The God Particle: If the Universe Is the Answer, What Is the Question?

by Leon Lederman

  Now comes an interesting pas de deux of theory-experiment. De Broglie's idea had no experimental support. An electron wave? What does it mean? The necessary support appeared in 1927, in, of all places, New Jersey—not a Channel island but an American state near Newark. Bell Telephone Laboratories, the famous industrial research institution, was engaged in a study of vacuum tubes, an ancient electronic device used before the dawn of civilization and the invention of transistors. Two scientists, Clinton Davisson and Lester Germer, were bombarding various oxide-coated metal surfaces with streams of electrons. Germer, working under Davisson's direction, noticed that a curious pattern of electrons was reflected from certain metal surfaces that had no oxide coating.

  In 1926 Davisson traveled to a meeting in England and learned about de Broglie's idea. He rushed back to Bell Labs and began to analyze his data from the point of view of wave behavior. The patterns he observed fit precisely with the theory of electrons behaving as waves whose wavelength was related to the energy of the bombarding particles. He and Germer rushed to publish. They were none too soon. In the Cavendish Laboratory, George P. Thomson, son of the famous J. J., was carrying out similar research. Davisson and Thomson shared the 1938 Nobel Prize for first observing electron waves.

  The filial affection of J. J. and G. P. is, incidentally, amply documented in their warm correspondence. In one of his more emotional letters, G. P. gushed:

  Dear Father,

  Given a spherical triangle with sides ABC ...

  [And, after three densely written pages of the same]

  Your son, George

  So now a wave is associated with an electron whether it is imprisoned in an atom or traveling in a vacuum tube. But what is there about this electron that waves?

  THE MAN WHO DIDN'T KNOW BATTERIES

  If Rutherford was the prototypical experimenter, Werner Heisenberg (1901–1976) qualified as his theoretical counterpart. He would have fit I.I. Rabi's definition of a theorist as one who "couldn't tie his own shoelaces." One of the most brilliant students in Europe, Heisenberg almost failed his Ph.D. orals at the University of Munich when one of his examiners, Wilhelm Wien, a pioneer in the study of black body radiation, took a dislike to him. Wien started asking practical questions, like how does a battery work? Heisenberg had no idea. Wien, after grilling him with more questions about experimentation, wanted to flunk him. Cooler heads prevailed, and Heisenberg got off with the equivalent of a gentleman's C.

  His father was a professor of Greek at Munich, and as a teenager Heisenberg had read the Timaeus, which includes all of Plato's atomic theory. Heisenberg thought Plato was nuts—his "atoms" were little cubes and pyramids—but he was fascinated with Plato's basic tenet that one can never understand the universe until the smallest components of matter are known. Young Heisenberg decided to devote his life to studying the smallest particles of matter.

  Heisenberg tried hard to picture the Rutherford-Bohr atom in his mind and kept coming up empty. Bohr's electron orbits were like none he could imagine. The cute little atom that would become the Atomic Energy Commission's logo for so many years—a nucleus with electrons circling in "magic" radii without radiating—just didn't make any sense. Heisenberg realized that Bohr's orbits were merely constructs that made the numbers come out right and got rid of or (better) finessed classical objections to the Rutherford model of the atom. But real orbits? No. Bohr's quantum theory didn't go far enough in discarding the baggage of classical physics. The unique way in which space in the atom permitted only certain orbits required a more radical proposition. Heisenberg came to realize that this new atom was fundamentally not visualizable. He developed a firm guide: do not deal with anything that can't be measured. Orbits can't be measured. Spectral lines, however, can be. Heisenberg wrote a theory called "matrix mechanics," based on mathematical forms called matrices. His methods were difficult mathematically, and even more difficult to visualize, but it was clear that he had made a major improvement in Bohr's old theory. In time, matrix mechanics repeated all the successes of the Bohr theory without the arbitrary magic radii. Heisenberg's matrices went on to new successes where the old theory failed. But physicists found the matrices hard to use.

  And then came the most famous vacation in the history of physics.

