Quantum Theory Cannot Hurt You - Marcus Chown (2007)
Part I. SMALL THINGS
Chapter 6. IDENTICALNESS AND THE ROOTS OF DIVERSITY
HOW THE BEWILDERING VARIETY OF THE EVERYDAY WORLD STEMS FROM THE FACT THAT YOU CANNOT TATTOO AN ELECTRON
I woke up one morning and all of my stuff had been stolen, and replaced by exact duplicates.
Steven Wright
They came from far and wide to see it—the river that ran uphill. It flowed past the fishing port, climbed through the close-packed houses, before meandering up the sheep-strewn hillside to the craggy summit overlooking the town. Startled seagulls bobbed on it. Excited children ran beside it. And at picnic tables outside pubs all along the river’s lower reaches, daytrippers sat transfixed by this wonder of nature as beer crept steadily up the sides of their beer glasses and quietly emptied itself onto the ground.
Surely, there is no liquid that can defy gravity like this and run uphill? Remarkably, there is. It’s yet another consequence of quantum theory.
Atoms and their kin can do many impossible things before breakfast. For instance, they can be in two or more places at once, penetrate impenetrable barriers, and know about each other instantly even when on different sides of the Universe. They are also totally unpredictable, doing things for no reason at all—perhaps the most shocking and unsettling of all their characteristics.
All of these phenomena ultimately come down to the wave-particle character of electrons, photons, and their like. But the peculiar dual nature of microscopic objects is not the only thing that makes them profoundly different from everyday objects. There is something else: their indistinguishability. Every electron is identical to every other electron, every photon is identical to every other photon, and so on.1
At first sight this may not seem a very remarkable property. But think of objects in the everyday world. Although two cars of the same model and colour appear the same, in reality they are not. A careful inspection would reveal that they differ slightly in the evenness of their paint, in the air pressure in their tires, and in a thousand other minor ways.
Contrast this with the world of the very small. Microscopic particles cannot be scratched or marked in any way. You cannot tattoo an electron! They are utterly indistinguishable.2 The same is true of photons and all other denizens of the microscopic world. This indistinguishability is truly something new under the Sun. And it has remarkable consequences for both the microscopic world and the everyday world. In fact, it is fair to say that it is the reason the world we live in is possible.
THINGS YOU CAN’T TELL APART INTERFERE
Recall that all the bizarre behaviour in the microscopic world, such as an atom’s ability to be in many places at once, comes down to interference. In the double slit experiment, for example, it is the interference between the wave corresponding to a particle going through the left-hand slit and the wave corresponding to the particle going through the right-hand slit that produces the characteristic pattern of alternating dark and light stripes on the second screen.
Recall also that if you were to set up some means of determining which slit each particle goes through—enabling you to distinguish between the two alternative events—the interference stripes disappear because of decoherence. Interference, it turns out, happens only if the alternative events are indistinguishable—in this case, the particle going through one slit and the particle going through the other slit.
In the case of the double slit experiment, the alternative events are indistinguishable just as long as nobody looks. But identical particles, such as electrons, raise the possibility of entirely new kinds of indistinguishable events.
Think of a teenage boy who plans to go out clubbing with his girlfriend, who happens to have an identical twin sister. Unbeknown to him, his girlfriend decides to stay in and watch TV and sends her twin in her place. Because the two girls appear identical to the boy (although they are not of course identical at the microscopic level), the events of going clubbing with his girlfriend and going clubbing with his girlfriend’s sister are indistinguishable.
Events such as this one, which are indistinguishable simply because they involve apparently indistinguishable things, have no serious consequences in the wider world (apart from allowing identical twin girls to run rings around their boyfriends). However, in the microscopic world, they have truly profound consequences. Why? Because events that are indistinguishable—for any reason whatsoever—are able to interfere with each other.
