Figure 31: Representation of the allowed quantum state solutions to the Schröedinger equation for an electron in an atom as a set of seats in a classroom. The Pauli principle indicates that each seat can accommodate two electrons provided they have opposite spins. Shown from left to right are the occupied quantum states for atoms containing one, two, three, six, and thirteen electrons, corresponding to hydrogen, helium, lithium, carbon, and aluminum, respectively.
Consider an atom such as aluminum, with thirteen electrons, shown in Figure 31e. All but one of these electrons are arranged in +ħ/2 and -ħ/2 pairs, and thus only this last electron can participate in chemical bonds. The other twelve electrons are chemically inert and form the inner core of the aluminum atoms. Not that we can’t make use of these inner electrons. The Pauli principle forces the electrons to reside in higher and higher energy states, equivalent to having some students sit in rows far from the front of the classroom, even when the atom is in its lowest energy configuration. If we could knock one of the electrons out of the ground state (a row close to the front of the lecture hall), then an empty position suddenly would open up, as if we had ejected a student sitting in a front-row seat. A student sitting in one of the upper rows could then jump down into the newly vacant seat. Just such a situation can arise when a high-energy beam of electrons strikes an atom. In that case, when one of the outer electrons falls down to occupy the lower energy state, it can emit an X-ray photon during the transition. This is in fact a very efficient way to generate X-rays, and most dental X-ray machines employ electron currents striking a copper target to create the penetrating radiation.
How do the last few electrons that are not residing in paired quantum states, and are thus available to participate in chemical bonds, combine with those from neighboring aluminum atoms to hold all trillion trillion atoms together in a solid piece of aluminum? How do the last unpaired electrons between carbon atoms in diamond combine to bind this rigid insulator? In both cases, the electrons arrange themselves to satisfy the Pauli exclusion principle, though the resulting material properties in aluminum and diamond could not be more different.
One easy way to satisfy the Pauli principle is to never let the electrons be at the same place at the same time. If I have a line of atoms, and next to each atom is a barrier, then I can place an electron inside each theoretical box (we’ll see soon what this “box” really is), and all these negative charges can be in the same quantum state. This does not cause any problem, for by creating a series of containers for each electron, I have in principle made them distinguishable. I can tell which electron is in the box on the right and which on the left, just as I could tell apart the two stones that I tossed into the pond. The walls of the boxes prevent the de Broglie waves of each electron from overlapping with those of its neighbors, so the trillion trillion electrons can all be in the same quantum state, as the total wave function is just the one electron function repeated a trillion trillion times. Each electron is described by its own ribbon, as shown in Figure 30, and no ribbon is used for more than one electron. When I calculate the average energy of each electron in a box, assuming the width of the box is the spacing between atoms in my solid, I arrive at a number of about three electron Volts (the exact value obviously depends on all sorts of details of how the atoms in the solid are arranged—termed the crystalline configuration). This is the energy I would have to give to an electron to remove it from the box. Of course, as each electron has two possible values of spin, I can actually put two electrons in each box (each box contains a love seat), as long as they have intrinsic angular momentum +ħ/2 and -ħ/2.
Consider carbon, shown in Figure 31d. Carbon can easily “rearrange the seats in the rows” of the four upper electrons, mixing the quantum mechanical wave functions to form differing configurations of quantum states that allow for a variety of chemical interactions. Carbon can form strong bonds in a straight line, in proteins and DNA; it can form graphite, with three strong bonds in a plane and one weak bond above or below the plane, which is why graphite can be easily peeled apart when used in a pencil, for example; and when the “seats” are configured to form four equally strong bonds, we call this form of carbon “diamond.” In each case, the carbon atom has four electrons that are capable of participating in chemical bonds, represented by four boxes, each of which holds one electron. The Pauli principle tells us that each box can hold a second electron, provided it has an opposite spin from the first. When two carbon atoms come close enough to each other that the quantum states containing these unpaired electrons overlap, the two electrons can be represented by a two-electron wave function. It turns out that each of the unpaired electrons can lower its energy if the electrons fill up the love seats in each box (Figure 32a). That is, I must add energy to the atoms to remove the electrons from these boxes, and restore each one to its unpaired state. The overlapping electron wave functions form a chemical bond between the atoms, holding them together in the crystalline solid. The Pauli principle is satisfied by the localization of the electrons in space. In a diamond crystal, each carbon is surrounded by four other carbon atoms in a tetrahedral arrangement, and their unpaired electrons can occupy the second seat in the first atom’s boxes (Figure 32b).
