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by Michio Kaku

  Thus Einstein, who was the father of the “old quantum theory” of the photon, became the godfather of the “new quantum theory,” based on these Schrödinger waves. (Today, when high school chemistry students have to memorize the funny football-shaped “orbitals” that surround the nucleus, with strange labels and “quantum numbers,” they are actually memorizing the solutions to the Schrödinger wave equation.) The breakthroughs in quantum physics now accelerated enormously. Realizing that the Schrödinger equation did not incorporate relativity, just two years later Dirac generalized the Schrödinger equation to a fully relativistic theory of electrons, and once again the world of physics was dazzled. While Schrödinger’s celebrated equation was nonrelativistic and only applied to electrons moving at slow velocities compared to light, Dirac’s electrons obeyed the full Einstein symmetry. Furthermore, Dirac’s equation could automatically explain some obscure properties of the electron, including something called “spin.” It was known from earlier experiments by Otto Stern and Walter Gerlach that the electron acted like a spinning top in a magnetic field, with angular momentum given by 1/2 (in units of Planck’s constant). The Dirac electron yielded precisely the spin 1/2 given by the Stern-Gerlach experiment. (The Maxwell field, representing the photon, has spin 1, and Einstein’s gravity waves have spin 2. With Dirac’s work, it became clear that the spin of a subatomic particle would be one of its important properties.)

  Then Dirac went one step further. By looking at the energy of these electrons, he found that Einstein had overlooked a solution to his own equations. Usually, when taking the square root of a number, we introduce both positive and negative solutions. For example, the square root of 4 can be either plus 2 or minus 2. Because Einstein ignored a square root in his equations, his famous equation E = mc2 was not quite correct. The correct equation was E = ±mc2. This extra minus sign, argued Dirac, made possible a new kind of mirror universe, one in which particles could exist with a new form of “antimatter.”

  (Strangely, just a few years earlier in 1925, Einstein himself had entertained the idea of antimatter when he showed that by reversing the sign of the electron charge in a relativistic equation, one can get identical equations if one also reverses the orientation of space. He showed that for every particle of a certain mass, there must exist another particle with opposite charge but identical mass. Relativity theory not only gave us the fourth dimension, it was now giving us a parallel world of antimatter. However, Einstein, never one to quibble over priority, graciously never challenged Dirac.)

  At first, the radical ideas of Dirac met with fierce skepticism. The idea of an entire universe of mirror particles that arose from E = ±mc2 seemed like an outlandish idea. Quantum physicist Werner Heisenberg (who with Niels Bohr had independently found a formulation of the quantum theory equivalent to Schrödinger’s) wrote, “The saddest chapter of modern physics is and remains the Dirac theory…. I regard the Dirac theory…as learned trash which no one can take seriously.” However, physicists had to swallow their pride when the antielectron, or positron, was finally discovered in 1932, for which Dirac later received the Nobel Prize. Heisenberg finally admitted, “I think that this discovery of anti-matter was perhaps the biggest jump of all the big jumps in our century.” Once again, the theory of relativity yielded unexpected riches, this time giving us an entirely new universe made of antimatter.

  (It seems strange that Schrödinger and Dirac, who developed the two most important wave functions in the quantum theory, were such polar opposites in their personalities. While Schrödinger was always accompanied by some lady friend, Dirac was painfully shy with women and was a man of remarkably few words. After Dirac’s death, the British, honoring his contributions to the world of physics, had the relativistic Dirac equation engraved into stone in Westminster Abbey, not far from Newton’s grave.)

  Soon, physicists at every institute on this planet struggled to learn the strange, beautiful properties of the Schrödinger and Dirac equations. However, for all their undeniable successes, quantum physicists still had to grapple with a troubling philosophical question: if matter is a wave, then precisely what is waving? This is the same question that had haunted the wave theory of light, which gave birth to the incorrect theory of the aether. A Schrödinger wave is like an ocean wave and eventually spreads out if left by itself. With enough time, the wave function eventually dissipates over the entire universe. But this violated everything that physicists knew about electrons. Subatomic particles were believed to be pointlike objects that made definite, jetlike streaks which could be photographed on film. Thus, although these quantum waves had near miraculous success in describing the hydrogen atom, it did not seem possible that the Schrödinger wave could describe an electron moving in free space. In fact, if the Schrödinger wave really represented an electron, it would slowly dissipate and the universe would dissolve.

