How to Classify Symmetric and Antisymmetric Wave Functions

You can determine what happens to the wave function when you swap particles in a multi-particle atom. Whether the wave function is symmetric or antisymmetric under such operations gives you insight into whether two particles can occupy the same quantum state.

Given that P_ij² = 1, note that if a wave function is an eigenfunction of P_ij, then the possible eigenvalues are 1 and –1. That is, for

an eigenfunction of P_ij looks like

That means there are two kinds of eigenfunctions of the exchange operator:

Now take a look at some symmetric and some antisymmetric eigenfunctions. How about this one — is it symmetric or antisymmetric?

You can apply the exchange operator P₁₂:

Note that because

is a symmetric wave function; that’s because

How about this wave function?

Again, apply the exchange operator, P₁₂:

Okay, but because

you know that

Here’s another one:

Now apply P₁₂:

How does that equation compare to the original one? Well,

Therefore,

is antisymmetric.

What about this one?

To find out, apply P₁₂:

All right — how’s this compare with the original equation?

Okay —

is symmetric.

You may think you have this process down pretty well, but what about this next wave function?

Start by applying P₁₂:

So how do these two equations compare?

That is,

is neither symmetric nor antisymmetric. In other words,

is not an eigenfunction of the P₁₂ exchange operator.