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I want to ask a question about showing whether a wavefunction is symmetric or antisymmetric.

Consider the wavefunctio below for the group state for Hydrogen, $H_2$

$\psi_A=1 \sigma_{g}\left(r_{1}\right) 1\sigma_{g}\left(r_{2}\right)[\alpha(1) \beta(2)-\beta(1) \alpha(2)]$

and the question I was asked was:

Is this wavefunction antisymmetric with respect to swapping coordinates of both electrons? Show working.

I can see that in this situation, we have a spin and a space part and I wanted to swap the electron coordinates $r_1$ and $r_2$ as such

$\psi_B=1 \sigma_{g}\left(r_{2}\right) 1\sigma_{g}\left(r_{1}\right)[\alpha(1) \beta(2)-\beta(1) \alpha(2)]$

and then I'd expect that the first wavefunction $\psi_A$ above would be the negative of the second wavefunction $\psi_B$.

However, from the working I've just shown above, $\psi_A$ and $\psi_B$ are the exact same and I'm confused on where to go on from here.

How can I proceed onwards?

vik1245
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1 Answers1

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You didn't specify what $1{\sigma_g}$, $\alpha$ and $\beta$ are, but it doesn't look like that is strictly needed.

The question asks if the wavefunction $\psi_A$ is antisymmetric w.r.t. swapping the coordinates (positions) of the electrons. This is a mathematical question, and you show that it is actually symmetric. I guess that you expected it to be antisymmetric because the electrons are fermions, but being a fermion doesn't mean that the state has to be antisymmetric under a formal interchange of the position coordinates, but it has to be antisymmetric under a (full) interchange of the electron identities (or labelings).

The wave function being symmetric in position implies that it has to be antisymmetric in the spin part, which indeed it is.

doetoe
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