The Amazing Story of Quantum Mechanics (24 page)

Insulators, on the other hand, such as diamond, have all the electrons tied into bonds between the atoms (the two electrons per box as discussed earlier). They are thus poor conductors of electricity and can conduct heat only by atomic vibrations (sound waves). The electrons in the box can assume only specific energy states, like electrons in atoms. Consequently, if you shine light on an insulator that does not correspond to an allowed transition, it will ignore the photons. This is why some insulators, such as diamond or window glass, are transparent to visible light. The details of the materials’ properties are very sensitive to the configurations of the boxes in which the electrons reside, which is why changes in crystal structure—say, when carbon transforms from graphite to diamond—can yield big variations in optical and electrical properties. The boxes here are the directional, rigid chemical bonds between the atoms (Figure 32b), and changes in the type of these bonds and the chemical constituents lead to big variations in crystal structure and rigidity (diamond is very stiff, while graphite is so soft, we use it for pencil lead). All the differences between insulators and metals can be understood at the most basic level by whether the last few unpaired electrons of the atoms in the solid satisfy the Pauli exclusion principle by localizing themselves in real space (insulators) or in momentum space (metals).

Thus, from playing with a ribbon, with one side black and the other side white, we see why the world is the way it is. Note that not everything has an intrinsic angular momentum equal to
ħ
/2. Some objects, such as helium atoms or photons, have spin values of either 0 or
ħ.
This seemingly small difference leads to superconductors and lasers.

Bert Hölldobler and Edward O. Wilson,
in
The Superorganism—The Beauty, Elegance, and Strangeness of Insect Societies,
propose that colonies of wasps, ants, bees, or termites can be considered as a single animal. They argue that each insect is a “cell” in the “superorganism”: foragers are the eyes and sense organs, the colony defenders act as the immune system, and the queen serves as the colony’s genitalia.
⁵⁰
An important difference between a superorganism and a regular animal is that the colony lacks a centralized brain or nervous system. Rather, each colony has its own rules for local interactions among the insects that govern its organization and size. In this way the colony can achieve levels of development that are far beyond the capabilities of the individual insects were they to act alone. As readers of science fiction pulps know, this is the mechanism (or at least one of them) by which humans defeat an alien invasion.

In Theodore Sturgeon’s 1958 science fiction novel
The Cosmic Rape
(an abridged version was simultaneously published in
Galaxy
magazine with the title
To Marry Medusa
), an alien intelligence applies an unconventional approach to its efforts to conquer Earth. (Spoiler alert!) The alien, in fact a cosmic spore, is capable of controlling the intelligence of a single human. It is surprised to discover that the people of Earth are not in mental contact with one another, given that the planet is covered with complex structures such as buildings, bridges, and roads. In the spore’s experiences conquering other planets that contained advanced infrastructure, such architecture is possible only when the primitive intelligences of the individual agents interact in a cooperative manner, as in a colony of ants or a hive of honeybees. The invading spore had never encountered a species for which a single agent is capable of designing a bridge or building on is or her own, and it thus assumes that a previously existing collective connection has been severed. The spore sets upon a plan to reestablish this connection and force all humans to think and work together in unison. Unfortunately for the alien entity, it succeeds in its plan. Once it finds itself dealing not with the intelligence of a single human, but with the collective consciousness of all several billion humans, the “hive mind” of humanity quickly devises an effective counterattack, destroying the alien spore. In this way Sturgeon has described the cooperative behavior of a Bose-Einstein condensate.

In Sturgeon’s novel, the alien spore creates from humanity a macroscopic quantum state, with a single wave function containing all the information about its constituent elements. Any change in one element, in Sturgeon’s case a human being, is instantly transmitted to every other element in the wave function, that is, the rest of humanity. Such situations occur frequently in the real world through quantum interactions between pairs of electrons in a superconductor or helium atoms in a superfluid. These collective states involve particles whose intrinsic angular momenta are multiples of
ħ,
rather than
ħ
/2. Particles with intrinsic angular momenta that are whole-number multiples of
ħ
are called bosons, as they obey a form of quantum statistics elaborated by Satyendra Bose and Albert Einstein, termed Bose-Einstein statistics.

In the previous chapter we discussed fermions, for which the angular momentum could be either +
ħ
/2 or -
ħ
/2, but not any other value. This is intrinsically asymmetric, as we can distinguish a top twirling clockwise from one rotating counterclockwise. We represented this situation, when the two fermions are so close that their wave functions overlap, with a ribbon with one side black and the other white. The significance of the two colors was that we could readily distinguish the spin = +
ħ
/2 electron from the spin = -
ħ
/2 electron, as we could the black and white sides of the ribbon. But experiments have revealed situations where the intrinsic angular momentum can have values of 0 or
ħ
or 2
ħ,
and so on, but not any fractional values. Let’s consider the case of quantum objects with a spin of zero first, and then turn to spin =
ħ
particles such as photons.

As the fundamental building blocks of atoms—electrons, protons, and neutrons—are all fermions, what sort of object would have spin of zero? One example is a helium atom. A helium nucleus has two protons and two neutrons, each having spin = +
ħ
/2 or -
ħ
/2. As the two protons are identical, in their lowest energy state in the nucleus they would pair up, +
ħ
/2 and -
ħ
/2, for a total spin of zero, as would the two identical neutrons. Similarly, the two electrons are spin paired, as indicated in the sketch in Figure 31b. Consequently, the total intrinsic angular momentum of a helium atom, when in its lowest energy configuration, has a spin value of zero.

