Einstein and the Quantum (40 page)

Born in 1887 and raised in an imperial Vienna that represented the flowering of art and culture at the turn of the century, Erwin Schrödinger was closer to Einstein's generation than he was to the rising cohort of brilliant young theorists (Heisenberg, Pauli, Dirac)
⁴
who would join him in driving the quantum revolution to completion. An only child, raised by a doting mother and aunts, he showed great intellectual talent from an early age. His father had studied chemistry at university, and pursued serious interests in art and botany, but contented himself with running the family linoleum business, while investing his son with his unrealized professional aspirations. Homeschooled until the age of eleven, Schrödinger then attended the elite Akademisches Gymnasium, Vienna's oldest secondary school, where he was the top student in his class for eight straight years. “
I was a good student
in all subjects, loved Mathematics and Physics, but also the strict logic of the ancient grammars (Latin and Greek),” he recalled. Unlike Einstein, the independent-minded Schrödinger managed to get along with his teachers and, in looking back, could “
only find words of praise
for my old school.” His intellectual facility astonished his classmates, one of whom recounted: “
I can't recall a single instance
in which our Primus
⁵
ever could not answer a question.”

When he matriculated at the University of Vienna in 1906, his brilliance was already widely known; a friend, Hans Thirring, recalls encountering a striking blond young man in the mathematics library and being told by a fellow student, sotto voce, “
das ist der Schrödinger
.”
⁶
Their first meeting instilled in Thirring the conviction that “this man is really somebody special … a fiery spirit at work.” By the time Schrödinger reached adulthood his erudition was legendary; he lectured comfortably in German, English, French, and Spanish, recited and wrote poetry (even publishing a volume late in life), and became a true expert in the philosophy of Schopenhauer and the Hindu spiritual texts, the Upanishads. Schrödinger “
would translate Homer
into English from the original Greek, or old Provencal poems into German,” and insisted throughout his life that study of the ancient Greek thinkers was not something for his “
hours of leisure
” but was “justified by the hope of some gain in the understanding of modern science.” It was said of Schrödinger's physics articles that “
if it were not for the mathematics
, they could be read with pleasure as literary essays.”

After settling on physics as his main focus at the end of his undergraduate years, Schrödinger went on to graduate work, primarily in experimental physics or in theoretical topics relating to the experimental work going on at the university. “
I learnt to appreciate
the significance of measuring. I wish there were more theoretical physicists who did.” However, by the end of this period, around 1914, when he obtained his habilitation, he had decided that he was personally unsuited to be an experimenter and that Austrian experimental physics was second rate. Nonetheless he continued to do some laboratory work, and his reputation as a broadly trained physicist, conversant with both experiment and theory, would be of great value when he began searching for academic positions.

Schrödinger was poised to dive into the rushing currents of change in theoretical physics in 1914, with Bohr's atomic theory newly hatched and Einstein's general relativity on the near horizon. But, as it did for de Broglie, the Great War intervened. Schrödinger was called into service as an artillery officer, and he served in that capacity for three years before being transferred to the meteorology service. In general Schrödinger's military assignments were not among the most challenging or dangerous, and he mainly suffered from boredom, and a certain degree of depression, during this period. However, early on in his tour of duty, in October of 1915, he was caught up in one of the major battles around the Isonzo River on the Italian front and received a citation for “
his fearlessness and calmness
in the face of recurrent heavy enemy artillery fire.”

During his war service he wrote to his many women friends, but only one visited him at the front, a young woman from Salzburg named Annemarie Bertel, whom he had met through friends in 1913. She admired and adored Schrödinger from their first meeting: “
I was impressed by him
because, first of all, he was very good-looking.” They would marry in 1920, and within a few years the marriage evolved into a close, but nonmonogamous, relationship, with both fairly openly engaging in affairs, although Erwin was certainly the more active in this regard. For Annie (as she was known), this was the price of involvement with a great man. “
I know it would be easier
to live with a canary bird than with a race horse. But I prefer the race horse.”

When Schrödinger returned full time to physics research in 1918, he was not particularly focused on the problems of quantum theory. He had learned theoretical physics at university from Fritz Hasenohrl, a leading disciple of the great Boltzmann, who along with Maxwell and Gibbs founded statistical mechanics. Boltzmann had died by suicide in 1906, the same year that Schrödinger began his studies; but his atomic worldview now prevailed; it had become a pillar of modern physics. “
No perception in physics
has ever seemed more important to me than that of Boltzmann,” Schrödinger recounted, “despite Planck and Einstein.”

During the war he had filled several notebooks with statistical calculations very much in the spirit of Einstein's early work on Brownian motion and diffusion. Upon returning to civilian life he published two papers based on these notes, the second of which, dealing with fluctuations in the rate of radioactive decay, is the longest article he ever produced, stretching to sixty journal pages. It was a tour de force of applied mathematics, and it announced to the world that he was to be taken seriously as a statistical physicist. In the same period he also published his first paper on quantum theory, focusing on further developments in Einstein's quantum theory of specific heat, as well as two short papers analyzing the equations of general relativity. In yet another nod to Einstein's work, in 1919, he performed an experiment trying to distinguish between the wave and particle theories of light, using a very small source. The experiment was similar in a general sense to the failed experiment that Einstein proposed in 1921 (his “monumental blunder”) and gave similarly equivocal results.

