QM: Changing indices of wavefunctions

[Niles]

Homework Statement

Hi all.

I am looking at a potential with two wells, where we denote the wells a and b. Now there are two electrons in this setup, which we label 1 and 2. I have the following innerproduct:

$$\left\langle {\phi _b (x_1 )} \right|\left\langle {\phi _a (x_2 )} \right|\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\frown}} \over H} \left| {\phi _b (x_1 )} \right\rangle \left| {\phi _a (x_2 )} \right\rangle,$$

where H = H₁+H₂+V_ee.

Now my question is that at a lecture, the professor suddenly said that it was OK to switch particle-indices of the wavefunctions (i.e. to change 1 and 2). Then he continued calculating, but he did not explain why this is so.

Can you tell me why? I can see that the wavefunction for well a is the same for particle 1 and particle 2, but I can't see why this justifies just changing the indices as one pleases.

Thanks in advance.

 

[Ben Niehoff]

The expectation value of the Hamiltonian must be invariant under the exchange of particles; after all, the particles are both electrons, so they're identical.

 

[Niles]

So you are saying that:

$$\left\langle {\phi _b (x_1 )} \right|\left\langle {\phi _a (x_2 )} \right|\hat H \left| {\phi _b (x_1 )} \right\rangle \left| {\phi _a (x_2 )} \right\rangle = \left\langle {\phi _b (x_2 )} \right|\left\langle {\phi _a (x_1 )} \right|\hat H \left| {\phi _b (x_2 )} \right\rangle \left| {\phi _a (x_1 )} \right\rangle = \left\langle {\phi _b (x_1 )} \right|\left\langle {\phi _a (x_1 )} \right|\hat H \left| {\phi _b (x_1 )} \right\rangle \left| {\phi _a (x_1 )} \right\rangle = \left\langle {\phi _b (x_2 )} \right|\left\langle {\phi _a (x_2 )} \right|\hat H \left| {\phi _b (x_2 )} \right\rangle \left| {\phi _a (x_2 )} \right\rangle$$

because the wavefunction is the same for both particle, so <H> is the same no matter what indices we use?

 

[Ben Niehoff]

The wavefunction is not invariant under particle exchange. In fact, because the electrons are fermions, the wavefunction picks up a minus sign when you exchange particles.

However, the wavefunction is not physically observable. All observables, however, must be invariant under any transformation which leaves the overall physical configuration the same.

 

[Niles]

My trouble is that I have an expression on the form:

$$<H> = \left\langle {\phi _a (x_1 )} \right|\left\langle {\phi _b (x_2 )} \right|H\left| {\phi _a (x_1 )} \right\rangle \left| {\phi _b (x_2 )} \right\rangle - \left\langle {\phi _b (x_1 )} \right|\left\langle {\phi _a (x_2 )} \right|H\left| {\phi _a (x_1 )} \right\rangle \left| {\phi _b (x_2 )} \right\rangle - \left\langle {\phi _a (x_1 )} \right|\left\langle {\phi _b (x_2 )} \right|H\left| {\phi _b (x_1 )} \right\rangle \left| {\phi _a (x_2 )} \right\rangle + \left\langle {\phi _b (x_1 )} \right|\left\langle {\phi _a (x_2 )} \right|H\left| {\phi _b (x_1 )} \right\rangle \left| {\phi _a (x_2 )} \right\rangle,$$

where I know that:

$$\left\langle {\phi _a (x_1 )} \right|\left\langle {\phi _b (x_2 )} \right|H\left| {\phi _b (x_1 )} \right\rangle \left| {\phi _a (x_2 )} \right\rangle =0.$$

According to what we've talked about, I am allowed to change the indices 1 and 2 as long as <H> is unchanged. But since I don't know what <H> is, how can I even start thinking about changing indices?

Thanks for helping me.