THE COLLISION OF IDENTICAL THINGS
Take two atomic nuclei that collide. Any such collision—and this particular point will have to be taken on trust—can be looked at from a point of view in which the nuclei fly in from opposite directions, hit, then fly back out in opposite directions. In general, the in and out directions are not the same. Think of a clock face. If the nuclei fly into the collision point from, say, 9:00 and 3:00, they might fly out toward 4:00 and 10:00. Or 1:00 and 7:00. Or any other pair of directions, as long as the directions are opposite each other.
An experimenter could tell which direction the two nuclei ricochet by placing detectors at opposite sides of the imaginary clock face and then moving them around the rim together. Say the detectors are placed at 4:00 and 10:00. In this case, there are two possible ways the nuclei can get to the detectors. They could strike each other with a glancing blow so that the one coming from 9:00 hits the detector at 4:00 and the one coming from 3:00 hits the one at 10:00. Or they could hit head on, so that the one coming from 9:00 bounces back almost the way it came and hits the detector at 10:00 and the one coming from 3:00 bounces back almost the way it came and hits the detector at 4:00.
The directions of 4:00 and 10:00 are in no way special. Wherever the two detectors are positioned, there will be two alternative ways the nuclei can get to them. Call them events A and B.
What happens if the two nuclei are different? Say the one that flies in from 9:00 is a nucleus of carbon and the one that flies in from 3:00 is a nucleus of helium. Well, in this case, it is always possible to distinguish between events A and B. After all, if a carbon nucleus is picked up by the detector at 10:00, it is obvious that event A occurred; if it is picked up by the detector at 3:00, it must have been event B instead.
What happens, however, if the two nuclei are the same? Say each is a nucleus of helium? Well, in this case, it is impossible to distinguish between events A and B. A helium nucleus that is picked up by the detector in the direction of 10:00 could have got there by either route, and the same is true for a helium nucleus picked up in the direction of 4:00. Events A and B are now indistinguishable. And if two events in the microscopic world are indistinguishable, the waves associated with them interfere.
In the collision of two nuclei, interference makes a huge difference. For instance, it is possible that the two waves associated with the two indistinguishable collision events destructively interfere, or cancel each other out, in the direction of 10:00 and 4:00. If so the detectors will pick up no nuclei at all, no matter how many times the experiment is repeated. It is also possible that the two waves constructively interfere, or reinforce each other, in the direction of 10:00 and 4:00. In this case, the detectors will pick up an unusually large number of nuclei.
In general, because of interference, there will be certain outward directions in which the waves corresponding to events A and B cancel each other and certain outward directions in which they reinforce each other. So if the experiment is repeated thousands of times and the ricocheting nuclei are picked up by detectors all around the rim of the imaginary clock face, the detectors will see a tremendous variation in the number of nuclei arriving. Some detectors will pick up many nuclei, while others will pick up none at all.
But this is dramatically different from the case when the nuclei are different. Then there is no interference, and the detectors will pick up nuclei ricocheting in all directions. There will be no places around the clock face where nuclei are not seen.
This striking difference between the outcomes of the experiment when the nuclei are the same and when they are different is not because of the difference in masses of the nuclei of carbon and helium, although this has a small effect. It is truly down to whether collision events A and B are distinguishable or not.
If this kind of thing happened in the real world, think what it would mean. A red bowling ball and a blue bowling ball that are repeatedly collided together would ricochet in all possible directions. But everything would be changed merely by painting the red ball blue so the two balls were indistinguishable. Suddenly, there would be directions in which the balls ricocheted far more often than when they were different colours and directions in which they never, ever ricocheted.
This fact, that events involving identical particles in the microscopic world can interfere with each other, may seem little more than a quantum quirk. But it isn’t. It is the reason why there are 92 different kinds of naturally occurring atoms rather than just 1. In short, it is responsible for the variety of the world we live in. Understanding why, however, requires appreciating one more subtlety of the process in which identical particles collide.
