The God Equation

by Michio Kaku

  Quantum Revolution

  Meanwhile, a new theory was being born that could explain all these mysterious discoveries. This theory would eventually unleash a revolution that would challenge everything we knew about the universe. It was called quantum mechanics. But what is the quantum anyway, and why is it so important?

  The quantum was born in 1900 when German physicist Max Planck asked himself a simple question: Why do objects glow when hot? When humans first harnessed fire thousands of years ago, they noticed that hot objects glow with certain colors. Pottery makers had known for centuries that, as objects reach thousands of degrees, they change color, going from red to yellow to blue. (You can see this for yourself by simply lighting a match or candle. At the very bottom, the flame is hottest, and its color might be bluish. It is yellowish in the center and coolest at the top, where the flame is reddish.)

  But when physicists tried to calculate this effect (called blackbody radiation) by applying the work of Newton and Maxwell to atoms, they discovered a problem. (A blackbody is an object that perfectly absorbs all radiation that falls on it. It is called black because the color black absorbs all light.) According to Newton, as atoms get hotter, they vibrate more rapidly. And according to Maxwell, vibrating charges, in turn, can emit electromagnetic radiation in the form of light. But when they calculated the radiation emitted from hot, vibrating atoms, the result defied expectations. At low frequencies, this model fit the data quite well. But at high frequency, the energy of light should eventually become infinite, which was ridiculous. To a physicist, infinity is just a sign that the equations aren’t working, that they don’t understand what is happening.

  Max Planck then posited an innocent hypothesis. He supposed that energy, instead of being continuous and smooth as in Newton’s theory, actually occurred in discrete packets he called quanta. When he adjusted the energy of these packets, he found that he could reproduce precisely the energy that radiated from hot objects. The hotter the object, the higher the frequency of radiation, corresponding to different colors on the light spectrum.

  This is why a flame changes from red to blue as the temperature increases. This is also how we know the temperature of the sun. When you first hear that the surface of the sun is about 5,000 degrees Celsius, you may wonder: How do we know that? No one has ever been to the sun with a thermometer. But we know the temperature of the sun because of the wavelength of light it is emitting.

  Planck then calculated the size of these packets of light energy, or quanta, and measured them in terms of a small constant h, Planck’s constant, which is 6.6 × 10–34 Joule-seconds. (This number was found by Planck by adjusting the energy of these packets by hand, until he could perfectly fit the data.)

  If we let Planck’s constant gradually go to zero, then all the equations of the quantum theory reduce to the equations of Newton. (This means that the bizarre behavior of subatomic particles, which often violate common sense, gradually reduces to the familiar Newtonian laws of motion as Planck’s constant is manually set to zero.) That is why we rarely see quantum effects in daily life. To our senses, the world seems very Newtonian because Planck’s constant is a very small number and only affects the universe on the subatomic level.

  These small quantum effects are called quantum corrections, and physicists spend entire lifetimes trying to calculate them. In 1905, the same year that Einstein discovered special relativity, he applied the quantum theory to light and showed that light was not just a wave but acted like a packet of energy, or a particle, that was called the photon. So light apparently had two faces: a wave as predicted by Maxwell, and a particle or photon as predicted by Planck and Einstein. A new picture of light was now emerging. Light was made of photons, which are quanta, or particles, but each photon created fields surrounding it (the electric and magnetic fields). These fields, in turn, were shaped like waves and obeyed Maxwell’s equations. So we now have a beautiful relationship between particles and the fields that surround it.

  If light had two faces, both as a particle and as a wave, then did the electron also have this bizarre duality? This was the next logical step, and it would have the most profound effect, shaking the world of modern physics and civilization itself.

  Electron Waves

  Physicists, to their shock, then found that electrons, which were once considered to be hard, point-like particles, could also act like waves. To demonstrate, take two parallel sheets of paper, one behind the other. You drill two slits in the first sheet, and then fire an electron beam at it. You would normally expect to find two spots on the second sheet, where the electron beams hit. Either the electron beam goes through the first or the second slit. Not both. That’s just common sense.

  But when the experiment is actually done, the pattern of dots on the second sheet appears to be arranged in a band of vertical lines, which is a phenomenon that occurs when waves interfere with each other. (The next time you take a bath, gently splash the surface at two places in synchronization, and you will see this interference pattern emerge, resembling a network of spiderwebs.)

  But this means, in some sense, that the electrons went through both slits simultaneously. This was the paradox: How can a point particle, the electron, interfere with itself, as if it had traveled through two separate slits? In addition, other experiments on electrons showed they vanished and reappeared somewhere else, which is impossible in a Newtonian world. If Planck’s constant were considerably larger, affecting things at a human scale, then the world would be a totally unrecognizable, bizarre place. Objects could disappear and reappear in a different location and could be two places at the same time.

