Chapter 7
Solid Objects: The Energy of Uncertainty
I slide a couple of slices of bread into the toaster oven, jiggling the rack when it sticks a bit, and lean against the counter while I wait …
Almost 2,500 years ago, the philosopher Zeno of Elea published a large number of paradoxes, attempting to demonstrate the absurdity of “common sense” ideas about reality. One of the most famous purports to show that motion is an illusion, because all motion should take infinite time. In order to walk across the room, for example, I first must walk halfway across the room, which takes some amount of time. Then I need to walk half of the remaining half (that is, to the three-quarters mark), which also takes some amount of time. And then I need to walk half of the remaining distance (that is, from three-quarters to seven-eighths), which takes some time as well. This process can continue forever, leading to the conclusion that moving any distance at all requires an infinite number of steps, each taking finite time. Which suggests that it should take an infinite amount of time to move anywhere, and thus motion is impossible.
Most people react to this in more or less the same way as that attributed to Diogenes the Cynic: by standing up and walking away from the philosopher. Motion is such an obvious fact of our everyday existence that declaring it impossible seems ridiculous. More mathematically inclined thinkers address the paradox by pointing out that each time you halve the distance, you also halve the time needed to cross it. With the invention of calculus, we know that an infinite number of successively smaller terms can add up to a finite total (and more specifically, that the sum involved in the setup of this paradox is ½ + ¼ + ⅛ + … = 1). Philosophers, however, have continued to argue about subtle points of Zeno’s arguments to the present day.
The solidity of objects is another inescapable fact of our existence, something we experience every time we place one object atop another. Questioning the stability of solid matter, then, might seem like the sort of thing best left to philosophers or those who have had a bit too much of some controlled substance. Yet it turns out to be extremely difficult to prove, based on the principles of physics, that solid objects consisting of large numbers of interacting particles are actually stable.
The problem is most easily described in terms of energy: as a general matter, any physical system always tries to reduce its energy, so in order for a large number of particles making up a solid to be stable, there must be some minimum energy arrangement from which no further energy can be extracted. As we saw when we talked about the Bohr model, though, the attraction between positive and negative charges gives them a negative potential energy that dives to negative infinity when the two are right on top of each other. That infinite value suggests that any collection of particles can, in principle, lower its energy by packing all the components more closely together. It’s not immediately obvious, then, that the attractive interactions that pull particles together to make atoms, molecules, and solids can’t, under the right circumstances, pull all those particles into an infinitesimally small space, causing would-be solids to implode—and releasing an enormous amount of energy in the process. A slice of bread waiting to be toasted, in this view, is a potential atomic bomb.
Preventing this implosion to allow for the existence of solid objects requires some additional factor that increases the energy as particles pack more tightly. This would ensure a minimum energy at some size in much the same way that balancing the electromagnetic force against the force needed to hold an electron in a small orbit let Bohr find an optimum radius for the lowest-energy orbit in his atomic model. In the end, this energy comes from two core quantum ideas we’ve already discussed, specifically the wave nature of matter and the Pauli exclusion principle. To explain how, we’ll have to introduce one of the most famous consequences of quantum physics, something no book on the subject can get away without mentioning: the Heisenberg uncertainty principle.
The Certainty of Uncertainty
In the mid-1920s, confronted with the inability of the Bohr-Sommerfeld model to correctly describe some seemingly simple systems—like the ionized hydrogen molecule that Wolfgang Pauli struggled with during his PhD—a number of (mostly) young physicists began to abandon the semiclassical underpinnings of the “old quantum theory.” The first breakthrough came when Werner Heisenberg decided that the key to the problem was to dispense with the idea of well-defined electron orbits altogether.
Heisenberg was of the same generation as Pauli (who was a year older), and like Pauli studied under Arnold Sommerfeld in Munich. His thesis research was on the classical physics of turbulence, but like many physicists of the day, he developed an interest in the emerging quantum theory. After he completed his PhD, he moved to Göttingen to work with Max Born, and spent the winter of 1924–25 at Niels Bohr’s institute in Copenhagen. While in Denmark with Bohr, he tried to use “old quantum theory” to explain the intensity of spectral lines—that is, why atoms emit and absorb more readily at some of their characteristic frequencies than others. Einstein’s statistical model of light from Chapter 3 gave some very general rules for these, but getting the details right turned out to be exceedingly difficult.
