In the photoelectric effect, the source of the electron-whacking energy has to come from light. Now, for the moment, assume light is a wave. If so, buffeting light waves must deposit their energy in the electrons, giving them a bunch of energy. This energy induces the electrons to leap away from the metal atoms. If the waves don’t deposit enough energy into the electron, if the waves’ collective energy is below the required threshold, then the electrons will stay put. However, if the waves are sufficiently energetic, then they will induce the metal to spark. So far so good.

  In wave theory, though, there are two ways to increase the energy of an incoming bunch of waves. The first way is pretty easy to see: just make the waves bigger. One-foot ocean waves pack less punch than three-foot waves, and ten-foot waves can knock a swimmer senseless. A wave’s height is known as its amplitude; the bigger the wave, the larger the amplitude and the more energy it carries. With water waves, amplitude translates to physical height, but in other sorts of waves it might have a different interpretation. With sound waves, for example, amplitude is related to volume: the louder a sound, the greater the amplitude of the sound waves. And with light, amplitude is related to brightness. A bright yellow beam has a greater amplitude than a dim yellow beam.

  The second way to increase energy in a set of waves is a little more subtle: make the waves more frequent. If the wave crests are closer together—if more waves are striking the beach per minute—the waves transfer more energy to the shore. So, the greater the frequency of the waves, the more energy they contain. With light, frequency corresponds to color. The lower-frequency light—infrared, red, and orange light—contains less energy than yellow, green, blue, or violet light, which have greater frequencies. Ultraviolet light and X-rays have more energy still, as their frequencies are even greater than those of visible light.

  So, if there is an energy threshold to knock electrons out of a metal, there should be two ways to get light above that critical threshold. With a beam of given brightness, you can change the frequency of light from red to green to blue to ultraviolet, and at some point the electrons should start leaping out of the metal. Sure enough, this is what happens. Red light didn’t cause sparking in Hertz’s metal sheet, nor did green or blue. But when the light color became a high enough frequency—when the beam was ultraviolet—the sparking suddenly began.

  The second way to get the light above that energy threshold is to fix the frequency of the beam—keeping it, say, at the same shade of yellow—but increasing the brightness of the beam. If you start off with a dim beam of yellow, it won’t have enough energy to induce the electrons to leap out of the metal. But as you brighten the beam, it gets more and more energetic. When the beam finally gets bright enough, when the amplitude of the beam gets big enough, the electrons should suddenly start getting knocked loose and the sparking should begin. This is not what happened.

  No matter how bright a yellow beam was, it never freed electrons from the metal. Worse yet, even the dimmest ultraviolet beam—which, according to the wave theory of light, shouldn’t have enough energy in the metal to free electrons—caused sparking. Just as it made no sense for a single particle of light to make an interference pattern, it made no sense for a dim wave of ultraviolet light to be able to knock electrons loose while a bright yellow beam could not. In wave theory, there should be an amplitude threshold for the photoelectric effect just as there was a frequency threshold. But Hertz’s experiment showed that only frequency seemed to matter. This contradicted the wave equations of light that scientists had long since accepted.

  Physicists were stuck. They couldn’t explain interference with the particle theory of light, and they couldn’t explain the photoelectric effect with the wave theory. It took nearly twenty years to figure out what was wrong, and when Einstein did—the same year, 1905, that he formulated the special theory of relativity—he destroyed the wave theory of light forever. In its place was a new theory, quantum theory. It was Einstein’s explanation of the photoelectric effect that earned him his Nobel Prize and placed quantum theory firmly in the mainstream of physics.

  Einstein’s work breathed life into a rogue idea that had been born five years earlier when Max Planck, a German physicist, came up with a method for resolving a mathematical dilemma. This dilemma, too, had to do with the behavior of light and matter. The equations that described how much radiation a hot chunk of matter emits—the ones that describe why a blacksmith’s iron glows red and the filament in a lightbulb glows white—were not working. These equations broke down under certain conditions, shredding the theory in a cloud of mathematical infinities. Planck came up with a solution, but it came at a price.

  Planck made an assumption that seemed physically absurd. He assumed that under certain circumstances matter could only move in certain ways: it is quantized. (Planck coined the term quantum, after the Latin word for “how much.”) For example, the quantization of an electron’s energy around an atom meant that an electron could only take on some energies and not others. This sort of thing doesn’t happen in everyday life. Imagine what would happen if your car’s speeds were quantized: if it could go 20 and 25 miles an hour, but couldn’t drive at 21 or 23 or any other speed in between. If you were driving at 20 miles an hour and you pressed down on the accelerator, absolutely nothing would happen for a while. You would keep tooling along at 20 miles an hour…20 miles an hour…20 miles an hour…and then, suddenly, pop! You would instantly be driving at 25 miles an hour. Your car would have skipped all the speeds in between 20 and 25. This obviously doesn’t happen. Our world is smooth and continuous, not jerky and jumpy. Planck himself called his quantum hypothesis an “act of desperation.” However, as strange as this idea—the quantum hypothesis—was, it banished the infinities that plagued the radiation equations.