  Something was terribly wrong. Finally, Einstein’s lifelong friend Max Born proposed one of the most controversial solutions to this puzzle. In 1926, Born took the decisive step, claiming that the Schrödinger wave did not describe the electron at all, but only the probability of finding the electron. He declared that “the motion of particles follows probability laws, but probability itself propagates in conformity with the laws of causality.” In this new picture, matter indeed consisted of particles, not waves. The markings captured on photographic plates are the tracks left by pointlike particles, not waves. But the chance of finding the particle at any given point was given by a wave. (More precisely, the absolute square of the Schrödinger wave represents the probability of finding the particle at a specific point in space and time.) Thus, it did not matter if the Schrödinger wave spread out over time. It simply meant that if you left an electron by itself, over time it would wander around and you would not know precisely where it was. All the paradoxes were now solved: the Schrödinger wave was not the particle itself, but represented the chance of finding it.

  Werner Heisenberg took this one step further. He had agonized endlessly with Bohr over the puzzles of probability infesting this new theory, often getting into heated arguments with his older colleague. One day, after a frustrating night of grappling with the question of probabilities, he took a long stroll down Faelled Park, behind his university, constantly asking himself how it was possible that one could not know the precise location of an electron. How can the location of an electron be uncertain, as claimed by Born, if you can simply measure where it is?

  Then, it suddenly hit him. Everything became clear. In order to know where an electron was, you had to look at it. This meant shining a light beam at it. But the photons in the light beam would collide with the electron, making its position uncertain. In other words, the act of observation necessarily introduced uncertainty. He reformulated this question into a new principle of physics, the uncertainty principle, which states that one cannot determine both the location and the velocity of a particle at the same time. (More precisely, the product of the uncertainty in position and momentum must be greater than or equal to Planck’s constant divided by 4p). This was not just a by-product of the crudeness of our instruments; it was a fundamental law of nature. Even God could not know both the precise position and momentum of an electron.

  This was the decisive moment when the quantum theory plunged into deep, totally uncharted waters. Up to then, one could argue that quantum phenomena were statistical, representing the average motions of trillions of electrons. Now, even the motions of a single electron could not be definitively determined. Einstein was horrified. He almost felt betrayed, knowing that his good friend Max Born was abandoning determinism, one of the most cherished ideas in all of classical physics. Determinism states, in essence, that you can determine the future if you know everything about the present. For example, Newton’s great contribution to physics was that he could predict the motion of comets, moons, and planets via his laws of motion once he knew the present state of the solar system. For centuries, physicists had marveled at the precision o
f Newton’s laws, that they could predict the position of celestial bodies, in principle, millions of years into the future. In fact, up to that time, all of science was based on determinism; that is, a scientist can predict the outcome of an experiment if the scientist knows the position and velocities of all particles. Followers of Newton summarized this belief by comparing the universe to a gigantic clock. God wound up this clock at the beginning of time and it has been steadily ticking ever since according to Newton’s laws of motion. If you knew the position and velocity of every atom in the universe, then you can, via Newton’s laws of motion, calculate the subsequent evolution of the universe with infinite precision. However, the uncertainty principle negated all of this, stating that it is impossible to predict the future state of the universe. Given a uranium atom, for example, one could never calculate when it will decay, only the likelihood of its doing so. In fact, even God or a deity did not know when the uranium atom would decay.

  In December 1926, responding to Born’s paper, Einstein wrote, “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 any closer to the Old Man’s secret. For my part, at least I am convinced that He doesn’t throw dice.” When commenting on Heisenberg’s theory, Einstein remarked, “Heisenberg has laid a big quantum egg. In Göttingen they believe in it (I don’t).” Schrödinger himself disliked this idea intensely. He once said that if his equation represented only probabilities, then he regretted having anything to do with it. Einstein chimed in that he would have become a “cobbler or employee in a gaming house,” if he had known that the quantum revolution he helped to initiate would introduce chance into physics.

  Physicists were beginning to divide into two camps. Einstein led one camp, which still clung to a belief in determinism, an idea that dated back to Newton himself and had guided physicists for centuries. Schrödinger and de Broglie were allies. The other, much larger camp was led by Niels Bohr, who believed in uncertainty and championed a new version of causality, based on averages and probabilities.

  Bohr and Einstein, in some sense, were polar opposites in other ways. While Einstein as a child shunned sports and was glued to books on geometry and philosophy, Bohr was renowned throughout Denmark as a soccer star. Whereas Einstein spoke forcefully and dynamically, wrote almost lyrically, and could exchange banter with journalists as well as royalty, Bohr was stiff, had a horrible mumble, was often inarticulate and inaudible, and would often repeat a single word endlessly when engrossed in thought. While Einstein could effortlessly write elegant and beautiful prose, Bohr was paralyzed when he had to write a paper. As a high school student, he would dictate all his papers to his mother. After he married, he would dictate them to his wife (even interrupting his honeymoon to dictate one long and important paper). He would sometimes involve his entire laboratory in rewriting his papers, once over a hundred times, completely disrupting the work. (Wolfgang Pauli, once asked to visit Bohr in Copenhagen, replied, “If the last proof is sent away, then I will come.”) Both were, however, obsessed with their first love, physics. Bohr, in fact, would scribble equations on the goal post of a soccer game if he had an inspiration. Both would also sharpen their thoughts by using others as sounding boards for their ideas. (Strangely, Bohr could only function if he had assistants around him to bounce off ideas. Without an assistant whose ear he could borrow, he was helpless.)

