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Expectation Value of Momentum for Wavepacket

  1. Jan 16, 2014 #1
    1. The problem statement, all variables and given/known data

    What is the average momentum for a packet corresponding to this normalizable wavefunction?

    [itex]\Psi(x) = C \phi(x) exp(ikx) [/itex]

    C is a normalization constant and [itex]\phi(x) [/itex] is a real function.

    2. Relevant equations
    [itex]\hat{p}\rightarrow -i\hbar\frac{d}{dx}[/itex]


    3. The attempt at a solution

    [itex]\int\Psi(x)^{*}\Psi(x)dx = \int C^2 \phi(x)^{2}dx= 1 [/itex]

    Plugging in the momentum operator and using the chain rule:

    [itex]<\hat{p}> = \hbar k \int C^2 \phi(x)^2 dx - i \hbar \int C^2 \phi^{'}\phi dx [/itex]

    The second term is always imaginary since [itex]\phi(x)[/itex] is real, so I said the momentum is [itex]\hbar k[/itex] which I think might be right, but for the wrong reasons? I didn't think Hermetian operators could give imaginary expectation values...
     
  2. jcsd
  3. Jan 16, 2014 #2

    strangerep

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    Try working on the 2nd term a bit more. Hint: use integration by parts.
     
  4. Jan 16, 2014 #3
    Are you saying that the second term must be zero since [itex]\phi[/itex] vanishes at ±∞ and the integral evaluates to [itex]\phi^2(x)/2[/itex]? That makes sense to me.
     
  5. Jan 16, 2014 #4

    strangerep

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    That's the idea.
     
  6. Jan 16, 2014 #5
    Awesome. Thanks for the help.
     
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