Part 5 (1/2)
Indeed it was not. With the rise of psychoa.n.a.lysis scientists began to look into how they had come up with their discoveries. Einstein wrote, ”There is no logical path to these laws; only intuition, resting on sympathetic understanding of experience, can reach them.” In this way, he added, scientists could glimpse the ”pre-established” harmony of the universe. Logical empiricists, however, saw this as aimless babble conjured up by scientists years after the fact.
In their view scientists constructed theories by moving logically-mathematically-from experimental data to a theory. They churned out equation after equation until they had solved the problem at hand. Einstein considered this wrongheaded. Scientists were unanimous in agreeing that their methods of research bore no resemblance to the proposals of positivists and logical empiricists. The key point for creative scientists such as Einstein was the delicate balance they had to maintain between the information obtained from experimental data and the laws of the theory as expressed in mathematics.
Pauli undoubtedly read Einstein's views as well as the famous polemic in the first decade of the twentieth century between Mach and the discoverer of quantum theory, Max Planck, whose opinions were similar to Einstein's. Planck accused Mach of degrading physics whereupon Mach simply withdrew in disgust: ”I cut myself off from the physicist's mode of thinking.”
Einstein believed, as did many scientists, in a world beyond perceptions in which electrons actually existed. Philosophers called this view ”scientific realism.” There were scores of hybrid philosophies besides scientific realism and positivism which a.s.serted that in fact there was nothing ”out there.” Pauli counted himself a ”'heretic,' not bowing down to any G.o.d, authority or 'ism.'”
As a philosophical opportunist, Pauli saw that positivism offered a way out of the mora.s.s of 1925, when Bohr's theory of the atom had collapsed with nothing to replace it. He thus advised Heisenberg to drop the unmeasurable concept of electron orbits and focus instead on measurable concepts like energy and momentum. This meant dropping the rea.s.suring visual image of the atom as a solar system. Pauli's belief was that once the ”systems of concepts are settled,” that is, once the new atomic theory had been worked out, then ”will visual imagery be regained,” as he wrote to Bohr. At Bohr's Inst.i.tute, Heisenberg and Bohr shared all correspondence from Pauli and eagerly awaited it.
Quantum mechanics-the new atomic physics.
Heisenberg's quantum mechanics identified individual electrons within atoms by the radiation they emitted while jumping between different stationary states, that is, the condition of an electron characterized by four quantum numbers as well as its momentum and energy, measurable as spectral lines. The transitions, or jumps, of the electrons maintained the flavor of the discontinuous quantum jumps in Bohr's theory of the atom. Discontinuities were a fact of life in the world of the atom, especially in a theory based on electrons as particles.
Pauli was convinced that Heisenberg's quantum mechanics would make it possible to solve problems that he had been unable to solve with the old Bohr theory. Late in 1925, he set out to calculate the stationary states for the simplest atom-hydrogen-using quantum mechanics. It involved juggling very complex mathematics but he came up with the answer with amazing speed.
Werner Heisenberg in 1925, when he discovered quantum mechanics.
Bohr applauded Pauli's ”wonderful results.” Heisenberg complained he was ”a bit unhappy” that he had not solved the problem himself, but was full of admiration and surprise that Pauli had done it ”so quickly.”
Irked that Pauli had stolen a march on him, just a month later Heisenberg, along with Pascual Jordan, another brilliant young physicist, tried applying quantum mechanics to the problem that had driven everyone to despair-the anomalous Zeeman effect. Just as Kepler's ellipses had eliminated the c.u.mbersome circles moving on circles, so Pauli's new concept of spin-part of Heisenberg's quantum mechanics-at a stroke swept away the concepts of ma.s.sive inert cores with their two-valuedness and strange forces which had cluttered up Bohr's theory. The problem had finally been put to rest, and the solution also helped set Heisenberg's quantum mechanics on a firm basis. This time they had the theory right.
Physicists applauded these calculational breakthroughs. But no one-including Bohr, Heisenberg, and Pauli-really understood the theory itself, because the properties of atomic ent.i.ties were so impossible to imagine. Not only was it unfamiliar and difficult to use, the mathematics of Heisenberg's quantum mechanics lacked any helpful visual image. Being a hybrid version of Bohr's virtual oscillators, it was like trying to visualize infinity. Its fundamental particles were also unvisualizable. But this was fine with Heisenberg, who felt the time was not ripe for a return to visual images which, in the past, had always turned out to be misleading.