In the summer of 1925, Heisenberg returned to Göttingen and continued to struggle with the problem of spectral lines. At the same time, he was also struggling with allergies, and to escape an intense bout of hay fever, fled to the remote island of Heligoland, where he could breathe pollen-free air and concentrate on his work. While there, he had an epiphany, realizing that it was a waste of time to try to figure out details of the classical orbits followed by electrons. No conceivable experiment could hope to track the motion of the electron in its orbit, so there was no point in worrying about the finer points of that motion. Instead, he looked for a way to formulate quantum theory solely in terms of experimentally observable quantities.
After an extended period of furious mathematical effort, Heisenberg found the answer he was seeking. This took the form of laboriously calculated tables of numbers describing the measurable properties of quantum jumps; these values were sorted into rows and columns based on the particular pairs of initial and final electron states involved. As with Schrödinger’s wave equation in the previous chapter, Heisenberg’s theory deals in probabilities—but in terms of allowed states, rather than particular positions for the electron. The problem of the intensity of spectral lines that motivated Heisenberg was a matter of determining the probability of an electron in one allowed state making a jump to another; the higher the probability, the brighter the line. Making these calculations involved combining results from his tables of numbers, according to rules that he slowly worked out.
On returning to Göttingen, Heisenberg showed his work to Born, who noticed a similarity between Heisenberg’s calculations and the matrices—indexed tables of numbers with special rules governing their manipulation in calculations—studied by colleagues in mathematics. Born and Heisenberg and another of Born’s assistants, Pascual Jordan, reformulated Heisenberg’s results in the language of matrices, leading to the first relatively complete theory of quantum physics: “matrix mechanics.”1
The initial reception of matrix mechanics was not all that enthusiastic, as physicists of the day were mostly not trained in the mathematics of matrices; when Erwin Schrödinger found his wave equation the following winter,2 many physicists reacted with relief. The two approaches are mathematically equivalent, though, and these days physicists learn a mix of both. The wavefunctions calculated with the Schrödinger equation are regularly described using mathematical terms borrowed from matrix mechanics, while Heisenberg’s insights are often described in terms of waves when that offers a more intuitive way of understanding what’s going on. Actual calculations are done using whichever approach is easiest for the problem at hand.
Heisenberg is best known outside of the physics community for his uncertainty principle, which is one of the ideas from quantum physics that has escaped into popular culture. The most famous formulation of this says that it is impossible to know both the position and the momentum of a particle to arbitrary precision. Both of these quantities are necessarily uncertain, and the product of their uncertainties must be greater than some minimum value—in other words, when uncertainty in one decreases, the uncertainty in the other must increase by at least the same factor. If you know exactly how fast something is moving, you lose your ability to know where it is, and vice versa.
The uncertainty principle is often described as a measurement phenomenon, with the act of attempting to measure the position disturbing the momentum, and vice versa. While this does get to the right basic relationship, it’s slightly misleading in that it leaves the impression that there is a “real” position and momentum associated with a quantum particle, and we just don’t know what it is. Quantum uncertainty is more fundamental than that, though. One of the words Heisenberg used when working out his theory is arguably better translated as “indeterminacy” than “uncertainty,” which offers a more useful way of thinking about the issue. Quantum indeterminacy follows directly from Heisenberg’s original epiphany on Heligoland: the idea of formulating the theory only in terms of measurable quantities implies the existence of other quantities that cannot be determined. The uncertainty principle isn’t about deficiencies in measurement—it reflects the fact that it simply does not make sense to speak of a well-defined position or momentum for a quantum particle.
To understand this indeterminacy, though, and how it helps produce the energy we need to keep our breakfasts from imploding, we must first circle back and make use of Schrödinger’s wave picture. We need to look more closely at what it means for a particle to behave like a wave, and vice versa.
Zero-Point Energy
The Heisenberg uncertainty principle is the best known of the weird consequences of quantum physics, but to explain it, we need to look at another phenomenon that’s just as deeply counter to our intuition. That idea is “zero-point energy,” which tells us that a confined quantum particle is never not in motion, and it follows directly from the wave nature of quantum particles. This will turn out to have profound consequences for the stability of matter.
