What Happens to a Wave Function When You Swap Two Particles

When you swap the order of operations for two particles in a multi-particle atom, this affects their wave function. Whether the wave function is symmetric under such operations gives you insight into whether the two particles can occupy the same quantum state.

Take a look at the general wave function for N particles:

image0.png

In this case, you can think of symmetry in terms of the location coordinate, r, to keep things simple, but you can also consider other quantities, such as spin, velocity, and so on. That wouldn’t make this discussion any different, because you can wrap all of a particle’s quantum measurements — location, velocity, speed, and so on — into a single quantum state, which you can call

image1.png

Doing so would make the general wave function for N particles into this:

image2.png

But this example just considers the wave function

image3.png

to keep things simple.

Now imagine that you have an exchange operator, Pij, that exchanges particles i and j. In other words,

image4.png

And Pij = Pji, so

image5.png

Also, note that applying the exchange operator twice just puts the two exchanged particles back where they were originally, so

image6.png

Here’s what that looks like:

image7.png

However, in general, Pij and Plm

image8.png

do not commute. That is,

image9.png

Therefore,

image10.png

For example, say you have four particles whose wave function is

image11.png

Apply the exchange operators P12 and P14 to see whether P12 P14 equals P14 P12. Here’s

image12.png

And here’s what

image13.png

looks like:

image14.png

Okay. Now take a look at

image15.png

And here’s what

image16.png

looks like:

image17.png

As you can see by comparing

image18.png

and this last equation,

image19.png

In other words, the order in which you apply exchange operators matters.

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