Then the French physicist Louis de Broglie suggested that electrons might be waves-in other words, that material objects, such as ourselves, might be considered as waves. His inspiration was Einstein's discovery, made some two decades previously, that light-traditionally thought of as a wave-could also be a particle, dubbed a light quantum. Perhaps electrons as well as light might be both wave and particle at the same time-simply beyond imaginable.
In spring 1926, the flamboyant Erwin Schrodinger, at the University of Zurich, burst on the scene. At thirty-nine, Schrodinger was an outsider in age, temperament, and thought to the group of impetuous twenty-something quantum physicists who cl.u.s.tered around Bohr in Copenhagen, Sommerfeld in Munich, and Born in Gottingen.
Schrodinger had found the equation that converted de Broglie's vision of matter as waves into a coherent theory. His version of atomic physics, which he called wave mechanics, was based on treating light and electrons as waves. ”My theory was inspired by L. de Broglie,” he wrote. ”No genetic relation whatever with Heisenberg is known to me. I knew of his theory, of course, but felt discouraged, not to say repelled, by the methods of the transcendental algebra, which appeared very difficult to me, and by the lack of visualizability.”
Schrodinger's wave mechanics sprang from a preference for a mathematics that was more familiar and beautiful, as opposed to what he referred to as Heisenberg's ugly ”transcendental algebra.” The ”Schrodinger equation” offered great advantages in calculations over Heisenberg's quantum mechanics, added to which it enabled the electron in an atom to be visualized as a wave surrounding the nucleus. It had taken Pauli twenty-odd pages to solve the hydrogen atom problem. Schrodinger did it in a page.
Schrodinger pointed out that the wave nature of matter promised a return to cla.s.sical continuity. The pa.s.sage of an electron between stationary states could be envisioned as a string pa.s.sing continuously from one mode of oscillation to another.
One year earlier there had been no viable atomic theory. Now there were two: Heisenberg's quantum mechanics and Schrodinger's wave mechanics.
Heisenberg was furious about Schrodinger's work and even more so about its rave reviews from the physics community. ”The more I reflect on the physical portion of Schrodinger's theory, the more disgusting I find it,” he wrote to Pauli. ”What Schrodinger writes on the visualizability of his theory I consider c.r.a.p.”
Heisenberg saw wave functions-that is, the solutions to Schrodinger's wave equation-as nothing more than a means to expedite calculations. To demonstrate this he applied them to the problem that had driven Born, Pauli, and himself to despair: to find a mathematical way to describe the properties of the helium atom. No one had been able to deduce stable orbits, or stationary states, for the two electrons in the helium atom using Bohr's theory of the atom. This being the case, they could not move on to deduce spectral lines for the helium atom because these resulted from its electrons dropping down from a higher to a lower orbit. Instead, the electrons' orbits remained unstable, meaning that an electron could be knocked out of the helium atom by the smallest of disturbances.
But in Heisenberg's quantum mechanics there were no orbits. The problem became one of deducing the atom's spectral lines from its stationary states expressed directly in terms of the electrons' energy and momentum in the atom. If the spectral lines turned out to be wrong, it would show that there were serious problems with the way quantum mechanics defined stationary states, that is, the energy levels of electrons in atoms. The spectral lines of the helium atom were particularly interesting to physicists because, as had been observed in the laboratory, they fell into two distinct groups. But why should this be the case?
Insight into the exclusion principle.
The helium atom has two electrons. Using his quantum mechanics Heisenberg showed how the two sets of spectra arise. To elucidate his result and speed up his calculation of numerical values for the spectral lines, he used Schrodinger's wave functions-the solutions to the Schrodinger equation-for both the spins and positions of these two electrons. The total wave function is the result of multiplying these two wave functions together. But there are many possible ways of constructing the total wave function for these two electrons.
Heisenberg found that only one sort produced the two distinct groups of spectral lines characteristic of the helium atom. This particular wave function had a unique property. It changed its sign when the spins and positions of the electrons were swapped. It was antisymmetrical, which also meant that it went to zero if the electrons had the same spins or positions.*
What was nature's selection device for choosing these two sets of wave functions for the two spectra out of the several possible ones? Heisenberg was stumped. Something strange was going on here. Perhaps it related to Pauli's exclusion principle, according to which no two electrons could have the same spin and position. If they did then one of the two wave functions that make up the total wave function-either for their positions or for their spins-would have to become zero. Perhaps that was the way nature selected the wave function suitable for a particular system of electrons. Thus Heisenberg realized that the exclusion principle was related to the symmetry property of the wave functions for a collection of electrons, in this case two electrons. It was a step forward in exploring its implications beyond making sense of the periodic table of elements.
