Section 6.5 Bosons
Now let's turn to bosons, the particles that form symmetric combinations \(|\psi_S\rangle\) in their two-particle states. These particles do not exhibit a Pauli exclusion principle. In fact, it's quite the opposite: bosons like to be in the same state, relative to distinguishable particles. 1
Demonstrating this effect is a little bit tricky. Let us consider the following scenario: imagine that there are three different states, which we'll call \(|\alpha\rangle\text{,}\) \(|\beta\rangle\text{,}\) and \(|\gamma\rangle\text{,}\) and there are two particles which will be put into those states. How many possible two-particle states are there?
For classical distinguishable particles, each particle can be in one of three possible states, so there are \(3\times 3=9\) possible two-particle states. These are illustrated in the chart on the left in Figure 6.2. Since the particles are distinguishable, we use different symbols for them: “\(\bullet\)” for particle 1 and “\(\circ\)” for particle 2.
Now consider indistinguishable bosons, shown in the middle chart of Figure 6.2. Now any two states that differ by an exchange of particles only gets counted once. For example, the classical case of having the \(\bullet\) particle in state \(\alpha\) and the \(\circ\) particle in state \(\beta\) (row 4) is different from the case where the two particles are switched (row 7). But for indistinguishable bosons these are the same state, so it only appears once (row 4 for the bosons).
Finally, for fermions it is pretty straightforward. They are indistinguishable, so we just use \(\bullet\) for both particles, and we add in the Pauli exclusion principle, which eliminates the cases where both particles are in the same state. This gives the chart on the right in Figure 6.2.
What do we learn from these charts? Let's consider the case where the three states, \(|\alpha\rangle\text{,}\) \(|\beta\rangle\text{,}\) and \(|\gamma\rangle\text{,}\) all have the same energy. If these quantum states are in thermal equilibrium, we can appeal to ergodicity to argue that every two-particle microstate is equally probable. And this gives us something interesting.
Look at the probability of finding two particles in the same state. For the distinguishable classical particles, there are 3 cases out of 9 where the particles are in the same state, so it's a probability of \(1/3\text{.}\) But for the bosons, there are 3 cases of out 6 where they are in the same state, so a probability of \(1/2\text{.}\) Indistinguishability has given the bosons an enhanced probability of being in the same state. It is almost as if the bosons have an attractive force pulling them into the same state, but really it's just a property of their statistics.
And this turns out to have quite an impact. When there are a large number of identical bosons in a system, the statistical “attraction” becomes even more powerful and it is sometimes possible to get a macroscopic fraction of the particles to cram together into the same quantum state. This state is called then a Bose condensate. When we achieve this, we end up with a material that exhibits quantum mechanical properties on our everyday macroscopic scale, which is truly bizarre.
We have managed to make Bose condensates with a number of different types of bosons. When we do this with photons, we get a laser. When we do this with helium-4 atoms, we get a superfluid. When we do this with electrons paired up to be bosons (more below), we get a superconductor. Most recently, physicists have figured out how to get a gas of large atoms, such as rubidium, to form a Bose-Einstein condensate. These are all very different systems with very different properties, and we devote the remainder of the chapter to describing some of them.