The Amazing Story of Quantum Mechanics
Page 5
Any solid the size of a sugar cube, such as a sugar cube, contains a little under a trillion trillion atoms. How these atoms are arranged, their chemical composition, and the nature of the connections to their neighbors determines whether the solid in question conducts electricity and is shiny (that is, reflects light), like a metal, or does not conduct electricity and is transparent to visible light, like a diamond. Consider a carbon atom, which chemically prefers to have four chemical bonds. There are many different ways a collection of carbon atoms can chemically bond to one another, and if one brings the atoms together in a haphazard, random manner, one such resulting configuration is “soot.”12 An alternative bonding scheme would have each carbon atom located carefully in relation to its neighbors, so that all four chemical bonds for each carbon atom have their ideal strength and location. One such uniform, periodic arrangement of carbon atoms is termed “diamond.” Chemically diamond and soot are identical as they both consist of carbon atoms bonded to each other, and yet they have very different structural properties (diamond is hard, while soot is soft), electrical conduction (soot is a pretty good conductor of electricity, while diamond is an excellent insulator), optical characteristics (soot absorbs visible light, which is why it appears black, while diamond is transparent in the visible portion of the spectrum), and financial (diamond is expensive precisely in proportion to its scarcity—and don’t try to give a soot ring to your beloved13). If soot and diamond are identical chemically, then all of these differences must be due to the arrangements of the carbon atoms in the two substances. A deep understanding of why carbon atoms would form certain types of chemical bonds in one circumstance and very different bonds in another would not arrive until the full, formal theory of quantum mechanics was developed by Schrödinger and Heisenberg. I describe how quantum mechanics accounts for all of chemistry later on—for now we are interested in the fact that for certain solids the atoms are arranged in periodic arrays, like the oranges stacked in a grocery store display, which enables large-scale three-dimensional uniform crystalline solids.
These crystalline arrangements of atoms can be used as atomic-scale “oil slicks” for interference experiments, as shown in Figure 6, providing uniform layers that reflect electron beams striking them, with each layer being one atom thick, which is just the right fraction of the length of the de Broglie wavelength of our electrons. Thus, if we send in a beam of electrons traveling at the right speed, their momentum will be such that their corresponding “pilot wave” will have a de Broglie wavelength commensurate to the spacing between atomic layers in our crystal. The incoming electrons will be repelled by the electrons around each atom in the crystal—as identical electrical charges experience a repulsive force. As any given collision between the electron beam and the crystal’s electrons is random, one would expect that the intensity of scattered electrons would be fairly uniform, regardless of the direction one looks. But thanks to quantum mechanics, this is not what is seen.
Just as in the case of the light scattered from an oil slick, where all colors are present in the incident white light, but only certain colors constructively interfere, one finds that the intensity of scattered electrons is not uniform. Rather, there are regions where a high intensity of scattered electrons are found, and other regions devoid of electrons, with a pattern exactly as one would expect for interfering waves, rather than colliding particles. Figure 7 shows strikingly similar interference patterns when green light from a laser passes through a fine metal mesh and when an electron beam passes through a graphite crystal. The wavelength of green light is much longer than that of the electron beam’s de Broglie waves, so the spacing between wires in the metal screen is correspondingly larger than the separation between atoms in the carbon crystal. The intensity of scattered electrons from uniform layers of atoms in a crystal (when the electrons have a suitable momentum so that their de Broglie wavelength is equal to the spacing between crystal planes) displays an identical interference pattern as is seen when X-rays, that also have a wavelength of the same size as the atomic spacing, are reflected from the same crystal. This interference pattern holds not only for reflected electrons but also for those passing through the thin crystal, as in Thompson’s experiments summarized in the February 1930 issue of Science Wonder Stories.
Figure 6: Cartoon sketch of de Broglie matter waves for electrons scattering from the planes of atoms in a crystal. If the separation between atomic planes in the solid is commensurate with the de Broglie wavelength of the electrons, then interference of the scattered electrons will be observed. The intensity of electrons will be high in directions where the matter waves constructively interfere and there will be no observed electrons in directions for which destructive interference occurs.
As in the case of photons, described in the previous chapter, this interference effect is not a result of large numbers of electrons behaving in a collective fashion like a wave. Consider the electrons passing through the crystal in Figure 7, detected by striking a chemically coated screen that emits a flash of light whenever an electron strikes it. You’re probably familiar with this—it’s an old-style cathode-ray television tube. (Modern flat-panel liquid crystal display models work differently.) Decreasing the current of the incoming beam of electrons striking the crystal, we can arrange it so that only one electron strikes the crystal every few seconds. We would then not see a full interference pattern, but a series of individual flashes of light on the TV detector screen. The more electrons we send in, the more flashes of light. If we recorded the location of each flash and at the end of the day added them all together, instead of a uniform coverage over the screen—as would be expected if the hard-sphere electrons collided with the electrons in the crystal’s atoms, sending them randomly in all directions—we see an interference pattern as in Figure 7.
