Problem10.1, Introductory QM,Liboff.(adsbygoogle = window.adsbygoogle || []).push({});

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.

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# Homework Help: Introductory QM problem in 3D

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