What does it mean to say that the wave function has the properties of a wave, such as a vibrating string or the series of concentric circles created on the surface of a pond when a rock is tossed into the water? Waves are distinguished by having amplitudes that vary periodically in space and time. Consider the ripples created when a rock is tossed into a pond. At some points of the wave there are crests, where the height of the water wave is large and positive (that is, the surface of the water is higher than normal); at some points there are troughs, where the height of the water is lower than normal; and in other regions the amplitude of the wave is zero—the height of the water’s surface is the same as it would be without the rock’s disturbance.
The amplitude of the peaks and valleys typically becomes smaller with distance from the source of the waves. This is why on the California shore we don’t notice if a rock is dropped into the center of the Pacific Ocean. Certain large disturbances can create tsunamis that maintain large amplitudes even when traveling great distances. Dr. Manhattan, presumably, is able to change the amplitude of his quantum mechanical wave function so that it can have an appreciable amplitude at some large distance away from him. This would be how he teleports, though in quantum mechanics we would say that he is “tunneling.”
Schrödinger’s equation enables us to calculate the wave function of an object as a function of the forces acting on it. If there are no net forces, the electron, for example, can have uniform straight-line motion, with a well-defined de Broglie wavelength determined by its momentum. If this electron strikes a barrier and it lacks sufficient energy to go over the obstacle, then the electron will be reflected, bouncing off the barrier and returning from where it came.
We are familiar with such wave phenomena whenever we use a mirror. Light waves move in straight lines, passing through the glass covering of the mirror, until they reach the silvered backing. Unable to penetrate the metal, the waves are reflected back in another straight-line trajectory, along a path that makes the same angle with a line perpendicular to the mirror’s surface as the incoming beam.
In fact, one does not need the metal backing to see this reflection effect. We all know that a single pane of glass can act like a mirror, when we look out the window from a well-lit room at night. In this case just the difference in optical media, glass and air, can cause light reflection, particularly when we look at the window at an angle. The reflection is more noticeable if the direction we are looking, relative to the glass surface, is larger than a particular angle that depends on the optical properties of glass and air. When we place our face against the glass, this reflection effect goes away, for then most of the light rays that we see from outside travel perpendicular to the surface. Light travels slower in glass than in air (more on this in a moment), and this difference in light velocities (characterized by the material’s index of refraction, for technical reasons) accounts for the reflection effect. This can occur during daytime as well but is less noticeable when more light comes into the room from the outside than goes out from the interior.
Suppose that we are looking at the window at night from the interior of a strongly lit room. The glass reflects our image as if it were a conventional mirror. Now imagine a second sheet of glass placed behind the first, as in a double-paned window, only the separation between the two sheets isn’t a quarter of an inch, but more like a millionth of a centimeter. In this case, even though the light would have been completely reflected without the second sheet of glass, the presence of the second pane enables some of the light to pass through both sheets of glass, even though they are not touching each other. This phenomenon is a hallmark of the wavelike properties of light (so for the sake of argument we will ignore for the moment that light is actually comprised of discrete photons). It turns out that the light wave is not completely reflected at the first glass-air interface, but instead a small amount of the oscillating electric and magnetic fields leak out into the air. The small leakage is limited to a thin region very near the interface and is normally not important. But when the second sheet of glass is brought very close to the first interface, some of the protruding light waves extend into the second piece of glass. In this case the wave is not reflected but rather propagates into the second glass. This “leakage effect” is not unique to light; it occurs for any wave—even those associated with matter!
Figure 18: Cartoon sketch illustrating a light wave, which is normally reflected at a glass/air interface and may have a small amplitude leaking into the air. If another piece of glass is placed near the first (the separation should be no more than a few wavelengths of the light), then the wave may be able to propagate into the second material. A similar phenomenon occurs with matter waves during quantum mechanical tunneling.
One of the most fantastic aspects of quantum mechanics, and one that provides dramatic confirmation that there is a wavelike aspect to the motion of matter, is that this “leakage effect” is observed for electrons, protons, and neutrons. Here, instead of light and a sheet of glass, consider an electron in a metal or semiconductor. Instead of a glass-air interface, there might be a barrier at the surface of the conductor, either the vacuum of empty space or some other electrical insulator. The electron would normally not have enough energy to leave the conducting material and would be reflected at this surface. If another conductor is placed on the other side of the barrier, and if this barrier is not too thick compared to the electron’s de Broglie wavelength, then there is a probability that the matter-wave can extend through the gap. Even though the electron does not have sufficient energy to jump or spark across the gap, as its quantum mechanical wave function leaks through the forbidden barrier into the second region, it can thus be found in the second conductor.
