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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

image0.png

an eigenfunction of Pij looks like

image1.png

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

image2.png

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

image3.png

You can apply the exchange operator P12:

image4.png

Note that because

image5.png

is a symmetric wave function; that’s because

image6.png

How about this wave function?

image7.png

Again, apply the exchange operator, P12:

image8.png

Okay, but because

image9.png

you know that

image10.png

Here’s another one:

image11.png

Now apply P12:

image12.png

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

image13.png

Therefore,

image14.png

is antisymmetric.

What about this one?

image15.png

To find out, apply P12:

image16.png

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

image17.png

Okay —

image18.png

is symmetric.

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

image19.png

Start by applying P12:

image20.png

So how do these two equations compare?

image21.png

That is,

image22.png

is neither symmetric nor antisymmetric. In other words,

image23.png

is not an eigenfunction of the P12 exchange operator.

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