It was a typical Heisenberg project. He chose a fundamental problem-in this case to understand the spectrum of the helium atom-and then let his intuition lead him into new terrain: the symmetry property of wave functions whether they are symmetric or antisymmetric. Thus he realized how essential the exclusion principle was for quantum mechanics: without it quantum mechanics could not be complete.
There was also the problem that had been Pauli's original bete noir from his PhD thesis, in which he showed that Bohr's theory of the atom failed to produce a stable hydrogen-molecule ion, H+2, even though it existed in nature. This problem vexed Born and Heisenberg as well. Pauli wrote to his friend Wentzel, ”In Copenhagen sits a gentleman who is calculating H+2 according to Schrodinger.” The ”gentleman” was the Danish physicist vind Burrau who, as Pauli pointed out, started directly with Schrodinger's wave mechanics as opposed to starting from the quantum mechanics as Heisenberg had done and used Schrodinger's wave mechanics only for calculations. As a result he was able to solve the problem simply. Heisenberg wrote to Pauli that, in his opinion, Burrau had straightened out the situation and mentioned the symmetry properties of the wave functions that Burrau had deduced. Perhaps Heisenberg had hoped to find a solution starting from his quantum mechanics. But these once-key problems had become mere calculations now that the correct atomic physics had been worked out.
Although problem after problem that had resisted solution using the old Bohr theory was now being solved by atomic physics, the meaning of the theories used-Heisenberg's quantum mechanics and Schrodinger's wave mechanics-was still not understood. And the tension between the two factions was growing.
To the Schrodinger faction Burrau's successful result, as well as Heisenberg's for the helium atom (despite his a.s.sertion that he had used Schrodinger's theory merely to facilitate calculations) was proof that Schrodinger's theory offered a solution to every problem of atomic structure, whereas Heisenberg's was daunting to use and ugly. This of course greatly pleased Schrodinger.
Heisenberg's uncertainty principle.
In fall 1926, Bohr summoned Heisenberg to his inst.i.tute in Copenhagen to hammer out a resolution to the dispute with Schrodinger. They struggled for days over numerous cups of tea and bottles of Carlsberg beer. Heisenberg wrote to Pauli: ”What the words 'wave' or 'particle' mean we know not any more; [we are in a] state of almost complete despair.”
The crux of the problem was this: how could ordinary language, with its visual connotations, be used to describe a realm of nature that defied the imagination?
While Bohr and Heisenberg were deliberating in Copenhagen, Pauli had an idea. He immediately wrote it up and mailed it to Heisenberg. It was based on an insight Born had had, looking into Schrodinger's wave equation.
Born had suggested that the wave function was a wave of probability for an electron moving between stationary states. Pauli pushed the idea further. He realized that the wave function gave the probability of an electron being detected in a certain region of s.p.a.ce. In his usual way, he didn't bother to write a paper to publish this idea and in the end Born took the credit.
But as he was working out the mathematics for Heisenberg, he came up with another extraordinary discovery: if he could determine a particle's position accurately, he could not determine its momentum with the same accuracy. Pauli was puzzled as to why this should be so. Why couldn't he determine both with the same degree of accuracy? Heisenberg was struck by the insight. He was, he wrote to Pauli, ”more and more inspired by the content of your last letter every time I reflect on it.”
By February 1927 Bohr and Heisenberg had hit an impa.s.se in their discussions on the deep meaning of the quantum theory, which seemed to be riddled with ambiguities. Bohr took a skiing break. Left to his own devices, Heisenberg set to work. The result was a paper that he called ”On the Intuitive Content of the Quantum-Theoretical Kinematics and Mechanics.” Hidden behind this daunting t.i.tle was one of the most earth-shattering discoveries of modern physics: the uncertainty principle.
Heisenberg realized that the supposed ambiguities of the quantum theory were essentially a problem of language. The issue was how to define words such as ”position” and ”velocity” in the ambiguous realm of the atom, a world in which ”things” can be both wave and particle at the same time. He used the term ”intuitive” in the t.i.tle of his paper, for his goal was to redefine the word in the world of the atom.
