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 He and others followed up this idea, but it was superseded by the concepts of Bose statistics.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, υ1 = υ0(1 –v2/c2)1/2.
7 Again from relativity theory, υ2 = υ0(1 –v2/c2)−1/2, so it is higher than υ0 by just the same factor that υ1 is lower.
8 Since any massive particle’s velocity must be less than the speed of light, the wave velocity, Vphase = (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 = mv, we find λ = h/mv. 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
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 th
e 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 Primus5 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.”
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