Einstein’s second paper about Bose-Einstein statistics, the one in which he cited Louis de Broglie’s work, sent ripples through the world of avant-garde physics. In Göttingen, it led to the realization that de Broglie’s prediction of electron waves had already been observed; yet that recognition led to no further study. The physicists of Göttingen did not seize the moment to look more deeply. In Munich, Arnold Sommerfeld raised the news of de Broglie’s thesis with his graduate students. They looked briefly but did not take the ideas seriously. It is often said that Einstein was viewed among physicists as a god and that even the greatest of them would grow humble and silent when he entered their presence. On some level, that is true, but it cannot be the full story because the history of Einstein is stuffed with incidents where Einstein’s hints are ignored. By itself, his authority was rarely enough to make a physicist think harder.
Only in Zürich did his colleagues say that they should try harder to understand the baffling Frenchman whom Einstein had seen fit to endorse. Two physicists in Zürich—Peter Debye and Erwin Schrödinger—together held parts of the job that had been offered years earlier to Einstein. The Yes-No answer that Einstein had tendered the Swiss led to his rejection of a coprofessorship at the University of Zürich and the Polytechnic. In his place, Debye had come to
the university while Schrödinger came to teach at the Polytechnic. Schrödinger was particularly eager to study de Broglie’s ideas, in part because he was already experienced in waves and statistics, and partly because, for Schrödinger, Einstein really was a god. Relativity, especially general relativity, was, according to a future colleague, “his chief love.” And like Einstein, Schrödinger thought that the existence of two different physical fields, gravitational and electromagnetic, occupying the same space at the same time was too ugly to endure, and he, too, longed for a unified field theory. So when his hero wrote that de Broglie’s ideas “involve more than merely an analogy,” Schrödinger decided to study this new theory, according to which “a moving corpuscle is nothing but the foam on a wave of radiation in the basic substratum of the universe.” Debye and Schrödinger agreed that Schrödinger would investigate the “undulatory” idea more fully and report his findings to a colloquium.
The problem with de Broglie’s waves was similar to the popular joke about the economist who recognizes a fact as being true in practice, but who doubts that it can be true in theory. De Broglie’s waves had been observed in the lab, but no theory described them. If there were waves in practice, there should be a wave equation in theory. Schrödinger would have to find the equation that defined nature’s wavy groove.
Any such equation that Schrödinger proposed would have to support the well-established laws of motion for large particles like planets orbiting the sun, or balls on a soccer field, or even pollen dust floating on water. In this regard, Schrödinger was like Einstein approaching general relativity. Einstein’s equation had needed to embrace gravity, relativity, and the new business about acceleration and light. Schrödinger expected that his equation would have to capture Newton’s laws of motion, plus special relativity, plus the established laws of quanta and electron function, plus the new ideas about particles surfing the waves of reality.
Whew! One begins to remember just what it was about Einstein’s ideas that made his colleagues grow silent. They were not too crazy, just too hard. Professional scientists cannot risk years of work by running down a likely dead end. Einstein spoke contemptuously of the
kind of physicist “who looks for the thinnest spot in a board and then drills as many holes as possible through it,” but of course that approach is eminently practical if you are concerned about your career.
Perhaps there was something in Switzerland’s mountain air, but Schrödinger, like Einstein, seems to have been a bold jumping horse. He gave himself to optimism and went hunting for a new wave equation. He was already well acquainted with the way sound waves function, and in particular he understood how wave groups or packets act together. That experience gave a path for approaching de Broglie’s mobiles. De Broglie’s paper had suggested a reason for the most arbitrary feature of an electron orbit—its use of fixed (or “quantized”) orbits.
Bohr had visualized electrons orbiting an atom’s nucleus, but if electrons really moved through the electric field of a nucleus, like planets orbiting the sun, the electrons would radiate energy and fall into the nucleus like a comet tumbling into the sun. Plainly, electrons do not fall that way. Why not? We know that before moving things can come to a halt they must lose their momentum. Things moving along a curve will stop only when they lose their angular momentum. To prevent electrons from coming to a halt, Bohr simply decreed that the electron’s angular momentum cannot decrease continuously. He took Planck and Einstein’s energy element, hυ, tossed out the frequency part (because electrons were assumed to be nonvibrating particles) and replaced it with the letter n, which could be any whole number (1, 2, 3, … and so on). Then he performed a little mathematics to change h. He gave no physical explanation for his math, but he divided Planck’s h by 2π. The quantum h was already extremely small. Dividing it by 2π made it even smaller, but it still was bigger than 0. Multiplying this new small number by n—whose least possible value is one, not zero—ensures that an electron’s angular momentum (and, therefore, its energy) cannot decrease forever. Bohr’s formula was arbitrary, but it worked.
