
I must refrain from drawing
a complete picture of my life, as I am not good at telling stories; besides, I would have to leave out a very substantial part of the portrait, i.e. that dealing with my relationships with women.” Thus we do not learn, for example, the name of the mystery woman (not his wife) who accompanied him on the Christmas ski vacation of 1925 during which the wave equation was discovered.
³