Quantum Theory Cannot Hurt You (8 page)

Why do particles with half-integer spin indulge in waveflipping, whereas particles with integer spin do not? This, of course, is a very good question. But it brings us to the end of what can easily be conveyed without opaque mathematics. Richard Feynman at least came clean about this: “This seems to be one of the few places in physics where there is a rule which can be stated very simply but for which no one has found an easy explanation. It probably means that we do not have a complete understanding of the fundamental principles involved.”

Feynman, who worked on the atomic bomb and won the 1965 Nobel Prize for Physics, was arguably the greatest physicist of the postwar era. If you find the ideas of quantum theory a little difficult,
you are therefore in very good company. It is fair to say that, 80-odd years after the birth of quantum theory, physicists are still waiting for the fog to lift so that they can clearly see what it is trying to tell us about fundamental reality. As Feynman himself said: “I think I can safely say that nobody understands quantum mechanics.”

Brushing the spin mystery under the carpet, we come finally to the point of all this—the implication of waveflipping for fermions such as electrons.

Instead of two helium nuclei, think of two electrons, each of which collides with another particle. After the collision, they ricochet in almost the same direction. Call the electrons A and B and call the directions 1 and 2 (even though they are almost the same direction). Exactly as in the case of two identical nuclei, there are two indistinguishable possibilities. Electron A could ricochet in direction 1 and electron B in direction 2, or electron A could ricochet in direction 2 and electron B in direction 1.

Since electrons are fermions, the wave corresponding to one possibility will be flipped before it interferes with the wave corresponding to the other possibility. Crucially, however, the waves for the two possibilities are identical, or pretty identical. After all, we are talking about two identical particles doing almost identical things. But if you add two identical waves—one of which has been flipped—the peaks of one will exactly match the troughs of the other. They will completely cancel each other out. In other words, the probability of two electrons ricocheting in exactly the same direction is zero. It is completely impossible!

This result is actually far more general than it appears. It turns out that two electrons are not only forbidden from ricocheting in the same direction, they are forbidden from doing the same thing, period. This prohibition, known as the Pauli exclusion principle, after Austrian physicist Wolfgang Pauli, turns out to be the ultimate reason for the existence of white dwarfs. While it is certainly true that an electron cannot be confined in too small a volume of space, this still does not explain why all the electrons in a white dwarf do not simply
crowd together in exactly the same small volume. The Pauli exclusion principle provides the answer. Two electrons cannot be in the same quantum state. Electrons are hugely antisocial and avoid each other like the plague.

Think of it this way. Because of the Heisenberg uncertainty principle, there is a minimum-sized “box” in which an electron can be squeezed by the gravity of a white dwarf. However, because of the Pauli exclusion principle, each electron demands a box to itself. These two effects, working in concert, give an apparently flimsy gas of electrons the necessary “stiffness” to resist being squeezed by a white dwarf ’s immense gravity.

Actually, there is yet another subtlety here. The Pauli exclusion principle prevents two fermions from doing the same thing if they are identical. But electrons have a way of being different from each other because of their spin. One can behave as if it is spinning clockwise and one as if it is spinning anticlockwise.
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Because of this property of electrons,
two
electrons are permitted to occupy the same volume of space. They may be unsociable, but they are not complete loners! White dwarfs are hardly everyday objects. However, the Pauli exclusion principle has much more mundane implications. In particular, it explains why there are so many different types of atoms and why the world around us is the complex and interesting place it is.

Recall that, just as sound waves confined in an organ pipe can vibrate in only restricted ways, so too can the waves associated with an electron confined in an atom. Each distinct vibration corresponds to a possible orbit for an electron at a particular distance from the central nucleus and with a particular energy. (Actually, of course, the orbit is
merely the most probable place to find an electron since there is no such thing as a 100 per cent certain path for an electron or any other microscopic particle.)

Physicists and chemists number the orbits. The innermost orbit, also known as the ground state, is numbered 1, and orbits successively more distant from the nucleus are numbered 2, 3, 4, and so on. The existence of these quantum numbers, as they are called, emphasises yet again how everything in the microscopic world—even the orbits of electrons—comes in discrete steps with no possibility of intermediate values.

Whenever an electron “jumps” from one orbit to another orbit closer to the nucleus, the atom loses energy, which is given out in the form of a photon of light. The energy of the photon is exactly equal to the difference in energy of two orbits. The opposite process involves an atom absorbing a photon with an energy equal to the difference in energy of two orbits. In this case, an electron jumps from one orbit to another orbit farther from the nucleus.

This picture of the “emission” and “absorption” of light explains why photons of only special energies—corresponding to special frequencies—are spat out and swallowed by each kind of atom. The special energies are simply the energy differences between the electron orbits. It is because there is a limited number of permitted orbits that there is a restricted number of orbital “transitions.”

But things are not quite this simple. The electron waves that are permitted to vibrate inside an atom can be quite complex three-dimensional things. They may correspond to an electron that is not only most likely to be found at a certain distance from the nucleus but more likely to be found in some directions rather than others. For instance, an electron wave might be bigger over the north and south poles of the atom than in other directions. An electron in such an orbit would most likely be found over the north and south poles.

Describing a direction in three-dimensional space requires two numbers. Think of a terrestrial globe where a latitude and longitude are required. Similarly, in addition to the numbers specifying its distance
from the nucleus, an electron wave whose height changes with direction requires two more quantum numbers to describe it. This makes a total of three. In recognition of the fact that electron orbits are totally unlike more familiar orbits—for instance, the orbits of planets around the Sun—they are given a special name: orbitals.