To gain some insight into wave nature and zero-point energy, it’s useful to go back to the simplest system in quantum physics: a single particle confined to a box.3 The basic idea is the same as the “stuff in a box” model we used to set up the black-body radiation problem, but in that case we were considering light waves. Now, armed with de Broglie’s idea of matter waves, we want to consider a material particle like an electron confined to a box. Our hypothetical “box” is impenetrable, so that while the electron can move about the interior freely, it can never escape.
Though the scenarios may seem quite different from the standpoint of everyday intuition, there’s very little mathematical difference between a light wave confined to a reflecting box and an electron with wave nature held in an impenetrable box. In both cases, the end result is a limited set of standing-wave modes, with the waves constrained to be zero at the ends of the box, and an integer number of half wavelengths fitting across the box’s length. Just as with the light waves, the longest possible wavelength for an electron inside the box is twice the length of the box.
When applied to light waves, this constraint didn’t seem problematic, but it has a very unusual implication when applied to an electron: namely, that the electron can never be truly at rest within the box. As de Broglie showed, the wavelength of an electron is related to its momentum—higher momentum means a shorter wavelength. Momentum is calculated by multiplying mass times velocity,4 and since the mass of the electron is fixed, the electron’s momentum is a reflection of its velocity. An electron at rest would have zero momentum, which would require an infinitely long wavelength. But a confined electron has a maximum possible wavelength—twice the length of the box—which means it has a minimum momentum that is not zero. Thus an electron confined to some region of space must always be moving.
In physics, velocity is a quantity that includes both a magnitude (speed) and a direction, and as a result momentum is also defined in terms of direction. Since an electron in a box can be moving in any direction, though, this makes it tricky to talk about confined particles in terms of momentum. To avoid the direction problem, it’s easier to discuss the confined electron in terms of its kinetic energy, which does not depend on where the particle is headed, only on how fast it’s moving. The standing-wave modes for an electron trapped in a box are states with well-defined kinetic energy, with the energy increasing proportionally to the square of the number of half wavelengths in the box—that is, the second state has four times the energy of the first, the third state nine times the energy of the first, and so on.
The critical feature here is that the lowest energy is not zero. This seems like a strange thing to say, from the perspective of classical physics—if I place a macroscopic everyday object like a marble inside a shoebox, I can perfectly well arrange for that object not to move at all relative to the box, and thus have zero kinetic energy. A quantum particle, on the other hand, can never be perfectly still, thanks to its wave nature. This minimum energy—zero-point energy—is unfortunately a rich source of material for scams. People with a little knowledge of quantum terminology and no scruples will sometimes pitch “free energy” schemes based on the idea of extracting this zero-point energy from empty space. As usual with promises that sound too good to be true, this is impossible: the zero-point energy is simply an inevitable consequence of the wave nature of matter and can never be extracted.
The electron’s minimum energy is set by the size of the box, and it depends on the inverse square of the length—that is, if you double the length of the box, the minimum energy of this larger box will be one-quarter that of the smaller. The more tightly you confine a particle, the shorter its maximum wavelength gets—and the higher its energy. This increase in energy is one of the crucial elements we need to understand the stability of matter.
The Uncertainty Principle
The wave nature of matter, then, ensures that a confined particle has some minimum energy, but it’s probably not obvious how this relates to the uncertainty principle. Why does the wave nature of matter make it impossible to know the position and momentum of a single particle at a given time?
The key idea is hiding a few paragraphs back, where we switched to talking about the energy of states. The standing-wave states of a confined electron are states with a definite energy but an indefinite momentum, because, as mentioned, the momentum includes not only the speed of the particle’s motion, but also its direction. The simplest version of our “particle in a box” hypothetical is a one-dimensional “box”—somewhat like a string, in that the electron is able to move in only two directions. An electron confined to a one-dimensional box is equally likely to be moving either to the left or to the right, giving it an uncertainty in momentum. For a one-dimensional system, we encode the direction of motion into the sign of the particle, so a leftward-moving particle has negative momentum and a rightward-moving particle has positive momentum. The spread in momentum, then, is twice the momentum associated with the fundamental wavelength of an electron. We can express this as an average momentum with some uncertainty: for instance, if the momentum can be either 5 or −5 units, the range of momentum would be 10 units, and we would say that the average momentum is 0, plus or minus 5 units.
