Einstein and the Quantum

by Stone, A. Douglas

  Indeed something had. This mode of thinking was simply orthogonal to everything physicists had been trying to do in atomic theory, and it broke the impasse. Heisenberg informed his friend Pauli, who was elated, saying that the idea gave him renewed “joie de vivre and hope … it’s possible to move forward again.” Heisenberg wrote up his initial ideas with the boldly stated goal of establishing a new quantum mechanics, “based exclusively on relationships between quantities which are in principle observable.” Born, with another talented student, Pascual Jordan, quickly realized that Heisenberg’s “lists of numbers” were objects that mathematicians refer to as matrices, and that the rule for combining them that Heisenberg had invented was the known rule for multiplying matrices. An odd thing about representing physical magnitudes by matrices is that when multiplying matrices, in general x times y is not equal to y times x. This curiosity would end up having a deep significance in the final theory. Within a few months Born, Heisenberg, and Jordan were able to put together a definitive paper announcing the rules for calculating observable quantities in the new quantum mechanics, which, in their version, would become known also as “matrix mechanics.”

  Einstein, despite Born’s endorsement, reacted suspiciously to the breakthrough from the beginning, writing to Ehrenfest in September 1925 with a typically earthy judgment: “Heisenberg has laid a big quantum egg. In Göttingen they believe in it (I don’t).” Despite his skepticism, he realized that a substantial advance had been made, telling Besso in December 1925 that matrix mechanics was “the most interesting thing that theory had produced in recent times”; but he could not resist a dig at its odd structure, “a veritable witches’ multiplication table … exceedingly clever and because of its great complexity safe against refutation.” Sarcasm notwithstanding, he studied the theory closely and discovered several technical objections, which he communicated to Jordan.2

  Bose recalled that upon his arrival in Berlin in the fall of 1925, “Heisenberg’s paper came out. Einstein was very excited about the new quantum mechanics. He wanted me to try to see what the statistics of light-quanta … would look like in the new theory.” But Einstein’s reservations were beginning to win out; early in 1926 he wrote to Ehrenfest, “more and more I tend to the opinion that the idea, in spite of all the admiration [I have] for [matrix mechanics], is probably wrong.” Just as he was hardening his negative view, in January the newly reenergized Pauli showed how to derive the basic hydrogen spectrum using matrix mechanics, an apparently decisive proof that the theory was on the right track. Of course Schrödinger was just at that time deriving the same result by the quite different method of his wave equation.

  Schrödinger’s approach was superficially much more congenial to the classical physics worldview, based as it was on a continuum wave equation in space and time, similar to that of Maxwell, and seeming to arrive at quantized energies via the familiar properties of vibrating waves. Einstein, Planck, Nernst, and Wien, the reigning royalty of German physics, all jumped on the Schrödinger bandwagon immediately. Born, now a bit under siege, later recalled that Schrödinger’s paper “made much more of an impression than ours. It was as though ours didn’t exist at all. All the people said now we have the real quantum mechanics.” However, Born would soon have a key ally; Niels Bohr had been moving toward a view that the conventional space-time picture of the atom was fatally flawed, and his force of personality would eventually prevail, although not without some further twists and turns.

  Initially the two sides believed that they were faced with a choice between two fundamentally different theories, so that Einstein, in the same letter to Ehrenfest in which he called matrix mechanics “probably wrong,” described Schrödinger’s innovation as “not such an infernal machine [as matrix mechanics], but a clear idea—and logical in its application.” And a few weeks later, in early May, he told Besso, “Schrödinger has come out with two excellent papers on the quantum rules, which present some profound truths.” But the period of either/or decision making was brief. A dramatic change in the debate occurred at one of the famed Berlin colloquia, where Einstein often presided. A young student, Hartmut Kallmann, recorded the events. “People were packed into the room as lectures on Heisenberg’s and Schrödinger’s theories were given. At the end of these reports Einstein stood up and said, ‘Now just listen! Up until now we have had no exact quantum theory, and now suddenly we have two. You will agree with me that these two exclude each other. Which theory is correct? Perhaps neither is correct.’ At that moment—I shall never forget it—Walter Gordon stood up and said: ‘I have just returned from Zurich. Pauli had proved that the theories are identical.’ ”3 Actually by mid-March Schrödinger, prior to Pauli (who never even bothered to publish his proof), was able to show that the equations of matrix mechanics followed from his wave equation and vice versa; matrix mechanics could be used to derive the Schrödinger equation. The two theories were indeed mathematically equivalent.