  Einstein solved the photoelectric puzzle by applying the quantum hypothesis to light. Contrary to what nearly all the physicists of the previous hundred years had assumed, Einstein postulated that light is not a smooth, continuous wave but chunky, discrete particles now known as photons. This was despite the evidence to the contrary, including Young’s interference experiment. In Einstein’s model, each particle carries a certain amount of energy proportional to its frequency; double the frequency of a photon and you double the energy it carries.1 Once you accept that idea, you can do a great job of explaining the photoelectric effect.

  In Einstein’s picture, each photon striking the metal can give the electron a kick, and the more energy the photon has, the bigger the kick. As before, the energy must meet a threshold. If the energy of the photon is too small, below the binding energy of the electron, the electron cannot escape. If the energy is large enough, though, the electron escapes. As did the wave theory of light, Einstein’s hypothesis explains the wavelength threshold: if the photons do not have enough energy, then they cannot knock the electrons away from the atoms. But unlike the wave theory, Einstein’s quantum theory of light also explains the lack of an amplitude threshold. It explains why merely increasing the brightness of the beam cannot make electrons start escaping the metal.

  If the beam is made up of individual particles of light, increasing the brightness just means that more of these particles are in the beam. Only one photon is likely to strike an atom at a time, and if that photon doesn’t have the required energy, it can’t knock the electron away—no matter how many photons there are in the surrounding neighborhood. It is one photon per atom, and if the incoming photon is too weak, nothing happens, regardless of the brightness of the beam.

  Einstein’s quantum theory of light explained the photoelectric effect in wonderful detail; the hypothesis completely explained the puzzling experimental observations that could not be explained by the wave theory of light.2 This was really puzzling to physicists at the time: Young showed that light behaves like a wave and not a particle, but Einstein showed that light behaves like a particle and not a wave. The two theories were in direct conflict, and they couldn’t both be right. Or could they?

  Just as with relativity theory, information was at the heart of the problem. In relativity theory, two different observers can gather information about the same event and get mutually contradictory answers. One might say that a spear was nine meters long while the other might say that it was fifteen meters long, and both can be right. In quantum theory, there is a similar problem. An observer, measuring a system in two different ways, might get two different answers. Do an experiment in one way and you might prove that light is a wave and not a particle. Do a similar experiment in a slightly different way and you can prove that light is a particle and not a wave. Which is right? Both—and neither. The way you gather the information affects the outcome of the experiment.

  Quantum theory can be cast in the language of information theory—in talk about transferring information (including the 1s and 0s of binary choices)—and when it is, it reveals a whole new depth to the paradoxes of the quantum world. The conflict between waves and particles is just the beginning.

  Einstein’s theory brought Planck’s quantum hypothesis into the mainstream, and over the next three decades Europe’s best physicists developed a theory that did a beautiful job of explaining the behavior of the subatomic world. Werner Heisenberg, Erwin Schrödinger, Niels Bohr, Max Born, Paul Dirac, Albert Einstein, and others built up a set of equations that explained with stunning precision the behavior of light and electrons and atoms and other very tiny objects.3 Unfortunately, though this framework of equations—quantum theory—always seemed to get the right answers, other consequences of those equations seemed to contradict common sense.

  The dictates of quantum theory, at first glance, are ridiculous. The strange, seemingly contradictory properties of light are par for the course. Indeed, they come directly from the mathematics of quantum theory. Light behaves like a particle under some conditions and like a wave under other conditions; it has some of the properties of each, yet is neither truly particle nor wave.

  It is not only light that behaves this way. In 1924, the French physicist Louis de Broglie suggested that subatomic matter—particles like electrons—should have wavelike properties as well. To experimentalists, electrons were obviously particles, not waves; any half-competent observer could see electrons leave little vapor trails as they streaked from one end of a cloud chamber to the other. These trails were clearly the tracks of little chunks of matter: particles, not waves. But quantum theory trumps common sense.

  Though the effect is much harder to spot with electrons than it is with light, electrons do show wavelike behavior as well as their more familiar particle-like behavior. In 1927, English physicists shot a beam of electrons at a crystal of nickel. As electrons bounce off regularly spaced atoms and zoom through the holes in an atomic lattice, they behave as if they have just passed through the slits of Young’s experiment. Electrons do interfere with each other, making an interference pattern. Even if you ensure that only a single electron at a time strikes the lattice, the interference pattern persists; the pattern cannot be caused by electrons bouncing off each other. This behavior is not consistent with what you would expect of particles: the interference pattern is an unmistakable sign of a smooth, continuous wave, rather than discrete, solid particles. Somehow, electrons, like light, have both wavelike and particle-like behavior, even though the properties of waves and of particles are mutually contradictory.

