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Introductory QM problem in 3D

  1. Oct 7, 2011 #1
    Problem10.1, Introductory QM,Liboff.
    1. The problem statement, all variables and given/known data
    If [itex]\psi (\mathbf{r},t)[/itex] is a free-particle state and [itex]b(\mathbf{k},t)[/itex] the momentum probability amplitude for this same state, show that
    [itex]\iiint \psi^* \psi d \mathbf{r}[/itex]=[itex]\iiint b^* b d \mathbf{k}[/itex]

    2. Relevant equations

    [itex]\psi_\mathbf{k} (\mathbf{r},t) = Ae^{i(\mathbf{k} \cdot \mathbf{r} - \omega t)}[/itex] (10.14)
    [itex]\hbar \omega = E_k[/itex]
    [itex]\delta (\mathbf{r} - \mathbf{r'}) = \frac{1}{(2 \pi)^3} \iiint e^{i \mathbf{k} \cdot (\mathbf{r} - \mathbf{r'})} d \mathbf{k}[/itex] (10.20)
    [itex]d \mathbf{k} = dk_x dk_y dk_z[/itex]
    [itex]\psi (\mathbf{r},t)[/itex]=[itex]\frac{1}{(2 \pi)^{3/2}}[/itex][itex]\iiint b(\mathbf{k},t)[/itex][itex]e^{i(\mathbf{k} \cdot \mathbf{r}-\omega t)} d \mathbf{k}[/itex] (10.22)
    [itex]b(\mathbf{k},t) = \frac{1}{(2 \pi)^{3/2}}[/itex][itex]\iiint[/itex][itex]\psi (\mathbf{k},t) e^{i(\mathbf{k} \cdot \mathbf{r} - \omega t)} d \mathbf{r}[/itex] (10.23)
    [itex]d \mathbf{r}=dxdydz[/itex]


    3. The attempt at a solution

    1.I substituted eq 22 into left-hand side of problem's equation. Then I don't know how to go further. I think there will be some manipulation on the equation but I'm lacking some knowledge how to do it.
     
    Last edited: Oct 7, 2011
  2. jcsd
  3. Oct 7, 2011 #2

    dextercioby

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    Your question is a simple case of Plancherel theorem. What is B equal to, if you're given its Fourier transformation ?
     
  4. Oct 7, 2011 #3
    Is this a fourier transform?
     
  5. Oct 7, 2011 #4

    dextercioby

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    Yes. Then what is b equal to ? Can you perform that integration, once you know psi ?
     
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