Certain concepts in quantum physics, Heisenberg claimed, such as ”position” and ”momentum” (ma.s.s times velocity), were ”derivable neither from our laws of thought nor from experiment.” Instead we would have to look into the peculiar mathematics of quantum mechanics, which should have alerted us all along that in the realm of the atom we would have to apply such words with great care.
In his paper Heisenberg made the amazing a.s.sertion that the more accurately we measure an electron's momentum in a certain experiment, the less accurately we can measure its position in that experiment. This quickly became known as the uncertainty principle. It was earth-shattering in that it questioned our understanding of the inherent nature of the physical world as completely as Einstein's relativity theory.
In the cla.s.sical physics of Newton we can measure the position and momentum of an object with the same degree of accuracy by observing how it moves. Using a telescope and a clock we can measure both the location of a falling stone and how fast it is moving with an accuracy limited only by the width of the telescope's crosshairs and the clock's mechanism. If we make these errors as small as possible we can deduce very precisely the stone's position and momentum. In principle, the product of the errors in measurements of position and momentum can both be zero. In quantum mechanics this is not possible.
Heisenberg wrote all this down in a detailed fourteen-page letter to Pauli. He asked for ”severe criticism” after all, it was Pauli who had given him the key idea. Pauli was elated. ”It becomes day in the quantum theory,” he declared.
Bohr's complementarity and beyond.
Bohr, however, was furious. He refused to let Heisenberg publish his paper on the subject, saying that Heisenberg had not provided any firm foundation for his argument. Furthermore, Heisenberg had based the argument entirely on the a.s.sumption that light and electrons behaved like particles.
Bohr insisted that electrons and light be understood as both wave and particle, even though this could not be imagined. One could visualize electrons and light as either a wave or a particle so long as one remembered the restrictions required by quantum mechanics, among them Heisenberg's uncertainty principle understood within the larger context of waves and particles.
This meant that electrons in experiments could exhibit one aspect or the other, but not both at once. If one experimented on an electron as if it were a wave, that was what it would be for the duration of the experiment, and similarly if one treated it as a particle. Bohr called this ”complementarity.”
Bohr was convinced that complementarity was relevant not only to physics but also to psychology and to life itself. Its basic idea, he wrote, ”bears a deep-going a.n.a.logy to the general difficulty in the formation of human ideas, inherent in the distinction between subject and object.” As in the Chinese concept of yin and yang, complementary pairs of concepts defined reality. There is nothing paradoxical about an electron having the characteristics of both a wave and a particle until an experiment is performed on it. It dawned on Bohr that in the weird quantum world there need not be only yes and no, an electron need not actually be either particle or wave. There could be in-betweens as well as ambiguities. An electron's wave and particle aspects complement each other, and their totality makes up what the electron is. Thus the electron is made up of complementary pairs-wave and particle, and position and momentum. Similarly it is the tension between complementary pairs-love and hate, life and death, light and darkness-that shapes our everyday existence.
Bohr sent the ma.n.u.script of his article on complementarity to Pauli for corrections and critical remarks. Pauli replied immediately. Apart from certain comments on details, he entirely agreed with Bohr's thesis.
Only the more philosophically inclined scientists took complementarity seriously. Pauli was one. He began to look to complementarity as another way to study consciousness as in the various ways of ”knowing” practiced in the East and West. He was growing more and more interested in the conscious and the unconscious, the rational and the irrational, and in how physics could be used to understand these complementarities. He was beginning to suspect that this was to be his life's work. The only problem was how to approach it.
Paul Dirac and quantum electrodynamics.
The previous autumn the eccentric twenty-five-year-old English physicist Paul Dirac had visited the Bohr Inst.i.tute. Dirac had already made important contributions to atomic physics and was eager to rub shoulders with other physicists of his generation whose papers he had studied in detail, Heisenberg and Pauli among them.
Dirac had been privy to the intense conversations between Bohr and Heisenberg on the issue of whether light and matter could be both wave and particle. In 1927 he was able to provide the vital clarification through a mathematical method he had developed for moving between the two and thus brought about ”complete harmony between the wave and light quantum descriptions.” Dirac's mathematical method ultimately concerned the way in which electrons and light interact. It formed the basis for a whole new subject, which scientists dubbed quantum electrodynamics. Pauli and Heisenberg worked enthusiastically to develop this new field.