  This twofold wave-particle nature is true of atoms and even molecules just as it is true of electrons and light. Quantum objects can behave like waves as well as particles; they have wavelike properties and particle-like properties. At the same time, they have properties inconsistent with being a wave and with being a particle. An electron, a photon, and an atom are both particle and wave, and neither particle nor wave. If you set up an experiment to determine whether a quantum object is a particle, 1, or a wave, 0, you will get a 1 sometimes or a 0 sometimes, depending on the experiment’s setup. The information you receive depends on how you gather that information. This is an unavoidable consequence of the mathematics of quantum mechanics. It is known as wave-particle duality.

  Wave-particle duality has some really bizarre consequences—you can use it to do things that are absolutely forbidden by the classical laws of physics—and this seemingly impossible behavior is encoded in the mathematics of quantum mechanics. For example, the wavelike nature of the electron allows you to build an interferometer from electrons just as you can build one from light. The setup is pretty much the same in both cases. In the matter-wave interferometer, a beam of particles, such as electrons, shoots toward a beam splitter and goes off in two directions at once. When the beams recombine, they either reinforce each other or cancel out each other, depending on the relative sizes of the two paths. If you tune the interferometer properly, you should never, ever spot an electron at the detector, because the beams moving down the two paths can completely cancel each other. This cancellation works no matter how dim the electron beam is, no matter how few electrons are striking the beam splitter. In fact, if you set up your apparatus correctly, even if a single electron enters the interferometer and hits the beam splitter you will never detect the electron emerging from the other side.

  Common sense would tell you that an indivisible particle like an electron would have to make a choice at the beam splitter: it would have to choose to take path A or path B, to go to the left or to go to the right, but not both. It should be a purely binary decision; you can assign a 0 to path A and a 1 to path B. The electron would travel down its chosen path. It would make its choice, 0 or 1, and then, at the far end of the interferometer, it would strike the detector. Because only a single electron travels through the interferometer, there should be nothing to interfere with it, no other particles to block it. Regardless of whether the electron chooses path A or path B, it should emerge on the other side at the detector unhindered. The interference pattern should disappear. There are no other particles that can interfere with the electron. But that is not what happens; common sense fails.

  Even when a single electron at a time enters the interferometer, there’s an interference pattern. Somehow, something is blocking the electron. Something prevents the electron from emerging from the beam splitter in certain ways and striking the detector in some places, but what could that something be? After all, the electron is the only thing in the interferometer.

  The answer to this seeming paradox is hard to accept, and you will have to suspend your disbelief for a moment, as it sounds impossible. The laws of quantum mechanics reveal the culprit. The object that blocks the electron’s motion is the electron itself. When the electron hits the beam splitter, it takes both paths at once. It doesn’t choose to take path A or path B; instead, it goes down both paths simultaneously, even though the electron itself is indivisible. It goes left and right at the same time; its choice is simultaneously a 0 and a 1. Faced with two mutually exclusive choices, the electron chooses both.

  In quantum mechanics, this is a principle known as superposition. A quantum object like a photon or an electron or an atom can do two (classically) contradictory things or, more precisely, be in two mutually exclusive quantum states, simultaneously. An electron can be in two places at once, taking a left path and a right path at the same time. A photon can be polarized vertically and horizontally at the same time. An atom can be both right side up and upside down (more technically, its spin can be up and down) at the same moment. And in information-theoretic terms, a single quantum object can be a 0 and a 1 simultaneously.4

  This superposition effect has been observed many times. In 1996, a team of physicists at the National Institute of Standards and Technology laboratories in Boulder, Colorado, led by Chris Monroe and David Wineland, made a single beryllium atom sit in two different places at the same time. First, they set up a clever laser system that separated objects with different spins. When the lasers struck an atom with an upward spin, they pushed it a tiny bit in one direction, say, to the left; when they struck an atom with a downward spin, they pushed it in the opposite direction, a hair to the right. The physicists then took a single atom, isolated it carefully from its surroundings, and bombarded it with radio waves and lasers, putting it in a state of superposition. It was in both a spin-up state and spin-down state, both 1 and 0, at the same time. Then they turned on the laser separating system. Sure enough, the same atom, spin up and spin down at the same time, moved to the left and to the right simultaneously! The atom’s spin-up state moved to the left, and its spin-down state moved to the right: a single beryllium atom was in two places at once. A classical, indivisible atom could never be both1 and 0 at the same time, but the Colorado team’s data indicated that the atom was simultaneously in two positions, fully 80 nanometers—about ten atomic widths—away from each other. The atom was in a (dramatic) state of superposition.5

  Superposition explains how a single electron can make an interference pattern even though a single classical object never could. The electron interferes with itself. When the electron hits the beam splitter, it enters a state of superposition; it takes path A and path B at the same time, it chooses both 0 and 1. It is as if two ghostly electrons travel down the two sides of the interferometer, one on the left and one on the right. When the two paths rejoin each other, the ghostly electrons interfere with each other, canceling out each other. The electron enters the beam splitter but never emerges, never strikes the detector, because the electron takes two paths simultaneously and cancels itself out.

 

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