More tightly confined particles must have shorter wavelengths, and thus greater momentum and energy, so we can reduce the momentum, and in turn the momentum uncertainty, by increasing the size of the box. When we do that, though, we necessarily increase the uncertainty of the position of the particle, which is something like half the size of the box—on average, the particle is in the middle, and could be up to half the length away in either direction.5 The product of these two uncertainties, though, is a constant: if we double the length of the box, we double the position uncertainty but cut the momentum uncertainty in half, so position uncertainty multiplied by momentum uncertainty remains unchanged.
So both the position and the momentum of a particle in a box must be uncertain in the way that the Heisenberg uncertainty principle demands. It may not be as obvious, though, that this applies to the case of a particle outside a box, one that’s free to move around as it pleases. To understand that, we need to think through what it means for a quantum object to have both particle and wave nature, and what we’re asking for when we try to define both its position and momentum.
In order to talk about a quantum particle—that is, a particle with wave nature—having a well-defined momentum, we need to be able to specify its wavelength, which necessarily means it must extend over enough space for us to see it oscillate. But this is not compatible with having a perfectly well-defined position. The best compromise we can make is to have something like a “wave packet,” a function where you have wavelike behavior in only a small region of space, as shown in the illustration.
A wave packet, with an obvious oscillation only in a small region of space.
This function clearly has both particle and wave characteristics, but how would we make such a thing out of ordinary waves? We can take a cue from the case of the particle in a box, where the lowest energy state is the sum of two different waves, one corresponding to the particle moving to the left, and the other to the right. Rather than having the particle moving at the same speed in two directions, though, let’s look at what happens when we add together waves corresponding to two different possible speeds. In that case, we end up with a wavefunction that looks something like the following illustration.
Adding two waves of slightly different frequency gives a wavefunction with beat notes where the two waves cancel out. We square this to get the probability distribution at the bottom.
When two waves with different wavelengths are added together, there are places where the two line up in phase and combine to produce large waves, but as the waves move along, they get out of phase with one another. Some distance away, there will be a point where they cancel each other out almost perfectly, leading to no waves at all. This is referred to as “beating,” because it’s a familiar phenomenon in music, leading to a discordant ringing noise when two slightly out-of-tune instruments try to play the exact same note.
With only two waves added together, we end up with only narrow regions of no waving, but if we add more waves, the regions where the waves cancel get broader, and the regions where there are waves become narrower and more well-defined. The more wavelengths you include, the more the resulting wavefunction resembles a wave packet that describes a particle. Each additional wavelength added, though, corresponds to a possible momentum. As you add wavelengths, you introduce a probability of finding the particle at each particular momentum; you get a smaller wave packet with a more well-defined position, but this process necessarily increases the uncertainty of the particle’s momentum.
Adding (bottom to top) one, two, three, and five wavelengths to make successively narrower wave packets.
This is why quantum uncertainty is perhaps better described as “indeterminacy”: the tension between particle and wave properties means that it’s simply impossible to define both the position and the momentum of a particle at the same time. Making a narrow wave packet to better define the position necessarily means adding wavelengths and increasing the uncertainty in momentum. On the other hand, reducing the number of possible wavelengths to better define the momentum will necessarily lead to a wider wave packet, with a greater uncertainty in position. Quantum uncertainty is not a practical limit on our ability to measure things, but a fundamental limit on what sorts of properties a quantum particle can have.
The Stability of Atoms
So, how do zero-point energy and the uncertainty principle help ensure the stability of matter? To understand this, we need to discard the simple but rather artificial particle-in-a-box model in favor of the more realistic situation of an electron bound to the nucleus of an atom.
An electron bound to an atom is clearly a more complicated situation than an electron confined to a box, but similar considerations come into play. A bound electron, more or less by definition, is restricted to a small region of space around the nucleus, and just as in the case of the electron in the box, the size of that region determines a minimum kinetic energy that the electron must have.