The precise shape of electron orbitals turns out to be crucially important in determining how different atoms stick together to make molecules such as water and carbon dioxide. Here, the key electrons are the outermost ones. For instance, an outer electron from one atom might be shared with another atom, creating a chemical bond. Where exactly the outermost electron is clearly plays an important role. If, for example, it has its highest probability of being found above the atom’s north and south poles, the atom will most easily bond with atoms to its north or south.

The science that concerns itself with all the myriad ways in which atoms can join together is chemistry. Atoms are the ultimate Lego bricks. By combining them in different ways, it is possible to make a rose or a gold bar or a human being. But exactly how the Lego bricks combine to create the bewildering variety of objects we see in the world around us is determined by quantum theory.

Of course, an obvious requirement for the existence of a large number of combinations of Lego bricks is that there be more than one kind of brick. Nature in fact uses 92 Lego bricks. They range from hydrogen, the lightest naturally occurring atom, to uranium, the heaviest. But why are there so many different atoms? Why are they not all the same? Once again, it all comes down to quantum theory.

Electrons trapped in the electric force field of a nucleus are like footballs trapped in a steep valley. By rights they should run rapidly downhill to the lowest possible place—the innermost orbital. But if this was what the electrons in atoms really did, all atoms would be roughly
the same size. More seriously, since the outermost electrons determine how an atom bonds, all atoms would bond in exactly the same way. Nature would have only one kind of Lego brick to play with and the world would be a very dull place indeed.

What rescues the world from being a dull place is the Pauli exclusion principle. If electrons were bosons, it is certainly true that an atom’s electrons would all pile on top of each other in the innermost orbital. But electrons are not bosons. They are fermions. And fermions abhor being crowded together.

This is how it works. Different kinds of atoms have different numbers of electrons (always of course balanced by an equal number of protons in their nuclei). For instance, the lightest atom, hydrogen, has one electron and the heaviest naturally occurring atom, uranium, has 92. In this discussion the nucleus is not important. Focus instead on the electrons. Imagine starting with a hydrogen atom and then adding electrons, one at a time.

The first available orbit is the innermost one, nearest the nucleus. As electrons are added, they will first go into this orbit. When it is full and can take no more electrons, they will pile into the next available orbit, farther away from the nucleus. Once that orbit is full, they will fill the next most distant one. And so on.

All the orbitals at a particular distance from the nucleus—that is, with different directional quantum numbers—are said to make up a shell. The maximum number of electrons that can occupy the innermost shell turns out to be two—one electron with clockwise spin and one with anticlockwise spin. A hydrogen atom has one electron in this shell and an atom of helium, the next biggest atom, has two.

The next biggest atom is lithium. It has three electrons. Since there is no more room in the innermost shell, the third electron starts a new shell farther out from the nucleus. The capacity of this shell is eight. For atoms with more than 10 electrons, even this shell is all used up, and another begins to fill up yet farther from the nucleus.

The Pauli exclusion principle, by forbidding more than two electrons from being in the same orbital—that is, from having the same
quantum numbers—is the reason that atoms are different from each other. It is also responsible for the rigidity of matter. “It is the fact that electrons cannot get on top of each other that makes tables and everything else solid,” said Richard Feynman.

Since the manner in which an atom behaves—its very identity—depends on its outer electrons, atoms with similar numbers of electrons in their outermost shells tend to behave in a similar way. Lithium, with three electrons, has one electron in its outer shell. So too does sodium, with 11 electrons. Lithium and sodium therefore bond with similar kinds of atoms and have similar properties.

So much for fermions, which are subject to the Pauli exclusion principle. What about bosons? Well, since such particles are not governed by the exclusion principle, they are positively gregarious. And this gregariousness leads to a host of remarkable phenomena, from lasers to electrical currents that flow forever to liquids that flow uphill.

Say two boson particles fly into a small region of space. One hits an obstruction in its path and ricochets; the other hits a second obstruction and ricochets. It doesn’t matter what the obstructing bodies are; they may be nuclei or anything else. The important thing here is the direction in which they ricochet, which is the same for both.

Call the particles A and B, and call the directions they ricochet in 1 and 2 (even if they are almost the same direction!). There are two possibilities. One is that particle A ends up in direction 1 and particle B ends up in direction 2. The other is that A ends up in direction 2 and B in direction 1. Because A and B are schizophrenic denizens of the microscopic world, there is a wave corresponding to A going in direction 1 and to B in direction 2. And there is also a wave corresponding to A going in direction 2 and to B in direction 1.

If the two bosons are different particles there can be no interference between them. So the probability that a detector picks up the
two ricocheting particles is simply the square of the height of the first wave plus the square of the height of the second wave, since the probability of anything happening in the microscopic world is always the square of the height of the wave associated with it. Well, it turns out—and this will have to be taken on trust—that the two probabilities are roughly the same. So the overall probability simply is twice the probability of each event happening individually.

Say the waves have a height of 1 for both processes. This would mean that if they were squared and added to get the probability for both processes, it would be (1 × 1) + (1 × 1) = 2. Now a probability of 1 corresponds to 100 per cent, so a probability of 2 is clearly ridiculous! But bear with this. It is still possible to make a comparison of probabilities, which is where all this is leading.

Now, say the two bosons are identical particles. In this case, the two possibilities—A going in direction 1 and B in direction 2, and A going in direction 2 and B in direction 1—are indistinguishable. And because they are indistinguishable, the waves associated with them can interfere with each other. Their combined height is 1 + 1. The probability for both processes is therefore (1 + 1) × (1 + 1) = 4.

This is twice as big as when the bosons were not identical. In other words, if two bosons are identical, they are twice as likely to ricochet in the same direction as if they were different. Or to put it another way, a boson is twice as likely to ricochet in a particular direction if another boson ricochets in that direction too.