Einstein and the Quantum (42 page)

Heisenberg was not the only young genius to find his way to Born's research team in Göttingen. Six of Born's research assistants and one of his PhD students would go on to win the Nobel Prize,
¹
and three of them—Enrico Fermi, Wolfgang Pauli, and Heisenberg—would contribute cornerstones to the rising quantum edifice. Born, only three years younger than Einstein, was from Prussian Silesia, and was of Jewish descent (like many of Einstein's closest friends). He had been appointed associate professor at Berlin from 1915 to 1919, arriving just in time to observe Einstein's awe-inspiring success with general relativity theory. He, and his wife Hedwig, formed a lifelong friendship with Einstein, although Born maintained as well a certain reverence for his friend, whom he would refer to, after his death, as “
my beloved master
.” Born made seminal contributions to physics and eventually won the Nobel Prize himself, but he was not an imposing intellect, and he sometimes had trouble keeping up with his brilliant wards. Of Pauli, who was renowned for his critical brilliance, he said, “
I was from the beginning quite crushed
by him … he would never do what I told him to do.” Heisenberg, he recalled, was quite different: “
he looked like a simple peasant
boy, with short, fair hair, clear bright eyes and a charming expression” when he arrived, “
very quiet and friendly
and shy…. Very soon I discovered he was just as good in the brains as the other one.”

After a few months spent visiting Niels Bohr in the fall of 1924, Heisenberg returned to Göttingen with the germ of an idea for a completely new quantum theory of the atom, distinct from the old Bohr-Sommerfeld approach. This approach, while it worked for hydrogen and a few other atoms, appeared to be breaking down for more complicated atoms and molecules. In fact, by 1924 more than a decade had passed since Bohr's pathbreaking work, and a full quarter century since that of Planck; many physicists were beginning to wonder if the fundamental laws of the atom were simply beyond human ken. In May of 1925 the enfant terrible, Pauli, wrote despairingly to a friend, “
right now physics is very confused
once again—at any rate it's much too difficult for me and I wish I were a movie comedian or some such.” However, Heisenberg was just about to shake the field out of its malaise.

FIGURE 28.1.
Werner Heisenberg circa 1927. AIP Emilio Segrè Visual Archives, Segrè Collection.

Heisenberg's idea was to take the continuous trajectory of a particle, which in classical physics is represented by the three Cartesian coordinates
x
,
y
,
z
that vary continuously with time, and replace each coordinate with a list of numbers arrayed in rows and columns, rather like a Sudoku puzzle. Each number in the list is not fixed, but oscillates in time sinusoidally, with a characteristic frequency. When applied to electrons in an atom, the frequencies corresponded to the observable “transition frequencies” at which the atom would absorb and emit light. First, however, Heisenberg considered the most basic “toy problem” of mechanics, the familiar linear harmonic oscillator (mass on a spring). He was able to show that using his new definition for position, and a similar one for momentum, the energy of the oscillator was conserved; that is, it didn't change in time as long as the
energy took the special values found by Planck so long ago, quantized in steps of
hυ
(where
υ
is the frequency of the oscillator). So here, in Heisenberg's new arithmetic, the whole numbers of quantum theory also arose naturally from the math and were not imposed externally, just as they would later appear naturally in Schrödinger's wave approach. Heisenberg first discovered this while recovering from an allergy attack on the North Sea island of Helgoland, and he was so excited that he stayed up all night working, and then, lying on a rock watching the sun rise, he thought to himself, “well something has happened.”

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?