The case of an atom, however, is complicated by the attractive interaction between the negatively charged electron and the positively charged nucleus. The convention in physics is to describe such an interaction in terms of a bound electron’s negative potential energy, which adds with the positive kinetic energy to establish the total energy of the particle. As mentioned earlier, this gives us a simple way to determine whether an electron is bound or not—bound electrons have negative total energy. (This is why the electron energy in a Bohr orbit that we described in Chapter 4 is a negative number.) The law of conservation of energy tells us that this total energy is a constant, with kinetic energy increasing as potential decreases, and vice versa, to keep the sum the same.
While the potential energy of a bound electron is always negative, this energy varies with position. At large distances between the electron and nucleus, it’s nearly zero, and as they come closer together, it becomes more and more negative. Mathematically, the magnitude of this negative potential energy increases without limit—placing the electron exactly on top of the nucleus should result in a potential energy of negative infinity. This is what raises the uncomfortable prospect that the atom might be unstable against implosion—that is, that the electron could always lower its total energy by moving closer to the nucleus.
Happily, it’s not very difficult to show mathematically that the increase in kinetic energy that comes from confining the electron more tightly is enough to counter the increase in negative potential energy. In fact, this kinetic energy posed a significant historical problem for nuclear physics—atomic masses are always greater than the number of protons inferred from the charge of the nucleus, so prior to the discovery of the neutron, physicists assumed that the nucleus must contain some number of additional protons with tightly bound “nuclear electrons” to cancel their positive charge. The kinetic energy of a confined electron becomes so enormous, though, that it’s impossible to hold one inside the space of an atomic nucleus with the interactions known to physics. The “nuclear electron” model never worked all that well, and Ernest Rutherford, among others, believed for many years that the nucleus must also contain a heavy neutral particle. When Rutherford’s colleague James Chadwick demonstrated the existence of the neutron in 1932, following up on hints in a paper by Frédéric and Iréne Joliot-Curie, many physicists were grateful to be done with “nuclear electrons.”
Thanks to the increasing kinetic energy that comes from restricting the electron to a smaller space, the total energy of an electron orbiting a nucleus has a lower limit. The energy of the electron is negative, indicating that it is bound, but it can never be made infinitely negative, so its wavefunction must always extend over some range about the nucleus. An atom consisting of an electron bound to a positively charged nucleus is stable, and will not implode.
Pauli Exclusion and Solid Matter
The wave nature of matter is enough to guarantee the stability of atoms, so it might seem the philosophical question regarding the existence of macroscopic objects is settled. The fact that a single nucleus orbited by a single electron is stable, however, does not necessarily mean that a large collection of nuclei and electrons will be. The single-atom calculation is simple enough to be a homework problem for undergraduate physics students, but once you add even a third charged particle, it becomes impossible to perform an exact calculation of the energy with pencil and paper; only approximate solutions and numerical simulations are possible.
This is not a problem that’s unique to quantum physics. The classical “three-body problem” is similarly intractable, and it was troubling people long before Planck’s introduction of the concept of energy quanta. The problem of multiple interacting objects first became a serious concern when Isaac Newton introduced his law of universal gravitation in the late 1600s, and used it to explain the orbits of the planets in the solar system. The basic properties of these orbits can be determined by considering the interaction between a given planet and the sun—but, of course, there are also the gravitational forces between the planets to consider. These are much smaller, but not insignificant: in 1846, the French astronomer Urbain Le Verrier used a minute deviation between the predicted and observed orbits of the planet Uranus to infer the presence of another planet orbiting even farther from the sun. Le Verrier predicted the location of this new planet using approximate calculations with Newtonian gravity, and the German astronomer Johann Galle found the planet Neptune in almost exactly that spot on his first night of observing after receiving Le Verrier’s prediction.
Despite the success of approximate orbital calculations like Le Verrier’s, the lack of a definite solution to the three-body (or more) problem remained a headache, with troubling implications for human existence. While the forces between individual planets are quite small compared to the gravitational attraction of the sun, if they align in just the wrong way, they could conceivably destabilize the orbits of the planets, flinging the Earth into the sun, or out into the depths of interstellar space. In the absence of a definite solution to the many-body problem, there’s no guarantee that the solar system will continue to exist in its current configuration.