  The showdown finally came at the Sixth Solvay Conference in Brussels in 1930. What was at stake was nothing less than the nature of reality itself. Einstein hammered incessantly at Bohr, who reeled under the constant attacks but managed to ably defend his positions. Finally, Einstein presented an elegant “thought experiment” which, he thought, would demolish the “demon,” the uncertainty principle: Imagine a box containing radiation. There is a hole in the box with a shutter. When the shutter is opened briefly, it can release a single photon from the box. Thus, we can measure with great certainty the precise time at which the photon was emitted. Much later, the box can be weighed. Because of the release of the photon, the box weighs less. Because of the equivalence of matter and energy, we can now tell how much total energy the box contains, also to great accuracy. Thus, we now know both the total energy and the time of opening of the shutter to arbitrary accuracy, without any uncertainty, and hence the uncertainty principle is wrong. Einstein thought he had finally found the tool to demolish the new quantum theory.

  Paul Ehrenfest, one of the participants to this conference and a witness to this fierce battle, would write, “To Bohr, this was a heavy blow. At the moment he saw no solution. He was extremely unhappy all through the evening, walked from one person to another, trying to persuade them all that this could not be true, because if E was right this would mean the end of physics. But he could think of no refutation. I will never forget the sight of the two opponents leaving the university club. Einstein, a majestic figure, walking calmly with a faint ironical smile, and Bohr trotting along by his side, extremely upset.” When he talked to Ehrenfest later that evening, all Bohr could mumble was one word, over and over again: “Einstein…Einstein…Einstein.” But after an intense, sleepless night, Bohr finally found the defect in Einstein’s argument, and he used Einstein’s own theory of relativity to defeat him. Bohr noted that because the box weighed less than before, it would rise slightly in the earth’s gravity. But according to general relativity, time speeds up as gravity gets weaker (so that time beats faster on the moon, for example). Thus, any minuscule uncertainty in measuring the time of the shutter would be translated into an uncertainty in measuring the position of the box. You cannot, therefore, measure the position of the box with absolute certainty. Furthermore, any uncertainty in the weight of the box will be reflected in an uncertainty in its energy and also its momentum, and hence you cannot know the momentum of the box with absolute certainty. When everything is put together, the two uncertainties identified by Bohr, the uncertainty in position and uncertainty in momentum, agree precisely with the uncertainty principle. Bohr had successfully defended the quantum theory. When Einstein complained that “God does not play dice with the world,” Bohr reportedly fired back, “Stop telling God what to do.”

  Ultimately, Einstein had to admit that Bohr had successfully refuted his arguments. Einstein would write, “I am convinced that this theory undoubtedly contains a piece of definitive truth.” Commenting on the historic Bohr-Einstein debate, John Wheeler said it was “the greatest debate in intellectual history that I know about. In thirty years, I never heard of a debate between two greater men over a longer period of time on a deeper issue with deeper consequences for understanding this strange world of ours.”

  Schrödinger, who also hated this new interpretation of his equations, proposed his celebrated problem of the cat to poke holes into the uncertainty principle. Schrödinger wrote about quantum mechanics: “I don’t like it, and I’m sorry I had anything to do with it.” The most ridiculous problem, he wrote, was that of a cat sealed in a box, inside which there is a bottle of hydrocyanic acid, a poisonous gas, connected to a hammer, triggered by a Geiger counter that is connected to a piece of radioactive substance. There is no question that radioactive decay is a quantum effect. If the uranium does not decay, then the cat is alive. But if an atom decays, it will set off the counter, trigger the hammer, break the glass, and kill the cat. But according to the quantum theory, we cannot predict when the uranium atom will decay. In principle, it may exist in both states simultaneously, both intact and decayed. But if the uranium atom can exist simultaneously in both states, then it means that the cat must also exist in both states. So the question is, is the cat dead or alive?

  Normally, this is a silly question. Even if we cannot open the box, common sense tells us that the cat is either dead or alive. One cannot be both dead and alive simultaneously; this would violate everything we know about the universe and physical reality. However, the quantum theory gives us a strange answer. The fina
l answer is, we don’t really know. Before you open the box, the cat is represented by a wave, and waves can add, like numbers. We have to add the wave function of a dead cat to that of a live cat. Thus, the cat is neither dead nor alive before you open the box. Sealed inside the box, all you can say is that there are waves that represent the cat being both dead and alive at the same time.