  MATTER WAVES AND THE LADY IN THE VILLA

  A few months after Heisenberg completed his matrix formulation, Erwin Schrödinger decided he needed a holiday. It was about ten days before Christmas in the winter of 1925. Schrödinger was a competent but undistinguished professor of physics at the University of Zurich, and all college teachers deserve a Christmas holiday. But this was no ordinary vacation. Leaving his wife at home, Schrödinger booked a villa in the Swiss Alps for two and a half weeks, taking with him his notebooks, two pearls, and an old Viennese girlfriend. Schrödinger's self-appointed mission was to save the patched-up, creaky quantum theory of the time. The Viennese-born physicist placed a pearl in each ear to screen out any distracting noises. Then he placed the girlfriend in bed for inspiration. Schrödinger had his work cut out for him. He had to create a new theory and keep the lady happy. Fortunately he was up to the task. (Don't become a physicist unless you are prepared for such demands.)

  Schrödinger had begun his career as an experimenter but had switched to theory rather early on. He was old for a theorist, thirty-eight that Christmas. Obviously, there are lots of middle-aged, even elderly, theorists around. But they usually do their best work in their twenties, then retire, intellectually speaking, in their thirties to become "elder statesmen" of physics. This shooting-star phenomenon was especially true during the heyday of quantum theory, which saw Paul Dirac, Werner Heisenberg, Wolfgang Pauli, and Niels Bohr all crafting their finest theories as very young men. When Dirac and Heisenberg went to Stockholm to accept their Nobel Prizes, they were, in fact, accompanied by their mothers. Dirac once wrote:

  Age is of course a fever chill

  That every physicist must fear.

  He's better dead than living still

  When once he's past his thirtieth year.

  (He won his Nobel for physics, not for literature.) Fortunately for science, Dirac didn't take his own verse to heart, living well into his eighties.

  One of the items Schrödinger took with him on vacation was de Broglie's paper on particles and waves. Working feverishly, he extended the quantum concept even further. He didn't just treat electrons as particles with wave characteristics. He came up with an equation in which electrons are waves, matter waves. A key actor in Schrödinger's famous equation is the Greek symbol psi, or ψ. Physicists are fond of saying that the equation thus reduces everything to psi's (sighs). ψ is known as the wave function, and it contains all we know or can know about the electron. When Schrödinger's equation is solved, it gives ψ as it varies in space and changes with time. Later the equation was applied to systems of many electrons and eventually to any system requiring a quantum treatment. In other words, the Schrödinger equation, or "wave mechanics," applies to atoms, molecules, protons, neutrons, and, especially important to us today, clusters of quarks, among other particles.

  Schrödinger was out to rescue classical physics. He insisted that electrons were truly classical waves, like sound waves, water waves, or Maxwell's electromagnetic light and radio waves, and that their particle aspect was illusory. They were matter waves. Waves were well understood, simple to visualize, unlike the electrons in the Bohr atom, jumping willy-nilly from orbit to orbit. In Schrödinger's interpretation, ψ (really the square of ψ, or ψ2) described the density distribution of this matter wave. His equation described these waves under the influence of the electrical forces in the atom. For example, in the hydrogen atom, Schrödinger's waves clump in places where the old Bohr quantum theory talked orbits. The equation gave the Bohr radii automatically, with no adjustments, and provided the spectral lines, not only for hydrogen but for the other elements as well.

  Schrödinger published his wave equation within weeks after he left the
villa. It was an immediate sensation, one of the most powerful mathematical tools ever devised to deal with the structure of matter. (By 1960, more than 100,000 scientific papers had been published based on the application of Schrödinger's equation.) He wrote five more papers in quick succession; all six papers were published in a six-month period that was among the greatest bursts of creativity in scientific history. J. Robert Oppenheimer called the theory of wave mechanics "perhaps one of the most perfect, most accurate, and most lovely man has discovered." Arthur Sommerfeld, the great physicist and mathematician, said Schrödinger's theory "was the most astonishing among all the astonishing discoveries of the twentieth century."

  For all of this, I personally forgive Schrödinger for his romantic dalliances which, after all, are of concern only to biographers, sociological historians, and envious colleagues.