TWO TRIBES OF PARTICLES
Recall the case in which the nuclei are different—a carbon nucleus and a helium nucleus—and consider again the two possible collision events. In one, the nuclei strike each other with a glancing blow, and in the other they hit head on and bounce back almost the way they came. What this means is that, for the nucleus that comes in at 9:00, there is a wave corresponding to it going out at 4:00 and a wave corresponding to it going out at 10:00.
The key thing to understand here is that the probability of an event is not related to the height of the wave associated with that event but to the square of the height of the wave. The probability of the 4:00 event is therefore the square of the wave height in the direction of 4:00 and the probability of the 10:00 event is the square of the wave height in the direction of 10:00. It is here that the crucial subtlety comes in.
Say the wave corresponding to the nucleus that flies out at 10:00 is flipped by the collision, so that its troughs become its peaks and its peaks become its troughs. Would it make any difference to the probability of the event? To answer this, consider a water wave—a series of alternating peaks and troughs. Think of the average level of the water as corresponding to a height equal to zero so that the height of the peaks is a positive number—say plus 1 metre—and the height of the troughs is a negative number—minus 1 metre. Now it makes no difference whether you square the height of a peak or the height of a trough since 1 × 1 = 1 and -1 × -1 also equals 1. Consequently, flipping the probability wave associated with a ricocheting nucleus makes no difference to the event’s probability.
But is there any reason to believe that one wave might get flipped? Well, the 10:00 collision and the 4:00 collision are very different events. In one, the trajectory of the nucleus hardly changes whereas in the other it is turned violently back on itself. It is at least plausible that the 10:00 wave might get flipped.
Just because something is plausible does not mean it actually happens. True. In this case, however, it does! Nature has two possibilities available to it: It can flip the wave of one collision event or it can leave it alone. It turns out that it avails itself of both.
But how would we ever know about a probability wave getting flipped? After all, the only thing an experimenter can measure is the number of nuclei picked up by a detector which depends on the probability of a particular collision event. But this is determined by the square of the wave height, which is the same whether the wave is flipped or not. It would seem that what actually happens to the probability wave in the collision is hidden from view.
If the colliding particles are different, this is certainly true. But, crucially, it is not if they are identical. The reason is that the waves corresponding to indistinguishable events interfere with each other. And in interference it matters tremendously whether or not a wave is flipped before it combines with another wave. It could mean the difference between peaks and troughs coinciding or not, between the waves cancelling or boosting each other.
What happens then in the collision of identical particles? Well, this is the peculiar thing. For some particles—for instance, photons—everything is the same as it is for identical helium nuclei. The waves that correspond to the two alternative collision events interfere with each other normally. However, for other particles—for instance, electrons—things are radically different. The waves corresponding to the two alternative collision events interfere, but only after one has been flipped.
Nature’s basic building blocks turn out to be divided into two tribes. On the one hand, there are particles whose waves interfere with each other in the normal way. These are known as bosons. They include photons and gravitons, the hypothetical carriers of the gravitational force. And, on the other hand, there are particles whose waves interfere with one wave flipped. These are known as fermions. They include electrons, neutrinos, and muons.
Whether particles are fermions or bosons—that is, whether or not they indulge in waveflipping—turns out to depend on their spin. Recall that particles with more spin than others behave as if they are spinning faster about their axis (although in the bizarre quantum world particles that possess spin are not actually spinning!). Well, it turns out that there is a basic indivisible chunk of spin, just like there is a basic indivisible chunk of everything in the microscopic world. For historic reasons, this “quantum” of spin is 1/2 unit (don’t worry what the units are). Bosons have integer spin—0 units, 1 unit, 2 units, and so on—and fermions have “half-integer” spin—1/2 unit, 3/2 units, 5/2units, and so on.
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.3 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.
WHY ATOMS AREN’T ALL THE SAME
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.
WHY ATOMS ARE NOT ALL THE SAME
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.
WHY BOSONS LIKE TO BE TOGETHER WITH THEIR MATES
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.