In 1887, in an attempt to settle the issue, the King of Sweden declared an international competition, with a prize for any mathematician who could find a solution to the many-body problem. This prize eventually went to Henri Poincaré, who invented an array of new analytical techniques for classifying the orbits of three or more objects interacting via gravity. Unfortunately, Poincaré’s solution was a negative one—his new techniques showed that, in fact, there is no guarantee that a system of many interacting objects will fall into, and continue in, regular orbits.6 Poincaré’s work is an early landmark in the mathematical study of chaos, and the techniques he invented are still among the standard tools for studying systems that are fundamentally unpredictable despite having relatively simple underlying physics. The long-term stability of the solar system remains in doubt, and thanks to Poincaré we know that this situation can never be resolved.
The situation of many interacting quantum particles is even more complicated than the many-body gravitational problem addressed by Poincaré: the electromagnetic force between charges has the same mathematical form as the gravitational force, which already makes stable orbits impossible, and on top of that it depends on the positions of the interacting particles, which we just showed can’t be defined. It’s impossible to find a pencil-and-paper solution for the allowed states of even a helium atom, with a single nucleus and two interacting electrons. It’s conceivable that some particularly unfavorable arrangement of a large number of nuclei and electrons might be fundamentally unstable. The complex interactions between such a system could end up flinging some particles out to very large distances, while the rest implode to an infinitesimally small point. Which again raises the disturbing possibility of a slice of toast imploding and releasing energy like an atomic bomb.
As with Zeno’s paradoxes of motion, of course, the ultimate answer is obvious: the fact is, we’re surrounded by enormous amounts of matter, in a variety of configurations, and it certainly appears to be stable. Demonstrating this mathematically, however, turns out to be a ferociously difficult problem. It was finally solved in 1967 by Freeman Dyson, who showed that there’s a lower limit on the total energy of a collection of electrons and nuclei, ruling out the possibility of implosion. Provided, that is, that the particles involved are subject to the Pauli exclusion principle.
It may not be obvious that the Pauli exclusion principle has anything to do with the energy of confined electrons, but we can see how it works out by looking in more detail at the mathematics of the situation. Pauli’s principle is, at a deeper level, a reflection of the fact that electrons are perfectly identical and cannot be distinguished from each other. This means that any labels we place on them for mathematical convenience—calling one A, the next B, and so on, or designating one direction in space as positive and the other negative—are arbitrary. The measurable properties of the many-electron state—including its total energy—cannot change if we swap the labels around. One unmeasurable property can change, though, and in fact is required to: the wavefunction must be “antisymmetric,” meaning that when you swap around the labels, it has to change sign from positive to negative. This is the formal mathematical requirement that leads to Pauli exclusion: a wavefunction with two electrons in precisely the same state cannot possibly change sign when you swap the labels, and thus such a state is forbidden.
The sign of the wavefunction doesn’t affect the energy directly—the wavefunction, remember, relies on the imaginary number i, so measurable properties can depend only on the square of the wavefunction—but this requirement restricts electrons to states that, in general, have a higher energy. We can see how the antisymmetry requirement leads to higher electron energies by considering a simple system that gives rise to two wavefunctions with different symmetries: a single electron shared between two atoms to form a molecule. This isn’t precisely identical to the multi-electron scenario, but it’s much easier to visualize, and demonstrates why antisymmetric states tend to have higher energies.
Wavefunctions for the two different states of an electron shared between two atoms.
A shared electron is attracted to both nuclei, so we expect that a slice through the probability distribution along the axis between the atoms should show two peaks, reflecting an increased chance of finding the electron near each of the nuclei. There are two different ways to make a wavefunction that leads to this sort of probability distribution though: one where the wavefunction is positive at both peaks, and one where the wavefunction changes from positive to negative as you move from one atom to the other.7
When thinking about the symmetry properties of these wavefunctions, we need to consider what happens when we swap the arbitrary labels of “left” and “right.” This is like reflecting the wavefunctions in a mirror, and we see immediately that the same-sign state is symmetric: both peaks in the wavefunction have the same sign, so if you swap left for right, nothing changes. The different-sign state, on the other hand, is antisymmetric: swapping left for right changes which peak is positive and which negative, which is the same as reversing the sign of the wavefunction.