  At this point the debate shifted to the question of the meaning of the new theory, and the aesthetic and conceptual merits of the two different formulations. Already, in his paper proving their equivalence, Schrödinger had slipped in a jibe against the matrix approach, saying that he was “discouraged, if not repelled” by the difficulty of its methods and its lack of transparency. And he repeatedly stated that his approach was the more “visualizable,” prompting a fed-up Heisenberg to declare in a letter to Pauli, “what [he] writes about Anschaulichkeit [visualizability] makes scarcely any sense…. I think it is crap.”

  Matters came to a head in July, when Schrödinger made a “victory tour” of the conservative physics centers of Berlin, where they had begun recruiting him to replace Planck, and Munich, where Wien and Sommerfeld were in charge. By coincidence Heisenberg was in Munich when Schrödinger spoke, and he raised some unresolved issues for wave mechanics in the question period at the end of the lecture. Before Schrödinger could respond, Heisenberg was almost “thrown out of the room” by Wien, who thundered, “young man, Professor Schrödinger will certainly take care of all these questions in due time. You must understand that we are now finished with all that nonsense about quantum jumps.” A shaken Heisenberg wrote immediately to Bohr, who responded by inviting Schrödinger to Copenhagen. A marathon session of conceptual arm wrestling ensued, ending with Schrödinger in bed exhausted and sick, but unconverted. The key point that Bohr insisted upon is that while Schrödinger’s wave equation appeared to restore a continuous description of nature, when applied to atoms it would inevitably lead back to the fundamental discontinuity of natural processes implied by quantum phenomena. At about the same time Einstein and his close friend Max Born were wrestling with exactly this issue.

  For Einstein, the mathematical equivalence of the two theories simply extended his doubts about matrix mechanics to wave mechanics. He was not immune to the exhilaration felt by his colleagues as the historic puzzles of atomic structure were being unraveled almost on a weekly basis. After another colloquium, at which the evidence for the newly discovered spin of the electron was presented, Bose ran into him on a streetcar: “we suddenly found him jumping [into] the same compartment where we were, and forthwith he began talking excitedly about the things we have just heard. He has to admit that it seems a tremendous thing, considering the lot of things which these new theories correlate and explain, but he is very much troubled by the unreasonableness of it all. We were silent, but he talked almost all the time; unconscious of the interest and wonder that he was exciting in the minds of the other passengers.”

  The unreasonableness that Einstein felt now focused mainly on the meaning of Schrödinger’s wavefunction, which somehow represented the behavior of electrons bound to atomic nuclei. Schrödinger originally tried to argue that his matter waves could accumulate in a localized region of atomic dimensions, carrying along a bump or “crest” that behaved like a particle. But further study soon showed that such a “wave packet” could not cohere over long times; the math was actually very
similar to that of light waves, and the failure of this idea reprised Einstein’s own failure to find particulate behavior in Maxwell’s wave equation back in 1910. It is likely that Einstein spotted this problem very quickly. A fallback position, taken by Schrödinger subsequently, was to assert that there simply are no electron particles; the “real electron” is a wave of electric charge density, spread out in space on dimensions somewhat larger than the atom. But there was a further basic problem with this picture. Einstein expressed this in June of 1926 in a letter to a colleague, Paul Epstein: “We are all here fascinated by Schrödinger’s new theory of quantum levels … strange as it is to introduce a field in q-space, the usefulness of the idea is quite astonishing.” What was this “q-space” that Einstein found so strange?