  A WAVE OF PROBABILITY

  Physicists loved Schrödinger's equation because they could solve it and it worked. Although Heisenberg's matrix mechanics and Schrödinger's equation both seemed to give the correct answers, most physicists seized on the Schrödinger method since this was a good old differential equation, a warm and familiar form of mathematics. A few years later it was shown that the physical ideas and numerical consequences of Heisenberg's and Schrödinger's theories were identical. They were just written in different mathematical languages. Today a mixture of the most convenient aspects of both theories is used.

  The only problem with Schrödinger's equation was that his interpretation of the "wave" was wrong. It turned out that the thing could not represent matter waves. There was no doubt it represented some sort of wave, but the question was, what's waving?

  The answer was provided by the German physicist Max Born, still in that eventful year 1926. Born insisted that the only consistent interpretation of Schrödinger's wave function is that ψ2 represents the probability of finding a particle, the electron, at various locations. ψ varies in space and time. Where ψ2 is large, the electron has a large probability of being found. Where ψ = 0, the electron is never found. The wave function is a wave of probability.

  Born was influenced by experiments in which a stream of electrons is directed toward some sort of energy barrier. This could be, for example, a wire screen connected to the negative terminal of a battery, say at −10 volts. If the electrons have an energy of only 5 volts, they should be effectively repelled by the "10-volt barrier" in the classical view. If an electron's energy is larger than that of the barrier it will penetrate the barrier like a ball thrown over a wall. If its energy is less than that of the barrier the electron is reflected, like a ball thrown against the wall. However Schrödinger's quantum equation indicates that some of the ψ-wave penetrates and some of the wave is reflected. This is typical light behavior. Pass a store window and you see the goodies displayed, but you also see a dim image of yourself. Light waves are both transmitted through and reflected by the glass. Schrödinger's equation predicts similar results. But we never see a fraction of an electron!

  The experiment goes as follows: we send 1,000 electrons toward the barrier. Geiger counters find that 550 penetrate the barrier and 450 are reflected, but in every case, it is an entire electron that is detected. The Schrödinger waves, when properly squared, give 550 and 450 as a statistical prediction. If we accept the Born interpretation, a single electron has a 55 percent probability of penetrating and a 45 percent chance of being reflected. Since a single electron never divides, Schrödinger's wave cannot be the electron. It can be only a probability.

  Born, along with Heisenberg, was part of the Gottingen school, a group of some of the brightest physicists of the age whose professional and intellectual lives revolved around the University of Gottingen in Germany. Born's statistical interpretation of Schrödinger's psi came from the Gottingen school's conviction that electrons are particles. They make Geiger counters click. They leave sharp tracks in Wilson cloud chambers. They collide with other particles and bounce off. So here is Schrödinger's equation, which gives correct answers but describes electrons as waves. How can it be converted to a particle equation?

  Irony is a constant companion of history, and the idea that changed everything was given (again!) by Einstein in a speculative paper of 1911 on the relationship of photons to Maxwell's classical field equations. Einstein had suggested that the field quantities guided the photons to places of high probability. Born's resolution of the particle-wave conflict is simply this: the electron (and its friends) act like particles at least when they are being detected, but their distribution in space between measurements follows the wavelike probability patterns that emerge from the Schrödinger equation. In other words, the Schrödinger psi quantity describes the probable location of the electrons. And this probability can behave like a wave. Schrödinger did the hard part, crafting the equation that lies at the heart of the theory. But it was Born, inspired by Einstein's paper, who figured out what the equation was actually predicting. The irony is that it was Born's probability interpretation of the wave function that Einstein never accepted.

  WHAT THIS MEANS, OR THE PHYSICS OF CLOTH CUTTING

  The Born interpretation of the Schrödinger equation is the single most dramatic and major change in our world view since Newton. It is not surprising that Schrödinger found the idea totally unacceptable and regretted inventing an equation that would involve such foolishness. However Bohr, Heisenberg, Sommerfeld, and others accepted it with little fuss because "probability was in the air." Born's paper made the eloquent assertion that the equation can only predict probability but that the mathematical form of probability is developed along perfectly predictable paths.