But there is another way to satisfy the Pauli exclusion principle. Let’s say that there are no boxes, and I let the electrons wander over the entire solid. In this case two electrons can be at the same place at the same time, so I have to ensure that they are each in different quantum states. How many different lowest-energy quantum levels are available for the last unpaired electron, such as the thirteenth electron for aluminum, shown in Figure 31e? As many as there are atoms in the solid. Since I have given up having any knowledge of where the electrons may be, I can compensate by having the electrons reside in states that have a well-defined momentum. I can have a matter wave with a very large wavelength, equal to the entire length of the solid. Since (momentum) × (wavelength) = Planck’s constant, this large wavelength corresponds to a very small momentum, and hence energy. At the other extreme, the smallest wavelength that can be constructed corresponds to the distance between atoms in the solid. This is very short, so the momentum of this matter-wave is high. The highest energy of this shortest wavelength is also about three electron Volts, again, depending on the details of the atomic configurations in the solid.
Figure 32: Sketch of the lowering in energy when two unpaired electrons from adjacent carbon atoms overlap and form a carbon-carbon bond (a). Also shown is a sketch of the configuration of carbon when in the diamond configuration, allowing four chemical binds with its neighbors, in a tetrahedral orientation (b).
So, whether the electrons are put inside boxes in each atom, or allowed to roam over the solid, we still wind up with an energy of about three electron Volts. However, in the second situation, where the electrons can move around the solid in discrete momentum states, three electron Volts is approximately the energy of the most energetic electron, while when the electrons are placed in boxes, three electron Volts is roughly the energy of each electron. The average energy of an electron in the free-to-roam case is less than three electron Volts, and in fact will be closer to 1.5 electron Volts (recall the class from our discussion of the Heisenberg uncertainty principle, where every student had a different exam score, from 0 to 100. In this case the average grade was 50 percent). For the electrons-in-a-box situation, every electron has the same energy, so the average energy is also three electron Volts (if every student scores a perfect 100 percent, then the class average is also 100 percent). Consequently, depending on its chemical composition, the solid as a whole can lower its energy by letting the electrons wander around the crystal. This won’t be true for all solids. Some materials will be able to lower their total energy by keeping every electron localized in boxes around each atom. We call the free-to-roam cases “metals,” and the electrons-in-a-box materials “insulators.”
And that’s how quantum mechanics explains solid-state physics. At very low tempera
tures, all solids either conduct electricity or they don’t. We call the first case metals, and the second are insulators (the distinction between insulators and semiconductors is most relevant around room temperature, and I defer for now a discussion of the differences between the two).
Metals such as aluminum are good conductors of electricity because the outermost electrons satisfy the Pauli principle by residing in momentum states and are free to move around the entire solid, while insulators keep theirs in boxes (bonds) around each atom. To remove a metal atom from the solid, I must first grab one of the free-range electrons and localize it on a positively charged atom—in essence, put it in a box so that I can pull the neutral atom out of the solid. But this costs me a few electron Volts of energy, and this can be considered the binding energy holding the atoms together in the metal. There are no directional bonds between atoms, so it is easy to move atoms past each other, which is why metals are easy to pull into wires or pound into thin sheets, without losing their structural coherence. If light is absorbed by the solid, there is always a free electron that can absorb its energy and reemit it back again, which is why metals are reflective and shiny. The sea of free electrons makes metals good conductors of both electrical current and heat.
Insulators, on the other hand, such as diamond, have all the electrons tied into bonds between the atoms (the two electrons per box as discussed earlier). They are thus poor conductors of electricity and can conduct heat only by atomic vibrations (sound waves). The electrons in the box can assume only specific energy states, like electrons in atoms. Consequently, if you shine light on an insulator that does not correspond to an allowed transition, it will ignore the photons. This is why some insulators, such as diamond or window glass, are transparent to visible light. The details of the materials’ properties are very sensitive to the configurations of the boxes in which the electrons reside, which is why changes in crystal structure—say, when carbon transforms from graphite to diamond—can yield big variations in optical and electrical properties. The boxes here are the directional, rigid chemical bonds between the atoms (Figure 32b), and changes in the type of these bonds and the chemical constituents lead to big variations in crystal structure and rigidity (diamond is very stiff, while graphite is so soft, we use it for pencil lead). All the differences between insulators and metals can be understood at the most basic level by whether the last few unpaired electrons of the atoms in the solid satisfy the Pauli exclusion principle by localizing themselves in real space (insulators) or in momentum space (metals).
Thus, from playing with a ribbon, with one side black and the other side white, we see why the world is the way it is. Note that not everything has an intrinsic angular momentum equal to ħ/2. Some objects, such as helium atoms or photons, have spin values of either 0 or ħ. This seemingly small difference leads to superconductors and lasers.
CHAPTER THIRTEEN
All for One and One for All
Bert Hölldobler and Edward O. Wilson, in The Superorganism—The Beauty, Elegance, and Strangeness of Insect Societies, propose that colonies of wasps, ants, bees, or termites can be considered as a single animal. They argue that each insect is a “cell” in the “superorganism”: foragers are the eyes and sense organs, the colony defenders act as the immune system, and the queen serves as the colony’s genitalia.50 An important difference between a superorganism and a regular animal is that the colony lacks a centralized brain or nervous system. Rather, each colony has its own rules for local interactions among the insects that govern its organization and size. In this way the colony can achieve levels of development that are far beyond the capabilities of the individual insects were they to act alone. As readers of science fiction pulps know, this is the mechanism (or at least one of them) by which humans defeat an alien invasion.
