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
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,
What about this one?
To find out, apply P12:
All right — how’s this compare with the original equation?
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?
is neither symmetric nor antisymmetric. In other words,
is not an eigenfunction of the P12 exchange operator.