  In this new interpretation, the equation deals with probability waves, ψ, which predict what the electron is doing, what its energy is, where it will be, and so on. However these predictions are in the form of probabilities. What "waves" about the electron is just these probability predictions. These wavelike solutions to the equations can pile up in one place to add up to high probability and cancel in other places to yield low probability. When one puts these predictions to the test, one in effect does the experiment a huge number of times. Indeed, in most of the trials, the electron ends up where the equation says the probability is high; only very rarely does it end up where the probability is low. There is quantitative agreement. What is shocking is that for two apparently identical experiments one can get two quite different results.

  The Schrödinger equation with Born's probability interpretation of the wave function has been enormously successful. It is the key to understanding hydrogen and helium and, given a big enough computer, uranium. It was used to understand how two elements combine to make a molecule, putting chemistry on a far more scientific footing. It allows one to design electron microscopes and even proton microscopes. In the period 1930–1950 it was carried into the nucleus and was found to be as productive there as in the atom.

  The Schrödinger equation predicts with a high degree of accuracy, but, again, what it predicts is probability. What does that mean? Probability in physics is similar to probability in life. It's a billion-dollar business, as executives from insurance companies, clothing manufacturers, and a good fraction of the Fortune 500 industries will assure you. Actuaries can tell us that the average white American nonsmoking male born in, say, 1941, will live to be 76.4 years old. But they can't tell you diddly about your brother Sal, who was born that same year. For all they know, he could be run over by a truck tomorrow or die of an infected toenail in two years.

  In one of my classes at the University of Chicago, I play garment-center mogul for my students. Being a success in the rag trade is similar to making a career in particle physics. In either case, you need a strong grasp of probability and a working knowledge of tweed jackets. I ask the students to sing out their heights while I plot each student's height on a graph. I have two students at 4 foot 8 inches, one at 4 foot 10, four at 5 foot 2, and so on. One guy is 6 foot 6, way outside the others. (If Chicago only had a basketball t
eam!) The average is 5 foot 7. After polling 166 students I have a nice, bell-shaped set of steps going up to 5 foot 7 and then stepping down toward the 6-foot-6 anomaly. Now I have a "distribution curve" of college freshman heights, and if I'm reasonably sure that choosing physics to fulfill the science requirement does not distort the curve, I have a representative sample of student heights at the University of Chicago. I can read percentages using the vertical scale; for example, I can figure out what percentage of students is between 5 foot 2 and 5 foot 4. With my graph I can also read that there is a 26 percent probability that the next student who shows up will be between 5 foot 4 and 5 foot 6, if this is something I want to know.

  Now I'm ready to make suits. If these students are my market (an unlikely prospect if I'm in the suit business), I can estimate what percentage of my suits should be size 36, 38, and so on. If I don't have a graph of heights, I have to guess, and if wrong, at the end of the season I have 137 size-46 suits left unsold (which I have to blame on my partner Jake, the schlemiel!).

  The Schrödinger equation, when solved for any situation involving atomic processes, generates a curve analogous to the distribution-of-student-heights curve. However, the shape may be quite different. If we want to know where the electron hangs out in the hydrogen atom—how far it is from the nucleus—we'll find some distribution that drops off sharply at about 10−8 centimeters, with about an 80 percent probability of finding the electron within the sphere of radius 10−8 centimeters. This is the ground state. If we excite the electron to the next energy level, we'll get a bell curve with a mean radius that's about four times as big. We can compute probability curves of other processes as well. Here we must clearly differentiate probability predictions from possibilities. The possible energy levels are very precisely known, but if we ask which energy state the electron will be found in, we calculate only a probability, which depends on the history of the system. If the electron has more than one choice as to which lower energy state to jump to, we can again predict probabilities; for example, an 82 percent probability of jumping to Ei, 9 percent into E2, and so on. Democritus said it best when he proclaimed, "Everything existing in the universe is the fruit of chance and necessity." The various energy states are the necessities, the only conditions that are possible. But we can only predict the probabilities of the electron being in any of these possible states. That's a matter of chance.