While there may not seem to be much difference between these, the energy of the antisymmetric state is slightly higher. To understand why, we need to look closely at the probability (shown below) of finding the electron at a point in the vicinity of the molecule—which we get, remember, by squaring the wavefunction, because there can be no negative probabilities.
Probability distributions for the two wavefunctions from the previous figure; inset is zoomed in on the midpoint between the two atoms.
These look nearly identical, save for a tiny zone midway between the two atoms. The same-sign (symmetric) state gives the electron some chance of being found exactly halfway between them, while the different-sign (antisymmetric) state has exactly zero probability of being found at the halfway point (since to get from positive to negative, you have to pass through zero). An electron in the antisymmetric state is excluded from a small region of space that an electron in a symmetric state would be free to occupy. That exclusion narrows the range of positions in which an electron might be found, and as we saw when we talked about the uncertainty principle, that necessarily increases the kinetic energy of the particle.
The above illustration involves single-electron wavefunctions, while the Pauli exclusion principle applies to systems with multiple electrons in multiple states, and considers the spin of the electron as well as its spatial distribution (we’ll look more closely at this in Chapter 9). The multi-electron problem is much more complicated than the simple one-electron state, but the conclusion carries over: antisymmetric wavefunctions are, in general, slightly higher in energy than symmetric ones, and Pauli exclusion tells us that the wavefunction for a collection of electrons must be antisymmetric. That means that electrons will be found in wavefunctions with a higher energy, so the total kinetic energy of a collection of electrons shared between two atoms increases more rapidly as you pack a number of electrons into a small area than it would if they were not subject to Pauli exclusion.
You can, with considerable mathematical effort, extend this same line of argument to more nuclei and more electrons, and find the same result. A collection of particles subject to the Pauli exclusion principle will always have a higher total energy than an identical number of particles that can occupy symmetric wavefunctions.8 And, in fact, this increase in energy is essential to prevent implosion. Just as adding an extra planet can destroy the stability of a solar system, adding additional particles can destroy the stability of a single atom. Without the extra kinetic energy arising from the need to be in antisymmetric states, a large collection of nuclei and electrons could always reduce its energy to a more negative value by packing more tightly, and solid matter would be inherently unstable.
The mathematical calculation underlying this is formidably complicated, and it wasn’t conclusively shown that the energy of a collection of matter had any lower limit at all until Freeman Dyson and Andrew Lenard managed it in 1967, some forty years after the introduction of the Pauli exclusion principle. Dyson and Lenard’s work left an uncomfortable amount of room for implosion—their lower limit still would have allowed matter to compress substantially and release tremendous energy, making every solid object a potential nuclear bomb. In subsequent years, Elliot Lieb and Walter Thirring substantially improved Dyson’s calculation, and these days we have very solid evidence that solid matter is, in fact, stable just as it is. It’s not a surprising result for anyone accustomed to the everyday world, but it is a great comfort to mathematical physicists.
Applications to Astrophysics
As a postscript to this discussion of the stability of matter, it’s interesting to note that when a star dies, the remnant it leaves behind is also held up by the Pauli exclusion principle.
While stars begin their lives with a truly enormous amount of hydrogen, that fuel supply is nonetheless finite and will eventually be exhausted. Once that happens (it can take place in a variety of ways, some more spectacular than others), a core is left behind that can no longer generate energy by fusion. Since the heat released in fusion is what holds an active star up against the attraction of gravity trying to collapse it, this core will shrink inward. As happened during the initial collapse, the energy gained from the inward fall and the electromagnetic repulsion between particles will increase the temperature. Since the 1930s, though, physicists have known that, in the absence of fusion, this increase can’t happen fast enough to stop the collapse. The question then is: What happens to the core?
For a smallish collapsing core—up to a mass a bit greater than our sun—Pauli exclusion comes to the rescue. The electrons and nuclei of the core are pulled in by gravity and packed tighter and tighter, to the point where their quantum character comes into play—when the spacing between them becomes comparable to the width of their wavefunctions. Then, just as in the case of solid matter, the fact that they’re subject to Pauli exclusion leads to a more rapid increase in kinetic energy than you could get from particles without that requirement. This “electron degeneracy pressure” is enough to withstand the pull of gravity, and the core becomes a white dwarf, an Earth-sized ball of incredibly dense matter held up by quantum mechanics—a one-centimeter cube of white-dwarf matter would weigh several hundred metric tons, compared to a few grams for a piece of rock the same size.