The fact that Bohr’s arbitrary rule worked so well gave him a special reputation among physicists. He did not take facts, explain them, and provide understanding. His genius seemed to rest on intuitions, deep insights based on more than normal powers. Instead of discover-
ing natural explanations as ordinary scientists do (or even imagining the explanations as geniuses do), Bohr intuited successful rules that allowed for the practical continuation of work in physics. Students like Heisenberg were delighted to find a clear pathway to progress in their research, but established physicists who wanted full explanations for their rules were deeply troubled. For them, de Broglie’s paper suggested that help was on the way.
De Broglie had proposed that the electron is not a particle orbiting like a planet but a mobile rippling around the atom like a wave. For physicists with a mathematical imagination this idea suggested a meaning for Bohr’s whole number, n. Square dancers will have an advantage at visualizing how numbers can determine the size of whirling systems. Imagine four dancers holding hands while swinging around a circle. Now suppose a fifth dancer joins the group. The size of the whirling circle must grow, too, to allow room for the new dancer’s arms. Suddenly two of the dancers leave. The remaining dancers must move into a tighter circle so that they can continue to hold hands while moving. De Broglie’s orbit is the equivalent of the dancer’s whirling circle, and each dancer is the equivalent of one full wave. As the number of waves in an orbit changes, so does it size. Also, just as you need a whole person to be a dancer, so an orbit needs a whole wave. You cannot have partial dancers or partial waves, so Bohr’s whole number rule suddenly makes physical sense.
Schrödinger, with his strong background in the physics of waves, immediately recognized a promising mathematical approach to the boundaries forced by whole waves. In this math Schrödinger was luckier than Einstein because, if he was right in his approach, he would not need to learn a new geometry before finding his equation.
After some struggle, Schrödinger derived a draft equation. It seemed to handle all the old cases, so he moved on to the newer problem of quanta and matter. He computed the motion of an electron orbiting a hydrogen atom but the result, regrettably, did not match the experimental data. Schrödinger had to find another approach; however, nothing occurred to him immediately.
Another way that Schrödinger resembled Einstein was in his fondness for and success with women. Possibly he even matched Einstein
in the way he grew more passionate as physics grew more frustrating. It is well known to science historians that during this period Schrödinger took a girlfriend to an alpine resort and between—or possibly during—erotic interludes he gained some new ideas. Sadly, we know nothing else about the woman and she must remain the phantom muse of quantum physics.
Schrödinger’s new work used an idea studied by his colleague Peter Debye in the years before the war. This was the wave packet, the bulge moving through the bullwhip. Schrödinger calculated that the speed of a moving particle and the speed of a wave packet matched. This kind of identity was the sort of thing that generally brought Einstein up on his toes because it hinted that behind the mathematical match there could be a real, physically meaningful identity. This match between the speed of particle and wave packet, Schrödinger declared, “can be used to establish a much more intimate connection between wave propagation and the motion of the representative point than was ever possible before.”
That talk of “the representative point” shows how profoundly Schrödinger’s understanding of nature was already changing. “The representative point” was an abstract term for what just a few months earlier he had called corpuscles or particles. Solid matter in his emerging theory was becoming more abstract, a representative of an underlying wave. In this view we can imagine God with a million arms and each of those arms is cracking a bullwhip with a wave packet moving along it. Only, for us mortals in the illusory world of gross sensations, God is invisible, the arms are invisible, and the bullwhips are invisible. All we can see are little points of matter—the merest representatives of the grand system of waves that invisibly comprise the universe’s secret reality.
“The true mechanical processes,” Schrödinger said, “will be realized and represented appropriately by the wave processes … and not by the motion of image points.…. Then we must proceed from the wave equation and not from the fundamental equation of mechanics, in order to include all possible processes.”
For the second time in 10 years Newton’s physics was being toppled. Oh, sure, as with gravity, the old equations were accurate
enough to keep using when solving practical problems, but they could no longer claim to express what was really going on in the universe. Waves, wave packets, and representational points were replacing the forces that pushed the indivisible atomic particles that Newton had believed in.
In the midst of this revolution, Schrödinger’s great stride came when he realized he did not have to be quite so revolutionary. He saw that if he altered his equation so that it continued to match Newton’s mechanics, but no longer agreed with special relativity, his quantum calculations matched the experimental results. Omitting relativity was no small thing, and Schrödinger, who loved relativity for its power and beauty, knew that a corner of the sheet was staying untucked, but he could see it was progress and took his step.