  Figure 7. Electrons passing through a double slit act as if they are a wave—that is, they interfere with one another on the other side, as if they are moving through two slits simultaneously, which is impossible in Newtonian physics but is the basis of quantum mechanics.

  As improbable as the quantum theory appeared to be, it began to have spectacular success. In 1925, Austrian physicist Erwin Schrödinger wrote down his celebrated equation that precisely described the motion of these particle waves. When applied to the hydrogen atom, with a single electron orbiting a proton, it gave remarkable agreement with experiment. The electron levels found in the Schrödinger atom exactly matched the experimental results. In fact, the entire Mendeleyev table could in principle be explained as a solution of the Schrödinger equation.

  Explaining the Periodic Table

  One of the spectacular achievements of quantum mechanics is its ability to explain the behaviors of the building blocks of matter, atoms and molecules. According to Schrödinger, the electron is a wave that surrounds the tiny nucleus. In figure 8, we see how only waves with certain discrete wavelengths can travel around the nucleus. Waves with an integral number of wavelengths fit nicely. But ones that do not have an integral number do not wrap fully around the nucleus. They are unstable and cannot form stable atoms. This means that electrons can only move in distinct shells.

  As we go farther away from the nucleus, this basic pattern repeats itself; as the number of electrons increases, the outer ring moves farther away from the center. There are more electrons the farther you move. This in turn explains why the Mendeleyev table contains regular discrete levels that repeat themselves, with each level mimicking the behavior of the shell below it.

  This effect is noticeable whenever you sing in the shower. Only certain discrete frequencies, or wavelengths, bounce off the walls and are magnified, but others that don’t fit are canceled, similar to the way electron waves circle the nucleus of an atom: only certain discrete frequencies work.

  Figure 8. Only electrons of a certain wavelength can fit inside an atom—that is, the orbit must be an integer multiple of the electron wavelength. This forces electron waves to form discrete shells around the nucleus. A detailed analysis of how electrons fill these shells can help to explain the Mendeleyev p
eriodic table.

  This breakthrough fundamentally changed the course of physics. One year, physicists were completely stumped when describing the atom. The next year, with Schrödinger’s equation, they could calculate the properties inside the atom itself. I sometimes teach quantum mechanics to graduate students, and try to impress upon them the fact that everything around them, in a sense, can be expressed as a solution of his equation. I mention to them that not only can atoms be explained by it, but one can also explain the bonding of atoms to form molecules and therefore the chemicals from which our entire universe is composed.

  No matter how powerful the Schrödinger equation was, however, it still had a limitation. It only worked for small velocities—that is, it was nonrelativistic. The Schrödinger equation said nothing about the speed of light, special relativity, and how electrons interact with light via Maxwell’s equations. It also lacked the beautiful symmetry of Einstein’s theory and was rather ugly and difficult to handle mathematically.

  Dirac Theory of the Electron

  Then a twenty-two-year-old physicist, Paul Dirac, decided to write a wave equation that obeyed Einstein’s special relativity by merging space and time. One of the things that was inelegant about the Schrödinger equation was that it treated space and time separately and hence calculations were often tedious and time-consuming. But Dirac’s theory combined the two and had a four-dimensional symmetry, so it was also beautiful, compact, and elegant. All the ugly terms in the original Schrödinger equation collapsed into a simple four-dimensional equation.

  (I remember when I was in high school, trying desperately to learn the Schrödinger equation, and struggling with all the ugly terms it contained. How could nature be so malicious, I thought, to create a wave equation that was so clumsy? Then one day, I stumbled upon the Dirac equation, which was beautiful and compact. I remember crying when I saw it.)

  The Dirac equation was a spectacular success. We saw earlier that Faraday had shown that a changing electric field in a coil of wire produced a magnetic field. But where did the magnetic field come from in a bar magnet, without any moving charges? This seemed like a total mystery. But according to Dirac’s equations, the electron was predicted to have a spin that created a magnetic field of its own. This property of the electron was built in from the very beginning in the mathematics. (This spin, however, is not the familiar spin we see around us—that is, as in a gyroscope—but is a mathematical term in the Dirac equation.) The magnetic field created by the spin matched precisely the field actually found surrounding electrons. This, in turn, helped to explain the origin of magnetism. So where does the magnetic field in a magnet come from? It comes from the spin of the electrons trapped inside the metal. Later, it was discovered that all subatomic particles have a spin. We will return to this important concept in a later chapter.

  Even more important, the Dirac equation predicted an unexpected new form of matter, called antimatter. Antimatter obeys the same laws as ordinary matter, except it has the opposite charge. So the anti-electron, called the positron, has a positive, not a negative, charge. In principle, it may be possible to create anti-atoms, made of anti-electrons circling anti-protons and anti-neutrons. But when matter and antimatter collide, they explode in a burst of energy. (Antimatter will become a crucial ingredient of a theory of everything, since all particles in the final theory must have an antimatter counterpart.)