The Pauli exclusion principle alone can’t resist gravity for heavier stars, though. Above about 1.4 times the mass of the sun,9 the gravitational pull of the core is great enough that the core continues to collapse. Electrons and nuclei are squeezed even tighter, until the distance between them becomes small enough for the weak nuclear interaction to come into play. The weak interaction only works on extremely small length scales, but when matter is dense enough, it lets an electron merge with an up quark, converting a proton into a neutron. In a collapsing core a bit larger than the limit for a white dwarf, the electrons and protons combine to form a mass that consists almost entirely of neutrons.
Neutrons, like protons and electrons, are particles subject to the Pauli exclusion principle. And while they’re electrically neutral and thus don’t repel each other, the requirement that they be in antisymmetric wavefunctions leads to a rapid increase of energy for sufficiently dense neutrons. This “neutron degeneracy pressure” can halt the collapse of a stellar core that’s too big to form a white dwarf. This forms a neutron star around ten kilometers in diameter, with a density around a million times that of a white dwarf.
Quantum degeneracy is an amazingly strong force, but in the end, gravity still wins. For a stellar core a bit more than twice the mass of our sun, not even Pauli exclusion can halt the collapse. The neutrons squeeze together more and more, until the whole thing becomes so compact that nothing, not even light, can escape from its surface. At that point, the core forms a black hole, and no further information about its fate is available to the outside universe.
Neutron stars and white dwarfs are some of the most exotic objects in the universe, very far removed from the experience of an ordinary morning. And yet, the quantum properties that keep those extreme astronomical bodies from ultimate collapse are the exact same properties that guarantee the continued existence of you and your breakfast.
Notes
1 Heisenberg won the 1932 Nobel Prize for developing matrix mechanics, though he wrote to Born afterward that he felt guilty over this, as the work had been done jointly by the three of them. Born eventually received the 1954 Nobel Prize; the delay is frequently attributed to politics, as Jordan was an early and enthusiastic supporter of the Nazis, and it took a while to find a way to honor Born without also including Jordan.
2 Like Heisenberg, Schrödinger’s breakthrough came while away from home, in his case on a ski holiday with one of his many mistresses. Ever since, physicists have tried to use these examples to argue for more vacation time.
3 You might think that the simplest possible system would be a particle by itself in free space, but that actually turns out to be much more complicated, as we’ll see in a little bit.
4 This is true for slow-moving particles, anyway; once speeds start to approach the speed of light, relativity changes the definition slightly, but for our present purposes “mass times velocity” is adequate.
5 The exact value for a one-dimensional box is a bit smaller than this because the probability distribution is peaked in the middle, but this gets at the basic idea.
6 Interestingly, Poincaré’s original conclusion was the opposite—he thought he had shown the ultimate stability of the many-body system. While his manuscript was being prepared for publication, though, one of the journal’s editors, the Swedish mathematician Lars Edvard Phragmén, pointed out what seemed to be a small gap in the proof; on closer examination, this turned out to totally reverse the conclusion. The original article had to be hastily retracted and rewritten, but Poincaré still got the prize.
7 As with standing waves on a string, way back in Chapter 2, the actual value of the wavefunction at one of these peaks oscillates through both positive and negative values over time. What matters is the relative sign of the two peaks: the same in the symmetric case, opposite in the antisymmetric one.
8 There’s a thriving branch of physics dedicated to studying the behavior of such particles, which is essential for understanding superconductivity. Most of the interesting phenomena in this area occur at temperatures within a few degrees of absolute zero, though, so they have little impact on a typical breakfast.
9 This is known as the “Chandrasekhar limit” after the Indian American physicist Subrahmanyan Chandrasekhar, who first calculated it on a steamship voyage to England in 1930. Chandrasekhar’s initial result met with a good deal of resistance, but he and others repeated and refined the calculations, and he was vindicated mathematically.