In Theodore Sturgeon’s 1958 science fiction novel The Cosmic Rape (an abridged version was simultaneously published in Galaxy magazine with the title To Marry Medusa), an alien intelligence applies an unconventional approach to its efforts to conquer Earth. (Spoiler alert!) The alien, in fact a cosmic spore, is capable of controlling the intelligence of a single human. It is surprised to discover that the people of Earth are not in mental contact with one another, given that the planet is covered with complex structures such as buildings, bridges, and roads. In the spore’s experiences conquering other planets that contained advanced infrastructure, such architecture is possible only when the primitive intelligences of the individual agents interact in a cooperative manner, as in a colony of ants or a hive of honeybees. The invading spore had never encountered a species for which a single agent is capable of designing a bridge or building on is or her own, and it thus assumes that a previously existing collective connection has been severed. The spore sets upon a plan to reestablish this connection and force all humans to think and work together in unison. Unfortunately for the alien entity, it succeeds in its plan. Once it finds itself dealing not with the intelligence of a single human, but with the collective consciousness of all several billion humans, the “hive mind” of humanity quickly devises an effective counterattack, destroying the alien spore. In this way Sturgeon has described the cooperative behavior of a Bose-Einstein condensate.
In Sturgeon’s novel, the alien spore creates from humanity a macroscopic quantum state, with a single wave function containing all the information about its constituent elements. Any change in one element, in Sturgeon’s case a human being, is instantly transmitted to every other element in the wave function, that is, the rest of humanity. Such situations occur frequently in the real world through quantum interactions between pairs of electrons in a superconductor or helium atoms in a superfluid. These collective states involve particles whose intrinsic angular momenta are multiples of ħ, rather than ħ/2. Particles with intrinsic angular momenta that are whole-number multiples of ħ are called bosons, as they obey a form of quantum statistics elaborated by Satyendra Bose and Albert Einstein, termed Bose-Einstein statistics.
In the previous chapter we discussed fermions, for which the angular momentum could be either +ħ/2 or -ħ/2, but not any other value. This is intrinsically asymmetric, as we can distinguish a top twirling clockwise from one rotating counterclockwise. We represented this situation, when the two fermions are so close that their wave functions overlap, with a ribbon with one side black and the other white. The significance of the two colors was that we could readily distinguish the spin = +ħ/2 electron from the spin = -ħ/2 electron, as we could the black and white sides of the ribbon. But experiments have revealed situations where the intrinsic angular momentum can have values of 0 or ħ or 2ħ, and so on, but not any fractional values. Let’s consider the case of quantum objects with a spin of zero first, and then turn to spin = ħ particles such as photons.
As the fundamental building blocks of atoms—electrons, protons, and neutrons—are all fermions, what sort of object would have spin of zero? One example is a helium atom. A helium nucleus has two protons and two neutrons, each having spin = +ħ/2 or -ħ/2. As the two protons are identical, in their lowest energy state in the nucleus they would pair up, +ħ/2 and -ħ/2, for a total spin of zero, as would the two identical neutrons. Similarly, the two electrons are spin paired, as indicated in the sketch in Figure 31b. Consequently, the total intrinsic angular momentum of a helium atom, when in its lowest energy configuration, has a spin value of zero.
Particles with zero value of spin are symmetric, in that we cannot describe the rotations as clockwise or counterclockwise. When two such particles are brought so close that their wave functions overlap, we will represent them by a ribbon whose sides are both white. I once again stress that the ribbon is employed as a metaphor for the resulting two-particle wave function, and as such, certain issues are being ignored here that would only distract from our discussion.
Let’s repeat the experiment with the ribbon from Chapter 12, only now using a ribbon with both sides the same color, for example, white (Figure 33). I can hold each end, and obviously a wh
ite side of the ribbon faces out (Figure 33a). Now, without letting go of either end, I will switch their positions, as before. The end of the ribbon on the left is now on the right, and vice versa. This procedure has introduced a half twist in the ribbon, as in Chapter 12 (Figure 33b). Of course, now both sides are still white. I can undo the half twist by flipping one end of the ribbon around, so that the back side turns outward (Figure 33c). When the ribbon had one side white and the other side black, this was a forbidden operation, as it changed the state of the ribbon (where before both ends had white facing out, one side would then have had a black side facing out). But if both sides of the ribbon are white, then this symmetry means I can flip one end of the ribbon and I have not changed anything except undoing the half twist. The important point is that a white ribbon can be restored to its original state following a single rotation, while the black/white ribbon requires two rotations to bring the original configuration back.
The Amazing Story of Quantum Mechanics Page 16
