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Particles Mean Momentum

  1. Jun 9, 2007 #1
    1. The problem statement, all variables and given/known data

    Particle in a 1D well of infinite depth, width 2a has wavefunction:
    [tex]\psi = Asin(k_{n}x)[/tex]
    Its prepared in state:
    [tex]\psi(x,t) = \frac{1}{\sqrt{2}} [\psi_{0}(x)+i\psi_{1}(x)][/tex]
    I need to find the mean momentum as a function of time ie. prove this:
    [tex] <\hat{p_{x}}(t)> = \frac{4\hbar}{3a} sin[\frac{3\hbar\pi^{2}}{8ma^{2}}t] [/tex]

    2. Relevant equations

    I've worked out [tex]k_{n}[/tex] and the relevant normalisation constants...But I don't think its important to state them for my question since I need to know how to attempt this problem.


    3. The attempt at a solution

    I thought about doing it like this:
    [tex]<\hat{p_{x}}>=\int\psi^*(x,t)\hat{p_{x}}\psi(x,t)\dx[/tex]
    where
    [tex]\hat{p_{x}}=-i\hbar\frac{d}{dx}[/tex]
    ...but this became far too tedious and I don't know where the 't' is meant to come in...
    I also tried doing:
    [tex]<\hat{p_{x}}>=m<\frac{d\hat{x}}{dt}>[/tex]
    (Ehrenfest's Theorem) but no luck...

    Any guidance will help...
     
  2. jcsd
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