When matter waves exhibit this leakage effect, it is termed “quantum mechanical tunneling” even though the electron does not, obviously, create a “tunnel” through the insulator or the vacuum of empty space. Recall that the square of the wave function tells us the probability per volume of finding the electron at some point in space and time. If the leakage of the wave function through the forbidden region is small, then there is a low probability of finding the electron in the second region on the other side of the barrier. But anything that has a probability larger than zero will happen if one tries enough times. If we send an electron moving toward the barrier of a particular height and width, examination of the wave function may show that there is a very large probability—at least 99.9999 percent—that the electron would be reflected at the interface. This means that there is only one chance in a million that the electron will show up in the second material. But if a million electrons approach the barrier, one may get through, and if a trillion electrons strike the barrier, then a million will pass through via tunneling to the other side. We don’t know which electrons will make it into the second conductor until we send them out, but based on the properties of the barrier, we can confidently predict how many on average will get through. As discussed in Section 5, many personal electronics devices employ the tunneling phenomenon to regulate the current in a circuit—putting this esoteric quantum mechanical effect to prosaic and reliable use.
Figure 19: An example from the graphic novel Watchmen—Dr. Manhattan’s physics remains unchanged when he multiplies his quantum mechanical wavefunction by a constant bigger than unity.
What an electron can do, Dr. Manhattan can do as well, at least in the pages of a comic book or a motion picture with an extensive special-effects budget! Presumably through his miraculous control of his quantum mechanical wave function, Dr. Manhattan is able to extend his de Broglie wavelength not just a few nanometers, as the electron in a tunneling diode does, but over thirty-six million miles.29 With a large enough amplitude at the remote location, the probability of Dr. Manhattan suddenly appearing at the new site becomes very large. He never is actually in the space between his starting point and final destination but is simply able to adjust his probability density to be a maximum where he wants to go—which
is certainly a savings in time and money compared to commercial air travel.
Dr. Manhattan is able to change his size at will (as shown in Figure 19) due to the fact that the Schrödinger equation is linear. In mathematics an equation is called “linear” if it depends only on the key variable (in the Schrödinger equation that would the wave function Ψ) and not on that variable squared or cubed, or the square root, and so on.30 A very simple linear equation is Ψ = Ψ, which is certainly a true statement. In fact, this equation is so simple that it is always true for any value of Ψ. So if Ψ = 1, then this equation tells us that 1 = 1 (which we already knew). In this case, if Ψ is ten times larger, then this simple equation tells us that 10 = 10, which is also a true statement. Given that the Schrödinger equation is linear, there is no change in the physics of the situation if we multiply Ψ by a constant, either a larger or smaller one. By multiplying the wave function by a constant (the “normalization” described in Chapter 6), we ensure that Ψ2 acts as a probability density and varies from 0 percent to 100 percent. In addition, the fact that the Schrodinger equation is linear means that if there are two possible solutions to the equation, such as ΨA and ΨB, then their sum ΨA + ΨB will also be a solution (this will be very important in Section 4). Presumably Dr. Manhattan is able to shrink himself down as well, multiplying his wave function by a value less than 1, though we never see him utilize this capability in the comics or the motion picture adaptation.
Jon Osterman, as shown in Figures 11 and 19, gained a bright blue pallor when he reassembled himself following the unfortunate “incident” in the intrinsic field chamber. As wave functions have no color, there are at least three possible explanations for his being blue: (1) always knowing what will happen in the future has taken all the joy out of life; (2) he’s depressed because he realizes that “nothing ever ends”; or (3) he’s emitting Cerenkov radiation.
Dave Gibbons, the artist of Watchmen, once stated in a radio interview that he elected to make Dr. Manhattan blue as a visual signifier in order to constantly remind the readers of Jon’s transformation. If Dr. Manhattan were red he would look like he was on fire, green was too close to the Hulk, and other colors would look too similar to actual skin tones on the printed comic page. Be that as it may, just because the color choice was one of casual necessity does not mean that we can’t obsessively discuss the underlying physics in great detail! For it turns out that given Dr. Manhattan’s origin, if he were to glow in any color of the optical spectrum, it would indeed be blue.
Figure 20: Image of a pencil (belonging to a certain fictional physicist) that appears broken at the air/water interface due to the different speeds of light in the two media.
When certain elements undergo radioactive decay, they may emit high-speed electrons as a by-product of their nuclear reaction (we’ll discuss the mechanism by which this occurs in the next section). When those electrons (also referred to as “beta rays”) travel faster than the speed of light in a material medium, they emit electromagnetic radiation in the blue-ultraviolet portion of the spectrum, which is known as Cerenkov radiation.
This last sentence is no doubt puzzling, for a central principle of Einstein’s Special Theory of Relativity is that nothing can travel faster than the speed of light. But this is in fact not strictly correct. The more accurate way to state this principle is that nothing can travel faster than the speed of light—in the vacuum of empty space! Light speed in a vacuum is three hundred million meters per second and is indeed the fastest velocity in the universe. However, light travels much slower than this when moving through denser media, such as water or glass.