  All the waves that were known to physicists at that time were represented as an oscillating field or disturbance in our normal three-dimensional space, even electromagnetic waves, which, according to Einstein, didn’t require a medium (the ether) to exist. The number of particles in a wave did not enter the equation except through the density of the medium, as for sound waves, or through the intensity of the wave, for electromagnetic waves. But electron waves could not be represented that way. Isolated free electrons could be studied, and their charge could be measured; it was –e, the same in magnitude (and opposite in sign) from that of the proton. The periodic table of elements requires that hydrogen have a single electron, helium two electrons, lithium three, and so on. So to describe electrons in helium, for example, one needed to have a wave equation for two electrons in different quantum states, with total electric charge adding up to 2 times –e. There was only one way to do this, mathematically speaking, using Schrödinger’s equation: the “two-electron” wavefunction had to “live” in a six-dimensional space, three dimensions for the first electron and three more for the second. Moreover, the wavefunction for electrons in a large atom such as uranium-235 had to live in a 705-dimensional space! This was “q-space,” an abstract space that “copies” our three-dimensional space N times in order to represent N electrons. Einstein recognized this strange feature in his very first letter to Schrödinger, and to him it was an enormous clue that a classical wave picture might not be restored through Schrödinger’s equation.

  Einstein was not the only person puzzling over how to interpret Schrödinger’s waves. Max Born also had grave reservations about the idea that the electron was a conventional wave. He worked closely with the noted experimentalist James Franck, who did measurements of electron beams colliding with atoms. “Every day,” he recalled, “I saw Franck counting particles and not measuring continuous wave distributions.” The moment he learned about the Schrödinger equation, he had an intuition about what its matter waves represented. He recalled Einstein’s idea that the electromagnetic field is a “ghost field” guiding the photons. “I discussed this with him very often. He said that as long as there was nothing better, one can [use this approach].” But now, for matter waves, Born felt this was the true picture: the Schrödinger wavefunction represented a guiding wave of probability. Mathematicians and physicists were already used to the idea of assigning probabilities to a continuous space, essentially by dividing the space into infinitesimal regions. Born argued that Schrödinger’s wavefunction represented such a probability density,4 which actually moved deterministically in space as a wave but simply described how likely it would be to find an electron particle in each particular region of space.

  By the end of June 1925 Born went public with his idea, submitting a paper titled “Quantum Mechanics of Collision Phenomena.” In it he formulates the problem of a directed matter wave (representing a stream of electrons in a beam directed at an atom) that interacts with the electric field of the atom and then “scatters” in all directions, just as water waves hitting a post send out circular waves in all directions. Could this really mean that each electron “breaks up” and goes in all directions like a smeared-out electrical “oil slick”? That is exactly the view that Schrödinger wants to take, but Born is having none of it. He insists that the expanding circular wave just determines the probability of finding a whole, pointlike electron emerging in particular direction. To test this idea you need to do the same experiment over and over again and count the number of electrons that go in each direction. “Here the whole problem of determinism arises. From the point of view of our quantum mechanics, there exists no quantity which in an individual case causally determines the effect of the collision…. I myself tend to give up determinism in the atomic world.” In his view we need to adopt a weaker form of determinism: “the motion of particles follows probabilistic laws, but the probability itself propagates according to the law of causality.”