One appeal of this near miss was the way it promised to get rid of what Schrödinger called “those damn jumps.” The electron’s habit, in Bohr’s account, of jumping from orbit to orbit was too spooky, like the ghost of Hamlet’s father—it’s here, it’s there, it’s nowhere in between. Schrödinger’s idea built on what he knew about sound waves. Musicians knew about them too. Violinists, for example, can produce a rhythmic beat without the use of percussion instruments, simply by creating two separate notes at the same time. Musical strings can vibrate simultaneously at many different frequencies, giving them the rich harmonic sound that people love. In the hands of a skilled performer these separate notes will produce an audible beat, the vibrato that Einstein’s own playing so notoriously lacked. A sensitive vibrato throbs like a heart rather than ticks like a clock. The sounds are produced by shifting frequencies. The sound reflects the differences between the wavelengths of the various, simultaneous notes, and the throbbing comes from the way the violinist keeps changing the string lengths so that the notes keep changing ever so slightly.
Schrödinger proposed that the changes in electron emissions could be understood as a kind of atomic vibrato. As the frequencies of an electron shifted slightly, they would create slightly different harmonic beats that would look like fluctuating pulses of energy. Schrödinger commented, “It is hardly necessary to point out how much more gratifying it would be to conceive of the quantum transition as an
energy change from one vibrational mode to another than to regard it as a jumping of electrons.”
Exciting as all this was, Schrödinger found that the more deeply he probed, the less physically real his waves became. His theory appeared in a series of four articles published between January and June, 1926. In each paper, the wave became a bit less part of nature and a shade more a part of mathematics. The theory’s basic problem was that it did not make good physical sense. What are these waves? Yes, the speed of a particle matches the speed of a wave packet, but where do all the elements of the basic wave form enter the story? The wave form is familiar to anybody who ever stood on a pier and watched the sea lapping below. Waves vary in height and length. Schrödinger’s problem was that none of these elements appear in his final equation for the particle-packet’s velocity. What kind of wave can there be when it has none of the wave form?
“If,” said Schrödinger in his final paper, “we regard the whole analogy [between optical waves and mechanical movement] merely as a convenient means of picturization, then the defect is not disturbing, and we would consider any attempt at improving upon it as idle trifling.” That view might satisfy some, but Einstein—and Schrödinger too, on days when he was not in love with an idea—always wanted to know what was really going on.
And Schrödinger saw a second problem if his system was taken literally. “Classical mechanics,” he maintained, “breaks down for the very small dimensions and very great curvatures of path.” There can hardly be a smaller dimension and tighter curve than those encountered by an electron buzzing around an atomic nucleus.
So Schrödinger no longer felt confident that his waves were real and he deliberately chose an abstract symbol—ψ, called psi, or p-sigh—to represent the solution to his equation. “One may, of course,” he commented, “be tempted to associate the function psi with a vibrational process in the atom, a process possibly more real than electronic orbits whose reality is being very much questioned nowadays. Originally, I, too, intended to lay the foundations for the new formulation of the quantum conditions in this more intuitive manner, but later I preferred to present them in … neutral mathematical form.”
For Schrödinger, the strength of this approach remained what had been its most compelling attraction when he first read de Broglie. It got rid of Bohr’s arbitrary rules for inserting whole number ns into quantum equations and replaced them with a coherent measure of waves occupying space. Yet even that space was no longer the kind of everyday space familiar to anybody who ever got lost in a woods. It was an abstract, mathematical “space” with its own rules and properties. As Schrödinger put it in his fourth and final paper on undulatory mechanics, “The psi function is to do no more and no less than to offer us a survey and mastery over [electrodynamic] fluctuations by [using] a single differential equation. It has been repeatedly pointed out that the psi function itself cannot and may not in general be interpreted directly in terms of three-dimensional space … because it is in general a function in [a mathematical] space and not in real space.”
Nevertheless, Schrödinger’s close attention to what was really going on was much appreciated by Einstein and his colleagues. Planck responded early to the articles, writing Schrödinger, “You can imagine the interest and enthusiasm with which I plunge into this study of these epoch-making works.”
Lorentz wrote, “If one can successfully explain the phenomenon [of fluctuating electron radiation] by connecting a definite frequency to the moving electrons, it would be much more beautiful” than in rival, more arbitrary theories.
Einstein wrote him too, “Professor Planck pointed your theory out to me with well justified enthusiasm, and then I studied it with the greatest enthusiasm…. The idea of your article shows real genius.”
This note delighted Schrödinger who promptly replied, “Your approval and Planck’s mean more to me than that of half the world. Besides the whole thing would certainly not have originated yet, and perhaps never would have (I mean, not from me), if I had not had the importance of de Broglie’s ideas really brought home to me by your second paper on gas degeneracy.”