What happens to wavefunction if you swap electrons?

[Kara386]

If you have a multi-electron atom and you swap two electrons around, what happens to the wavefunction? I think nothing happens because electrons are identical, but then they can have different spins, so would the wavefunction change if you swapped a spin up electron with a spin down one?

 

[phinds]

How do you propose to "swap" the electrons?

 

[Kara386]

How do you propose to "swap" the electrons?

I have no idea, it's a good question that I hadn't considered. Because isn't that what the whole symmetric-antisymmetric thing is about? It was explained to me in terms of swapping electrons, having some permutation of a pair, and I didn't really understand it so I thought I'd ask here! :)

 

[Kara386]

I may have completely misunderstood...

 

[PeterDonis]

isn't that what the whole symmetric-antisymmetric thing is about?

Sort of. Perhaps a concrete example will help. It actually works better as a simple example to think of things in the path integral formulation, rather than wave functions.

Suppose we run an experiment in which two particles get emitted from two different sources, and some time later two particles get detected by two different detectors. To keep things simple, we'll assume that the particles have the same spin throughout the experiment (because we've prepared them with the same spin and the apparatus doesn't change the spin), so we only need to consider their positions. We label the sources as S1 and S2, and the detectors as D1 and D2. Then the path integral that describes this process (leaving out all the higher order terms that arise in quantum field theory when we include virtual particles) will have two terms:

Term 1: a particle goes from S1 to D1, and a particle goes from S2 to D2.

Term 2: a particle goes from S1 to D2, and a particle goes from S2 to D1.

Now we can state "the symmetric-antisymmetric thing" easily: if the particles are bosons (symmetric), Term 1 and Term 2 have the same sign; but if the particles are fermions (antisymmetric, like electrons), Term 1 and Term 2 have opposite signs.

Note that, if D1 and D2 are the same, then for fermions the amplitude for this process is zero: both terms have the same magnitude (can you see why?), and opposite sign, so they cancel. This is the familiar Pauli exclusion principle. But if D1 and D2 are not the same, then the terms won't exactly cancel, because the distances will be slightly different, so there will still be a nonzero amplitude for this process to happen--but it will be smaller than in the case of bosons, where the amplitudes are the same sign.

 

[A. Neumaier]

If you have a multi-electron atom and you swap two electrons around, what happens to the wavefunction?

The wave function changes sign because of antisymmetry. That's all.