  When Schrödinger learned of Born’s interpretation, he was incensed and engaged him in an “acrimonious debate.” As Born recalled, “he believed that [matter waves] meant some continuous distribution of matter and I was very much opposed to it [because of Franck’s experiments]…. he was very offensive, as he always was when somebody objected to [his ideas].” Schrödinger’s opposition notwithstanding, the Born probabilistic interpretation of the wave function was widely adopted almost immediately, and was the basis of Born’s eventual Nobel Prize. However, the person who had inspired Born’s critical step, Einstein, was among the few holdouts. In November of 1926 Born wrote to his dear friend: “I am entirely satisfied, since my idea to look upon the Schrödinger wave field as a ‘[ghost field]’ in your sense proves better all the time…. Schrödinger’s achievement reduces itself to something purely mathematical; his physics is quite wretched.” But by this time Einstein’s reservations had solidified into an unshakable conviction. Just a few days later he sent Born his famous and crushing response: “Quantum mechanics calls for a great deal of respect. But some inner voice tells me that this is not the true Jacob. The theory offers a lot, but it hardly brings us closer to the Old Man’s secret. For my part, at least, I am convinced he doesn’t throw dice.”

  Some months earlier Einstein had met privately with Heisenberg to discuss quantum mechanics. Heisenberg had presented his view that the new theory should restrict itself to describing observable quantities, and not unobservable electron orbits. Einstein rejected this view, leading Heisenberg to rejoin, “isn’t that precisely what you have done with relativity theory.” Einstein responded, “possibly I did use this form of reasoning … but it is nonsense all the same.5 … It is the theory which decides what can be observed.” This conversation stuck with Heisenberg, and a year later, while pondering the meaning of quantum mechanics, it came back to him. “It must have been one evening after midnight when I suddenly remembered my conversation with Einstein, and particularly his statement, ‘it is the theory which decides what we can observe.’ I was immediately convinced that the key to the gate that had been closed so long must be sought right here.” Within days he had used the new quantum mechanics to prove his uncertainty principle. One could observe the position of an electron very accurately, or the momentum of an electron very accurately, but not both at the same time. That’s what the theory had decided. Even this realization, so fiercely opposed by Einstein, had been stimulated by his own insight.

  1 In addition to Heisenberg, Pauli, and Fermi, the others were Max Delbruck (PhD), Eugene Wigner, Gerhard Herzberg, and Maria Goeppert-Mayer, the second woman to win the prize in physics.

  2 Einstein wrote a number of letters to Heisenberg in this period, all of which have been lost. At least one, Heisenberg recalled, was signed “in genuine admiration.”

  3 Kallmann most likely erred in his memory of the city, since Gordon was working with Pauli in Hamburg at that time. Schrödinger was in Zurich.

  4 More precisely, it is the absolute square of the wavefunction that represents a probability density.

  5 Later, in replying to the same reproach from his friend Philipp Frank, Einstein responded with the pithy retort, “A good joke should not be repeated too often.”

  CHAPTER 29

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  All the fifty years of conscious brooding have brought me no closer to the answer to the question, “what are light quanta?” Of course today every rascal thinks he knows the answer, but he is deluding himself.

  —EINSTEIN TO BESSO, 1951

  “Here I sit in order to write, at the age of 67, something like my own obituary … [this] does … not come easy—today’s person of 67 is by no means the same as was the one of 50, of 30 or of 20. Every reminiscence is colored by today’s being what it is, and therefore by a deceptive point of view.” Einstein, in the autobiographical sketch he thus begins, confirms his initial disclaimer. Readers hoping to learn from the man himself amusing anecdotes or details of his personal life were disappointed; the article of forty-six pages is a rather dense treatment of his philosophy of science, the evolution of physical theory, and then his actual contributions to science, ending with a technical statement of his latest attempt at a unified field theory. However, his revolutionary work on light quanta, and his groundbreaking quantum theory of specific heat of 1905–1907, merit only one long sentence. His early discovery of wave-particle duality gets a bit less than one page, ending in a remark that the current quantum theoretical explanation for it is “only a temporary way out.” His foundational work on the quantum theory of radiation and the spectacular discovery of Bose-Einstein condensation get no mention at all. He devotes much more space to his critique of quantum mechanics than to his contributions thereto. In contrast, relativity theory, special and general, is laid out in beautiful and exacting detail.