 How to Classify Symmetric and Antisymmetric Wave Functions - dummies

# 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 Pij2 = 1, note that if a wave function is an eigenfunction of Pij, then the possible eigenvalues are 1 and –1. That is, for an eigenfunction of Pij 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  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. 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.