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  Previously, physicists considered symmetry to be an aesthetically pleasing but nonessential aspect of any theory. Now, physicists were staggered at the power of symmetry, that it could actually predict entirely new and unexpected physical phenomena (such as antimatter and electron spin). Physicists were beginning to understand that symmetry was an essential and inescapable feature of the universe at a fundamental level.

  What Is Waving?

  But there were still some nagging questions. If the electron had wavelike properties, then what was disturbing the medium in which the wave existed? What was waving? And how can it go through two different holes simultaneously? How can an electron be in two places at the same time?

  The answer was startling and incredible, and split the physics community right down the middle. According to a paper by Max Born in 1926, what was waving was the probability of finding an electron at that point. In other words, you could not know for certain precisely where an electron was. All you could know was the probability of finding it. This was codified in Werner Heisenberg’s celebrated uncertainty principle, which stated that you cannot know precisely the velocity and location of an electron. In other words, electrons are particles, but the probability of finding the particle at any given location is given by a wave function.

  This idea was a bombshell. It meant that you could not accurately predict the future. You could only predict the odds that certain things will happen. But quantum theory’s successes were undeniable. Einstein wrote that “the more successful the quantum theory becomes, the sillier it looks.” Even Schrödinger, who had introduced the concept of the electron wave in the first place, rejected this probabilistic interpretation of his very own equations. Even today, there are arguments among physicists debating the philosophical implications of the wave theory. How can you be two places at the same time? Nobel laureate Richard Feynman once said, “I think I can safely say that nobody understands quantum mechanics.”

  Ever since Newton, physicists believed in something called determinism, the philosophy that all future events can be accurately predicted. The laws of nature determine the motion of all things in the universe, making them ordered and predictable. To Newton, the entire universe was a clock, beating in a precise predictable fashion. If you knew the location and velocity of all the particles in the universe, you could deduce all future events.

  Predicting the future, of course, has always been an obsession of mortals. In Macbeth, Shakespeare wrote,

  “If you can look into the seeds of time

  And say which grain will grow and which will not,

  Speak, then, to me.”

  According to Newtonian physics, it is possible to predict which grain will grow and which will not. For several centuries, this view prevailed among physicists. So uncertainty was heresy, and shook modern physics to the core.

  Clash of Titans

  On one side of this debate were Einstein and Schrödinger, who helped to start the quantum revolution in the first place. On the other side were Niels Bohr and Werner Heisenberg, creators of the new quantum theory. It culminated in the historic sixth Solvay Conference in 1930 in Brussels. It was to be a debate for the ages, when the giants of physics would go head-to-head to battle for the meaning of reality itself.

  Paul Ehrenfest would write, “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.” Bohr could be heard muttering dejectedly to himself in the hallways, saying just one word, “Einstein…Einstein…Einstein.”

  Einstein led the charge, raising objection after objection to the quantum theory, trying to expose how absurd it was. But Bohr successfully countered each of Einstein’s criticisms one by one. When Einstein kept repeating that God does not play dice with the universe, Bohr reportedly said, “Stop telling God what to do.”

  Princeton physicist 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.”

  Historians agree for the most part that Bohr and the quantum rebels won the debate.

  Still, Einstein was successful in exposing the cracks in the foundation of quantum mechanics. Einstein showed that it was a towering giant standing on philosophi
cal feet of clay. These criticisms are heard even today, and they all center on a certain cat.

  Schrödinger’s Cat

  Schrödinger devised a simple thought experiment that exposed the essence of the problem. Place a cat in a sealed box. Put a piece of uranium in the box. When the uranium fires a subatomic particle, it triggers a Geiger counter that sets off a gun that fires a bullet at the cat. The question is: Is the cat dead or alive?

  Since the firing of a uranium atom is a purely quantum event, it means that you have to describe the cat in terms of quantum mechanics. To Heisenberg, before you open the box, the cat exists as a mixture of different quantum states—that is, the cat is the sum of two waves. One wave describes a dead cat. The other wave describes a live cat. The cat is neither dead nor alive but a mixture of both. The only way to tell if the cat is dead or alive is to open the box and make an observation; then the wave function collapses into a dead or live cat. In other words, observation (which requires consciousness) determines existence.

  To Einstein, all this was preposterous. It resembled the philosophy of Bishop Berkeley, who asked: If a tree falls in the forest and no one is there to hear it, does it make a sound? The solipsists would say no. But the quantum theory was even worse. It said that if there is a tree in the forest with no one around, the tree exists as the sum of many different states: for example, a burnt tree, a sapling, firewood, plywood, etc. Only when you look at the tree does its wave magically collapse into an ordinary tree.