Schrödinger was establishing his research style as a critic and polymath, one able to work expertly in many subfields at once, who took the ideas of others and either demolished them or clarified and extended them. Although his radiation experiment had not had been a major success, it resulted in an invitation from Sommerfeld to visit Munich, where he became enamored of the (old) quantum theory of atomic spectra, due to Bohr and elaborated in great detail by the “
beautiful work
” of the Sommerfeld school.
⁷
By 1920 he had been
appointed full professor at Breslau, and he threw himself into research on atomic spectra, something Einstein had never been willing to do. By January of 1921 he had produced a step forward in the theory of alkali atoms, leading to a correspondence with Bohr, who wrote: “
[your paper] interested me very much
… some time ago I made exactly the same consideration.”
⁸
He would continue to make respectable, but not decisive, contributions to the Bohr-Sommerfeld theory regularly, into the fateful year of 1925, when the old theory would be overthrown by two revolutions, one of his own making.

FIGURE 27.1.
Erwin Schrödinger circa 1925. AIP Emilio Segrè Visual Archives, Physics Today Collection.

By 1922 he had been recruited to Zurich and was a certified expert in both modern quantum theory and modern statistical physics, but still a virtuoso without a masterpiece of his own. Almost all his work for the next four years would be on either atomic spectra or the statistical mechanics of gases; surprisingly, it was the latter that led him to his great discovery, with more than a nudge from Albert Einstein. As we have seen, in February of 1925, shortly after the publication of Einstein's key paper on the quantum theory of the ideal gas and Bose-Einstein condensation, Schrödinger wrote to Einstein respectfully but firmly suggesting that his paper contained an error. When, in his reply, Einstein explained to him how the new statistics worked, the scales dropped from Schrödinger's eyes, and he was entranced by the “
originality of [Einstein's] statistical method
.” He immediately set out to deepen his understanding of this new form
of statistical physics, which he would soon describe as “a radical departure from the Boltzmann-Gibbs type of statistics.”

By July of 1925 he had produced a typically insightful but incremental response, a paper titled “Remarks on the Statistical Definition of Entropy for the Ideal Gas,” which contrasted Planck's definition of entropy for the gas with that of Einstein. Planck for some time had been suggesting a weaker form of indistinguishability of gas particles than that of Bose and Einstein,
⁹
which was sufficient to save Nernst's law but didn't lead to the weird statistical attraction that is implied by Bose-Einstein statistics. Schrödinger realized that Planck's method was illogical because it got rid of
too many
states. Recall that Bose-Einstein's new counting method, when applied to dice, would insist that the two dice “states” (4, 3) and (3, 4) are just one state, so that for each such unequal pair one should count only one state, not two, reducing the number of states and hence the entropy of the system. However, there is no such reduction for doubles (there is only one way to roll snake eyes); so there is no reduction in the number of double “states” for quantum versus classical dice. Yet Planck's method, once you understood it deeply, boiled down to counting each double as only
half
a state, which was clearly wrong. Schrödinger says exactly this: “
in order that two molecules
are able to exchange their roles, they must really have different roles … one is [then] almost automatically led to that definition of the entropy of the ideal gas which has recently been introduced by A. Einstein [Bose-Einstein statistics].” In a quaint custom of the time, this rather significant criticism of Planck was read to the Prussian Academy by Planck himself, on behalf of Schrödinger.

Einstein was impressed by this exegesis, which he himself apparently had not appreciated; in September of 1925 he wrote to Schrödinger again: “
I have read with great interest
your enlightening considerations on the entropy of ideal gases.” He then sketched for Schrödinger another approach to the ideal gas problem, which he had
worked through crudely, leading to results that he found puzzling. When Schrödinger wrote back to Einstein on November 3, in addition to applauding Einstein's development of Bose statistics he proposed to carry through Einstein's alternative approach in detail, which he was able to do in a scant few days. He was less troubled than Einstein by the answer he found, which confirmed Einstein's original argument, and proposed a joint publication: “
the basic idea is yours
… and you must decide about the further fate of your child…. I need not emphasize the fact that it would be a great honor for me to be allowed to publish a joint paper with you.”

A touching exchange ensued, in which Einstein insists he should not be a coauthor, “
since you have performed
the whole work; I feel like an ‘exploiter,' as the socialists call it.” Schrödinger immediately demurs: “
not even jokingly
would I have … thought of you as an “exploiter” … one might say: ‘when kings go building, wagoners have work.' ” On December 4 he sent Einstein a complete draft of the paper with the second author slot blank, but Einstein presented the paper to the Prussian Academy without his own signature. Einstein's intuition was again right; there were subtle errors in the reasoning, and the approach itself turned out to be unwieldy, in contrast to Einstein's first approach, which is still found in modern texts.

However, only a couple of weeks later, Schrödinger had yet another ideal gas paper in the works, which led directly to the wave equation. Just prior to his letter to Einstein in early November, Schrödinger had finally managed to get hold of de Broglie's thesis, which he had sought because of Einstein's strong endorsement of it in his work on the quantum gas. Having now got hold of the thesis, Schrödinger tells Einstein that “
because of it, section 8
[the wave-particle section] of your second [quantum gas] … paper has also become completely clear to me for the first time.” Two weeks later, writing to another physicist, he remarks that he is very much inclined toward a “
return to wave theory
.”