The more bosons there are the more significant the effect. If n bosons are present, the probability that one more particle will ricochet in the same direction is n + 1 times bigger than if no other bosons are present. Talk about herd behaviour! The mere presence of other bosons doing something greatly increases the probability that one more will do the same thing.
This gregariousness turns out to have important practical applications—for instance, in the propagation of light.
LASERS AND LIQUIDS THAT RUN UPHILL
All the processes so far considered have involved particles colliding and ricocheting in a particular direction. But that is not essential. The arguments used could apply equally well to the creation of particles—for instance, the “creation” of photons by atoms that emit light.
Photons are bosons, so the probability that an atom will emit a photon in a particular direction with a particular energy is increased by a factor of n + 1 if there are already n photons flying in that direction with that energy. Each new photon emitted increases the chance of another photon being emitted. Once there are thousands, even millions, flying through space together, the probability of new photons being emitted is enormously enhanced.
The consequences are dramatic. Whereas a normal light source like the Sun produces a chaotic mixture of photons of all different energies, a laser generates an unstoppable tide of photons that surge through space in perfect lockstep. Lasers, however, are far from the only consequence of the gregariousness of bosons. Take liquid helium, which is composed of atoms that are bosons.
Helium-4, the second most common atom in the Universe, is one of nature’s most peculiar substances.4 It was the only element to have been discovered on the Sun before it was discovered on Earth, and it has the lowest boiling point of any liquid, -269 degrees Celsius. In fact, it is the only liquid that never freezes to become a solid, at least not at normal atmospheric pressure. All these things, however, pale into insignificance beside the behaviour of helium below about -271 degrees Celsius. Below this “lambda point,” it becomes a superfluid.
Usually, a liquid resists any attempt to move one part relative to another. For instance, treacle resists when you stir it with a spoon and water resists when you try to swim through it. Physicists call this resistance viscosity. It is really just liquid friction. But whereas we are used to friction between solids moving relative to each other—for instance, the friction between a car’s tyres and the road—we are not familiar with the friction between parts of a liquid moving relative to each other. Treacle, because it resists strongly, is said to have a high viscosity, or simply to be very viscous.
Clearly, viscosity can manifest itself only when one part of a liquid moves differently from the rest. At the microscopic level of atoms, this means that it must be possible to knock some liquid atoms into states that are different from those occupied by other liquid atoms.
In a liquid at normal temperature, the atoms can be in many possible states in each of which they jiggle about at different speeds. But as the temperature falls, they become more and more sluggish and fewer and fewer states are open to them. Despite this effect, however, not all atoms will be in the same state, even at the lowest temperatures.
But things are different for a liquid of bosons such as liquid helium. Remember, if there are already n bosons in a particular state, the probability of another one entering the state is n + 1 bigger than if there were no other particles in the state. And for liquid helium, with countless helium atoms, n is a very large number indeed. Consequently, there comes a time, as liquid helium is cooled to sufficiently low temperatures, when all the helium atoms suddenly try to crowd into the same state. It’s called the Bose-Einstein condensation.
With all the helium atoms in the same state, it is impossible—or at least extremely difficult—for one part of the liquid to move differently from another part. If some atoms are moving along, all the atoms have to move along together. Consequently, the liquid helium has no viscosity whatsoever. It has become a superfluid.
In superfluid liquid helium there is a kind of rigidity to the motion of the atoms. It is very hard to make the liquid do anything because you either have to get all of its atoms to do the thing together or they simply do not do the thing at all. For instance, if you put water in a bucket and spin the bucket about its axis, the water will end up spinning with the bucket. This is because the bucket drags around the water atoms—strictly speaking, the water molecules—that are in direct contact with the sides, and these in turn drag around the atoms farther from the sides, and so on, until the entire body of water is turning with the bucket. Clearly, for the water to get to the state in which it is spinning along with the bucket, different parts of the liquid must move relative to each other. But as just pointed out, this is very hard for a superfluid. All the atoms move together or they do not move at all. Consequently, if superfluid liquid helium is put in a bucket and the bucket is spun, it has no means open to it to attain the spin of the bucket. Instead, the superfluid helium stays stubbornly still while the bucket spins.