Anyone who has noted that a straw or pencil in a glass of water appears to be “broken” at the water-air interface, as shown in Figure 20, has observed an optical effect that results from light moving slower in water than in air. In order to be seen, light must be reflected from the straw and detected by our eyes. The change in the speed of light at the water-air surface causes straight-line light rays to bend, in a phenomenon termed “refraction.” The light that bounces off the portion of the straw protruding from the water of course does not bend and travels in a straight line. When we observe the light from the straw in the air and the light that bent upon leaving the water, we interpret the image as a straw with a sharp discontinuity at the water’s surface.
Why does light travel slower in water and other media? It is because the electromagnetic waves interact with the electrons surrounding each atom in the material. When running through a swimming pool, you will move slower if you hold your arms out away from your body and increase the drag from the water. Light experiences an “electromagnetic drag” from the electrons that can slow its motion down markedly. The speed of light in water or glass is only 75 percent of what it is in a vacuum, which is still pretty fast. But high-speed electrons can move through these media with fewer interactions, and thus it is possible for an electron to travel in water faster than light can. When this happens, the electron (which does interact with the electrons surrounding the atoms in the material, only not as strongly as light does) generates an “electromagnetic sonic boom,” emitting light in the blue-ultraviolet region of the electromagnetic spectrum. This blue-light shock front is termed Cerenkov radiation, after Pavel Cerenkov, who discovered and explained this phenomenon in 1934 (for which he was awarded the Nobel Prize in Physics in 1958).
Air is much less dense than water or glass, and light slows down only slightly when moving through the atmosphere compared to its largest speed in a vacuum. Nevertheless, for the purposes of explaining the science underlying a fictional character in a comic book, let’s stipulate that it is possible to generate Cerenkov radiation from high-speed electrons jetting through the air. Let’s also suppose that when Dr. Manhattan reassembled himself following the removal of his intrinsic field, he did so in such a way that he is continually leaking high-speed electrons, giving him a healthy blue glow. There are always many electrons from the Earth that he can draw upon in order to maintain his charge neutrality. If he wanted to darken his hue (as he does at one point for the benefit of television cameras), he could simply change the speed at which the electrons escape.
Nuclear reactor piles at the bottom of deep pools of water31 give off a blue glow, and this Cerenkov light indicates that the pile is active and emitting beta rays. In Watchmen (spoiler alert!) a character frames Dr. Manhattan, so that he is accused of giving his close friends and an ex-girlfriend cancer. One way to inflict Osterman’s associates that would plausibly suggest him as the source of the disease is to surreptitiously expose these people to nuclear isotopes, such as strontium-90, that are known to be carcinogenic and are deadly precisely because of their beta radiation emissions.
Another striking characteristic of Watchmen’s Dr. Manhattan is his ability to experience the past, present, and future simultaneously. It is specified in the graphic novel that the post-intrinsic-field-removal Jon Osterman is able to see only his own future and thus would not know of events to come unless he either directly experiences or participates in them or is told about them. Again, if Dr. Manhattan did indeed have control over his macroscopic quantum mechanical wave function, then as the wave function contains all the information about the object’s probability density in space and time, this characteristic is plausible.
The fact that there is no other source of information about the future evolution of an object than what is contained in its wave function is significant. If all we have is the wave function, and the wave function can tell us only the probability per unit volume of finding the object in space and time, then, even in a perfect, idealized situation, we must resign ourselves to knowing only the odds as to the object’s location. When we deal with probabilities and statistics in other nonquantum situations in physics, it is simply to make our lives easier. We know that Newton’s laws of motion provide a nearly complete description of the interactions of the air molecules in the room in which you are reading this right now. However, to apply these equations to the ai
r would involve solving Newton’s laws for all trillion trillion molecules simultaneously. In this and similar situations, it is much more reasonable to describe the average pressure, for example, or introduce the concept of “temperature” (which represents the average kinetic energy per molecule) rather than deal with each molecule separately in turn. In contrast, in the quantum world, the emphasis on probability density is a matter of necessity, not convenience. Even with infinitely fast and infinitely precise observations, we can never know exactly where the object is, but only its average location.
This inability to do better than knowing the odds is a consequence of the wavelike nature of matter. Recall the discussion of the Heisenberg uncertainty principle from the preceding chapter. The wavelength of the matter-wave associated with the electron, for example, is directly connected to its momentum. A pure, single wave has only one wavelength, and thus we know exactly what its momentum is, but at the expense of having any information about where the electron is. The more we localize the electron, say, by ensuring that it will be found within the one-third of a nanometer that is the typical spatial extent of an atom, the less defined its momentum becomes. If we had perfect knowledge of its position (which is what physicists desired in order to put the “probability density” aspect to rest), then this would come at the cost of perfect ignorance about its momentum. It could in principle have any momentum between zero and infinity, and we would thus have to contend with a probability interpretation of its motion. As we need both positions and momenta to employ a traditional Newton’s law description of a system, probabilities are the best we can ever do.
The Amazing Story of Quantum Mechanics Page 10
