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QM: Changing indices of wavefunctions

  1. Mar 1, 2009 #1
    1. The problem statement, all variables and given/known data
    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:

    [tex]
    \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,
    [/tex]

    where H = H1+H2+Vee.

    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.
     
    Last edited: Mar 1, 2009
  2. jcsd
  3. Mar 1, 2009 #2

    Ben Niehoff

    User Avatar
    Science Advisor
    Gold Member

    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.
     
  4. Mar 1, 2009 #3
    So you are saying that:

    [tex]
    \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
    [/tex]

    because the wavefunction is the same for both particle, so <H> is the same no matter what indices we use?
     
  5. Mar 1, 2009 #4

    Ben Niehoff

    User Avatar
    Science Advisor
    Gold Member

    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.
     
  6. Mar 1, 2009 #5
    My trouble is that I have an expression on the form:

    [tex]
    <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,
    [/tex]

    where I know that:

    [tex]
    \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.
    [/tex]

    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.
     
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