The cooperative motion of atoms in superfluid liquid helium leads to even more bizarre phenomena. For instance, the superfluid can flow through impossibly small holes that no other liquid can flow through. It is also the only liquid that can flow uphill. Interestingly, helium has a rare, lightweight cousin. Helium-3 turns out to be a normal, boring liquid. The reason is that helium-3 particles are fermions. And superfluidity is a property solely of bosons.
Actually, this isn’t entirely true. The microscopic world is full of surprising phenomena. And in a special case, fermions can behave like bosons!
ELECTRIC CURRENTS THAT RUN FOREVER
The special case, when fermions behave like bosons, is that of an electric current in a metal. Because the outermost electrons of metal atoms are very loosely bound, they can break free. If a voltage is then applied between the ends of the metal by a battery, all the countless liberated electrons will surge through the material as an electric current.5
Electrons are, of course, fermions, which means they are antisocial. Imagine a ladder, with the rungs corresponding to ever higher energy states. Electrons would fill up the rungs two at a time from the bottom (bosons would happily crowd on the lowest rungs). The need for a separate rung for each pair of electrons means that the electrons in a metal have far more energy on average than might be naively expected.
But something really weird happens when a metal is cooled to close to absolute zero, the lowest possible temperature. Usually, each electron travels through the metal entirely independently of all other electrons. However, as the temperature falls, the metal atoms vibrate ever more sluggishly. Although they are thousands of times more massive than electrons, the attractive electrical force between an electron and a metal atom is enough to tug the atom toward it as the electron passes by.6 The tugged atom, in turn, tugs on another electron. In this way, one electron attracts another through the intermediary of the metal atom.
This effect radically changes the nature of the current flowing through the metal. Instead of being composed of single electrons, it is composed of paired-up electrons known as Cooper pairs. But the electrons in each Cooper pair spin in an opposite manner and cancel out. Consequently, Cooper pairs are bosons!
A Cooper pair is a peculiar thing. The electrons that make it up may not even be close to each other in the metal. There could easily be thousands of other electrons between one member of a Cooper pair and its partner. This is just a curious detail, however. The key thing is that Cooper pairs are bosons. And at the ultralow temperature of the superconductor all the bosons crowd into the same state. They therefore behave as a single, irresistible entity. Once they are flowing en masse, it is extremely difficult to stop them.
In a normal metal an electrical current is resisted by nonmetal, impurity atoms, which get in the way of electrons, obstructing their progress through the metal. But whereas an impurity atom can easily hinder an electron in a normal metal, it is nearly impossible for it to hinder a Cooper pair in a superconductor. This is because each Cooper pair is in lockstep with billions upon billions of others. An impurity atom can no more thwart this flow than a single soldier can stop the advance of an enemy army. Once started, the current in a superconductor will flow forever.
1 Since photons come with different wavelengths, we are of course talking here about photons with the same wavelength being identical to each other.
2 John Wheeler and Richard Feynman once came up with an interesting suggestion for why electrons are utterly indistinguishable—because there is only one electron in the Universe! It weaves backwards and forwards in time like a thread going back and forth through a tapestry. We see the multitude of places where the thread goes through the fabric of the tapestry and mistakenly attribute each to a separate electron.
3 Physicists call two alternatives spin “up” and spin “down.” But that is just a technicality.
4 Helium-4 has four particles in its nucleus—two protons and two neutrons. It has a less common cousin, helium-3, which has the same number of protons but one fewer neutron.
5 Why then doesn’t a metal fall apart? The full explanation requires quantum theory. But, simplistically, the stripped, or conduction, electrons form a negatively charged cloud permeating the metal. It is the attraction between this cloud and the positively charged electron-stripped metal ions that glues the metal together.
6 Strictly speaking, the atoms are positive ions, the name given to atoms that have lost electrons.