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Expectatoon value particle in superposition of momentum states

  1. Jan 9, 2012 #1
    1. The problem statement, all variables and given/known data

    Demonstrate the relation between the expectation value and the measurement outcomes of an observable of a particle by conisdering as an observable the kinetic energy operator
    E=p^/2m when the particle is in a superposition of 2 momentum eigenstates
    2. Relevant equations

    <O> = Int (from -inf -> inf) [(Psi*)O(Psi)] dx

    3. The attempt at a solution

    I am taking the superposition of 2 momentum eigenstates as

    Psi= square root (1/L) [ A*exp(ikx)exp(-iEt/Hbar) +B*exp(ikx)exp(-iEt/Hbar) ]

    And then putting this into the integral

    <O> = Int (from 0->L) [(Psi*)(-hbar/2m*d2/dx2(Psi)] dx

    However I end up with a very long equation for the expectation value whereas I thought the expectation value would be something along the lines of
    A2hbar2k12/2m + B2hbar2k22/2m as this looks like an eigenvalue
  2. jcsd
  3. Jan 14, 2012 #2
    I'd take out the [itex]\sqrt{1/L}[/itex] since there's no reason to consider any sort of box here (and besides you can absorb it into the [itex]A[/itex] and [itex]B[/itex] terms). Get rid of the time-dependent bit (since you're going to ignore it anyway - you're integrating over [itex]x[/itex]) and make sure you label your two [itex]k[/itex]s differently, like you have in your final suggestion: [itex]k_1[/itex] and [itex]k_2[/itex]. And then the approach you're using should work! You seem to have some idea what you expect to find, which is good - if you can't get there post where you get up to.
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