Particles with zero value of spin are symmetric, in that we cannot describe the rotations as clockwise or counterclockwise. When two such particles are brought so close that their wave functions overlap, we will represent them by a ribbon whose sides are both white. I once again stress that the ribbon is employed as a metaphor for the resulting two-particle wave function, and as such, certain issues are being ignored here that would only distract from our discussion.

Let’s repeat the experiment with the ribbon from Chapter 12, only now using a ribbon with both sides the same color, for example, white (Figure 33). I can hold each end, and obviously a white side of the ribbon faces out (Figure 33a). Now, without letting go of either end, I will switch their positions, as before. The end of the ribbon on the left is now on the right, and vice versa. This procedure has introduced a half twist in the ribbon, as in Chapter 12 (Figure 33b). Of course, now both sides are still white. I can undo the half twist by flipping one end of the ribbon around, so that the back side turns outward (Figure 33c). When the ribbon had one side white and the other side black, this was a forbidden operation, as it changed the state of the ribbon (where before both ends had white facing out, one side would then have had a black side facing out). But if both sides of the ribbon are white, then this symmetry means I can flip one end of the ribbon and I have not changed anything except undoing the half twist. The important point is that a white ribbon can be restored to its original state following a single rotation, while the black/white ribbon requires two rotations to bring the original configuration back.

This symmetry indicates that the two-particle wave function for spin = 0 particles, such as helium atoms, as well as spin =
ħ
photons, termed bosons, can be written as the sum of the two functions A and B, Ψ = A + B, rather than Ψ = A - B, for fermions.
⁵¹
As before, A and B depend on the product of the one-particle wave functions at positions 1 and 2. Now the two-particle wave function Ψ = A + B is unchanged if the positions of particles 1 and 2 are switched, in which case Ψ would be given by Ψ = B + A. But this is just the same as Ψ = A + B = B + A. When two particles for which the intrinsic angular momentum has values of spin = 0 or spin =
ħ
⁵²
are brought close enough to each other that their de Broglie waves overlap, the resulting two-particle wave function is just the sum of the functions A and B, which are in turn functions of the one-particle wave functions.

What is the consequence of writing the two-electron wave function as Ψ = A + B? Recall that for fermions such as electrons, the fact that the two-electron wave function is Ψ = A-Bmeant that the probability is exactly zero that both electrons would be in the same quantum state, for which A = B. For bosons, Ψ = A + B indicates that the probability is large exactly when both particles are in the same quantum state, when A = B. Because when A = B, then Ψ = A + A = 2A and the probability density Ψ
²
= (2A) × (2A) = 4A
²
. For a single particle in state Ψ = A the probability density would be Ψ
²
= A×A= A
²
. For two single particles the probability would be A
²
+ A
²
= 2A
²
. So just by bringing a second identical particle near the first, the probability that they would both be found in state A is double what it would be for the two particles separately. While the probability is not 100 percent that they will both be in the same state, it is enhanced compared to the single-particle situation. A larger probability of both particles being at the same location in the same quantum state indicates that it is more likely to occur than not.

As the temperature of a system is reduced, the particles will settle down into lower energy states. If we had particles that were somehow distinguishable, for example, if their wave functions did not overlap so we did not have to worry about Fermi-Dirac or Bose-Einstein statistics, then at low temperatures we would find many particles in the lowest energy state, some in the next available quantum level, a few more in the next higher level, and negligible occupation of very high-energy states. For fermions, such as electrons in a solid, only two electrons can occupy the lowest energy level (one with spin = +
ħ
/2 and the other with spin = -
ħ
/2), regardless of temperature. In contrast, bosons will have an enhanced probability of collecting into the lowest-energy ground state at low temperatures, relative to the distinguishable particle case. For these particles, the rule of one particle per spin orientation per seat (valid for fermions) is thrown out, and one can have many particles dog piling into a single state. These spin = 0 or spin =
ħ
particles obey statistics described by Bose and Einstein, and this settling into the ground state is termed Bose-Einstein condensation.

Why do we need to go to low temperatures to see this condensation? If the particles are very far apart, then there will be little or no overlap of their wave functions, and the whole issue of indistinguishable particles is irrelevant. Temperature is just a bookkeeping device to keep track of the average energy per particle, so the lower the temperature, the less kinetic energy and the lower the momentum. From de Broglie’s relationship, a low momentum corresponds to a long matter-wavelength. If the particles involved have long de Broglie wavelengths, it will increase the opportunity for the waves of different identical particles to overlap. Similarly, confining the particles to a small volume also increases the possibility for interactions among wave functions. Consequently, low temperatures and small volumes (achieved by squeezing the system at high pressures) help induce Bose-Einstein condensation.

What are the special attributes of a Bose-Einstein condensate? We have considered the case of two identical bosons whose wave functions overlap such that they can be described by a single, two-particle wave function. As the temperature of a gas of bosons is lowered, millions of identical atoms’ wave functions overlap, all in the same quantum state. We thus obtain one single wave function that describes the behavior of millions of atoms. In this way the individual indistinguishable bosons behave as a single entity, and whatever happens to one atom is experienced by many. The Bose condensate is not unlike the demonically possessed children in the 1960 science fiction film
Village of the Damned.
The fair-skinned, blond children play the role of indistinguishable particles, and the fact that knowledge gained by one child is instantly shared with all is a natural consequence of the multiparticle wave function that describes this collective phenomenon.