Figure 7: Examples of light diffraction (a) and electron diffraction (b). The image on the left is obtained by passing a green laser light through a fine-mesh metal screen (not unlike a screen door) and shining the light on a wall several feet from the screen. The light scattering from the metal wires, arranged in a periodic array, leads to a symmetric constructive (bright-green spots) and destructive (dark regions) interference pattern. On the right an electron beam in a cathode ray tube passes through a graphite crystal. The momentum of the electrons is chosen so that their de Broglie wavelength is on the order of the spacing between atoms in the crystal. The atoms in the crystal scatter the electrons in a similar manner as the wire mesh does to the laser beam.
X-rays (more about these later as well) correspond to electromagnetic waves with a wavelength roughly equal to an atomic diameter, just as for the electrons we considered in the scattering experiment. The X-rays scatter from the electrons in the crystal’s atoms, though the mechanism is a little more involved than simple electron-electron repulsion. But we can decrease the intensity of the light, so that one X-ray photon strikes the crystal every few seconds as well. Here again, a detector screen will record distinct flashes of light, and when all the flashes are added together from the scatter of many photons, the observed interference pattern is identical to that found using electrons.14 The “dual character” symmetry between particles and waves holds for both matter and light. This, we will see, is truly the most amazing science story of the twentieth century.
CHAPTER FOUR
It’s All Done with Magnets
Everything—light and matter—has an “intrinsic angular
momentum,” or “spin,” that can have only discrete values.
After years of unsuccessful attempts to land a newspaper distribution deal for a comic strip featuring their creation Superman, Jerry Siegel and Joe Shuster eventually sold their story, along with the rights to the character, to the comic book publisher National Allied Periodical for $130—a nice sum in 1938, but of course a pittance compared to what the character would soon be worth. Debuting in Action Comics # 1 in June 1938, the Man of Tomorrow would soon be selling millions of comics per month and st
arring in live-action and animated movie shorts, an extremely popular radio show, and a syndicated newspaper comic strip.
No doubt one of the strong appeals of Siegel and Shuster’s comic-book creation is the fantasy that mild-mannered Clark Kent, dismissed and underestimated by all, is in reality the most powerful person on the planet. As Jules Feiffer argued in The Great Comic Book Heroes, Bruce Wayne must don his Batman costume in order to become the Caped Crusader, and Lamont Cranston his cloak, slouch hat, and red scarf to fight crime as the Shadow, but Superman is who he is. When he wakes up in the morning, he is Superman, and Clark Kent is the disguise he elects to wear. In essence, Kent is a representation of how Superman views us: weak, bumbling, inept. Superman is the iconic role model for those who feel that the world does not see their true, hidden essence.
One of the surprising discoveries of quantum mechanics, described in the principle at the start of this chapter, is that electrons, protons, and neutrons, the building blocks of atoms, also have a secret identity. While physicists in the 1920s knew them to be mild-mannered subatomic particles, characterized by their mass and electrical charge, it turns out that they would soon discover that the particles possessed a hidden characteristic, a superpower if you will, that is termed “spin.”
Figure 8: In the 1960s, Dick Tracy comic strips predicted a future in which we traveled via personal flying garbage cans levitated by the power of magnetism.
It was proposed in 1925 that every fundamental particle behaves as if it is a spinning top,15 rotating about an internal axis—and this holds not just for matter, but for photons as well. This rotation is not associated with the “orbital motion” of electrons around the nucleus in an atom (Schrödinger would eventually show that the picture of electrons circling around the positively charged nucleus, a neat analogy to the planets orbiting the sun in our solar system, is not technically accurate). This internal rotation is present even if the subatomic particles are in free space, not bound in an atom or molecule.
This built-in rotation is called “spin,” for it is as if the electron is rotating about an axis passing through the particle itself—similar to a twirling ballerina. The fact that all subatomic particles have an internal rotation turns out to be pretty important. Without accounting for the spin of electrons we cannot make sense of chemistry and solid-state physics. One characteristic of all atomic particles, associated with their internal spin, is that electrons, protons, and neutrons all have an internal magnetic field that has nothing to do with the magnetic field generated by an electrical current. It is through this magnetic field that this “power,” that is, spin, first revealed itself to the world.
Science fiction pulps often cited magnetism as the basis for a variety of technological wonders. Magnets were frequently called upon as a catch-all explanation for levitating heavier-than-air ships, while “reverse magnetism” was often invoked for force beams or other offensive weaponry. Readers of the daily newspaper’s comic-strip page have known since the mid-1960s that personal flying devices would someday be a reality, thanks to magnetism. Figure 8 shows a panel from a 1960s Dick Tracy comic strip, where Tracy, in silhouette, and his partner, Detective Sam Catchem, are able to scout for criminals using magnetized flying garbage cans. (Tracy is also carrying on a conversation using a “two-way wrist radio,” an early form of the cell phone.) Magnetism was expected to usher in the world of tomorrow—as the panel reproduced in Figure 8 promised, “The nation that controls magnetism will control the universe.”
Figure 9: Angular momentum was a frequently invoked physics principle for futuristic weapons of war, as shown in the cover of the April 1930 Air Wonder Stories.
