Einstein and the Quantum (43 page)

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

NICHT DIESE TÖNE

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.

After the decisive year of 1926, in which he rejected the new quantum theory as the ultimate description of reality, he briefly sought to show, via his classic method of gedankenexperiments, that the theory contained internal contradictions. However, fairly soon he accepted the consistency of its logical structure with the comment “
I know this business
is free of contradictions, but in my view it contains a certain unreasonableness.” By September of 1931 he would graciously nominate both Heisenberg and Schrödinger for the Nobel Prize, with the comment “
I am convinced that this theory
undoubtedly contains a part of the ultimate truth.”

But despite this grudging endorsement, Einstein himself never applied the quantum formalism to a specific physics problem for the rest of his career, except in the context of a famous critical paper written in 1935 with younger collaborators, Podolsky and Rosen. The article drew attention through a thought experiment to the “spooky action at a distance” implied by quantum theory, which the authors claimed made the theory an incomplete description of reality. Modern realizations of this “EPR” experiment have fully confirmed the existence of this effect, a counter-intuitive correlation between distant particles. Such effects are referred to as “entanglement”; they form the basis of much of the new field of quantum information science. Many now consider Einstein's recognition and prediction of the EPR effect as his last major contribution to physics.

Not only Einstein but also de Broglie and Schrödinger, the two quantum pioneers whom he had championed, made little contribution to the further application of quantum theory, and both ended up joining Einstein in rejecting it on philosophical grounds. As a consequence, the history of the discovery/invention of quantum theory was told from the perspective of Bohr, Heisenberg, Born, and their legions of students and collaborators. (Einstein, de Broglie, and Schrödinger
had
no students or collaborators in their works on quantum theory.) The matrix mechanicians, whose approach was instantly devalued following Schrödinger's discovery of the “real quantum mechanics,”
simply appropriated that work and gave it the interpretation that fit their understanding (and, it must be admitted, the experimental evidence). Ironically, Schrödinger was correct; his method
was
much more intuitive and visualizable than that of Heisenberg and Born, and it has become the overwhelmingly preferred method for presenting the subject. But with Born's probabilistic interpretation of the wave-function, Heisenberg's uncertainty principle, and Bohr's mysterious complementarity principle,
¹
the “Copenhagen interpretation” reigned supreme, and the term “wave mechanics” disappeared; it was all quantum mechanics. The limitations on human knowledge of the physical world implied by these concepts were accepted by all practicing physicists. To this new generation Einstein became known primarily for relativity theory, admired by all, and secondarily for his stubborn refusal to accept the elegant new atomic theory of everything.

However, if one takes stock of the conceptual pillars of the new theory, in light of the historical record, a rather different picture emerges. Einstein surely shares with Planck the discovery of quantization of energy, as Planck never accepted that the quantum of action implied quantization of mechanical energy until many years after Einstein had become the first to proclaim it. It was Einstein who first realized that quantized energy levels explained the specific heat of solids, which justified the Third Law of thermodynamics and brought chemists such as Nernst into the quantum arena. Einstein, in his paper on light quanta, discovered the first force-carrying particles, photons, now the paradigm for all the fundamental forces. Following up on this, he discovered the wave-particle duality of light and, in 1909, based on his rigorously correct fluctuation argument, predicted that a “fusion theory” must emerge to reconcile the two views. In 1916 his quantum theory of radiation combined the ideas of Bohr, Planck, and his own light quanta to put Planck's blackbody law on a firm basis. Here he introduced, for the first time, the core concept of intrinsic randomness in atomic processes, which the mature
theory would accept as fundamental. He also introduced the notion of the
probability
to make a quantum jump, and he distinguished between spontaneous and stimulated transitions, ideas fundamental to, for example, the invention of the laser. And during 1924–1925 he elevated Bose statistics from obscurity, explained what it meant and why it had to be correct, and derived the mind-boggling condensation phenomenon it implied, something undreamt of by Bose himself. Finally, without ever publishing it, he developed the rule of thumb that electromagnetic wave intensity could be thought of as determining a probability to find photons in a certain region of space, the idea that stimulated Born's crucial interpretation of matter waves.