Bucknell Physics 212 Supplement

Section 6.2 Quantum Statistics

The whole phenomenon of indistinguishability results from considering pairs of quantum particles and asking the following:

In a two-particle system where particles 1 and 2 are of the same type, say, both photons or both electrons, is it possible to tell which is particle 1 and which is particle 2?

To be specific, let's consider the ground state of a helium atom. Helium has two electrons orbiting the nucleus and both electrons can be in the ground state energy level. In classical physics, the two electrons must be distinguishable: zoom in with a detailed enough probe and you could see which electron is where. You could name them electron-Fred and electron-George, and then follow their motion with a fast camera and keep track of which was Fred and which was George. So the classical physics answer to the question above is yes: with enough technology we can tell which is which.

But the microscopic world isn't classical, and in quantum mechanics things are different. Heisenberg's uncertainty principle has already demanded that the electrons in the ground state have some spread in position and spread in momentum. Since the two electrons are overlapping in this smeared out “Fred-George-blob” state, Heisenberg is prohibiting us from tracking the two particles separately, no matter how fast or high resolution our camera is. Therefore it is at least possible for quantum mechanical particles to be indistinguishable. But that just leaves the question open. To demonstrate quantum indistinguishability we will have to work out its implications and then test them against nature.

Let's develop some notation. If particle 1 is in state \(|\alpha\rangle\) and particle 2 in state \(|\beta\rangle\) (these might be spin states, or energy levels, or whatever), we can write the two-particle state as \(|\psi\rangle = |\alpha\,\beta\rangle\text{.}\) Here we read the particle 1 state as the first symbol and the particle 2 state as the second symbol.

But if these quantum particles are indistinguishable we cannot really have the state \(|\alpha\,\beta\rangle\text{,}\) since that notation says we know it's particle 1 in state \(|\alpha\rangle\text{,}\) and by assumption we can't know this. What we really want to say is “one of the particles is in \(|\alpha\rangle\) and the other particle is in \(|\beta\rangle\) and that's all we know.” We can achieve this by making a superposition:

The state \(|\psi_S\rangle\) truly makes no distinction between particle 1 and particle 2. You can see this by consider the effect of swapping the particle labels 1 and 2:

If we apply this “swap” operation on \(|\psi_S\rangle\text{,}\) we find that we get the same state back:

Therefore the state \(|\psi_S\rangle\text{,}\) which is the symmetric superposition of \(|\alpha\,\beta\rangle\) and \(|\beta\,\alpha\rangle\text{,}\) is a candidate state to describe indistinguishable particles.¹

There is another possibility, though. Recall that the overall sign of a wavefunction or state has no impact on any measurable quantity. All the measurements we can make amount to probabilities, which come from squaring the coefficients in front of the states. So we could also make indistinguishable particle states with the antisymmetric combination

This state picks up exactly an overall minus sign under the “swap” operation:

So we have two possibilities for quantum indistinguishability: the two-particle states can either be the symmetric \(|\psi_S\rangle\) or the antisymmetric \(|\psi_A\rangle\text{.}\) And now for a strong statement.

This includes photons, electrons, protons, neutrons, the quarks that make up protons and neutrons, and more exotic particles like the \(W\) and \(Z\) bosons. It seems to be a fundamental property of the quantum world obeyed by everything, just like wave-particle duality.

The particles that are symmetric under swap are called bosons,² and the particles that are antisymmetric under swap are called fermions.³ We will consider both cases in detail below.