Similarly, spinning is also a hallmark of the flying saucers and futuristic weapons (or sometimes both, as shown on the cover of the April 1930 issue of Air Wonder Stories in Figure 9). It would have to wait for the full relativistic form of quantum mechanics, developed by Paul Adrien Maurice Dirac in 1928, to show the fundamental connection between internal rotation and magnetism.
The third quantum principle states that everything, matter and photons, has an internal rotation about an axis that passes through the object, like a twirling figure skater. For ordinary matter, there is only one question about the rotation—clockwise or counterclockwise? In the previous chapter we discussed linear momentum defined as the product of an object’s mass and velocity. Since the objects in Chapter 3 were moving in straight lines, we could employ the linguistic shortcut and just refer to it as “momentum” rather than the more accurate term “linear momentum.” The greater an object’s momentum, the harder it is to change its motion. A baseball thrown at 100 miles per hour has more momentum than one thrown at 1 mile per hour; the latter may be arrested safely bare-handed, without a catcher’s mitt, while I wouldn’t recommend this method for the former (in fact, you’d need to stand pretty close to the pitcher in order to catch the slower ball before it fell to the ground).
Similarly, “angular momentum” is the rotational analog of “linear momentum.” The rotation may be about an axis passing through the object, as is seen in a spinning top, or about a distant axis, exemplified by the moon orbiting the Earth. In quantum physics, the spin of electrons or protons resembles a top or a ballerina more than it does an orbiting satellite. Moreover, the spinning of the particles within an atom is not arbitrary but must correspond to particular values of angular momentum. This is like saying that the linear momentum of a car can have two values, moving forward or backward at multiples of a given speed, such as 10 miles per hour. So the car could go 30 miles per hour forward or 30 miles per hour backward, but not, say, 13 miles per hour in either direction.
It turns out, based on experimental observation, that certain fundamental particles in the universe have an internal angular momentum that has a value of either 0 (a very special case) or Planck’s constant, h, divided by 2π. Photons, for example, have an intrinsic angular momentum of h/2π. Other fundamental particles, such as electrons, protons, and neutrons, can have an internal angular momentum of exactly one-half of this value of Planck’s constant, h, divided by 2π, that is, (1/2) × (h/2π). That’s it. Whether an object has an internal angular momentum that is either an integer multiplied by h/2π or a half-integer multiplied by h/2π will have a profound effect on how it interacts with other identical particles.
As 2π is just a number, if h/2π is a measure of angular momentum then Planck’s constant, h, is a unit of angular momentum. When Planck introduced the constant h as a fudge to account for the spectrum of light emitted by hot, glowing objects, he had hit upon a fundamental constant of the universe. There is a set of basic numbers that one must specify when setting up a universe, such as the mass of the electron and the speed of light. Things would look very different if the speed of light, for example, was a value a person could achieve while riding a bicycle, such as 15 miles per hour. One would then have an intimate, firsthand intuition about the consequences of the Special Theory of Relativity. Similarly, if Planck’s constant were a much larger number, we would have to deal with quantum phenomena in our daily lives.
In Isaac Asimov’s novel Fantastic Voyage II: Destination Brain, a team of scientists is reduced in size, smaller than a single cell, in order to travel within the body of an injured scientist (who has figured out a way to make miniaturization energy efficient!) and perform an operation. Asimov proposes that the mechanism underlying this shrinking process involves creating a field that reduces the magnitude of Planck’s constant. Considering an atom to be a sphere, Bohr calculated its radius to be a few times ro, where ro = h/ [(2π)mec α] and me is the mass of the electron, c is the speed of light, and α is termed “the fine structure constant” that involves another collection of fundamental constants (such as h, c, and the charge of the electron). If one could tune Planck’s constant at will, making it larger or smaller, then one could enlarge or shrink any object by changing the fundamental size of its atoms.16 The fact that we cannot do this in reality reflects the fact that fundamenta
l constants are just that—constant and unchanging.
It was unnerving to physicists when Albert Einstein suggested in 1905 that there was no velocity faster than light speed, but the universe and its laws indeed ensure that nothing can move faster than the speed of light in a vacuum. Apparently, with the discovery that subatomic particles have internal angular momentums whose values are multiples of either h/2π or 1/2 of h/2π but not any other values, the universe also cares about rotation.
The electron is the basic unit of negative charge, while the proton has an equal charge, but of an opposite sign (by convention the proton’s charge is termed “positive” while the electron’s is “negative”). It has been known since the 1820s that moving electric charges, that is, an electrical current, create a magnetic field. This is the basic physical principle underlying electromagnets and motors. If an electrically charged sphere rotates about a line passing through its center, like a wheel about its center spoke, then there are certainly electrical charges in motion, and these currents will generate a magnetic field. If there is an intrinsic angular momentum, it shouldn’t be surprising that as a consequence of this rotation every electron and proton has its own internal magnetic field. In fact, the quantized internal angular momentum aspect was proposed to account for the experimentally observed internal magnetic fields inside atoms. That is, the observation of the magnetic field came first, and later, in an attempt to account for it, the argument about intrinsic angular momentum was put forward.