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Quantum harmonic potential problem

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

    Consider a particle of mass m in a harmonic potential:
    gif.latex?V(x)%20%3D%20%5Cfrac%7B1%7D%7B2%7Dm%5Comega%5E2x%5E2.gif

    If the particle is in the first excited state (n = 1), what is the probability of finding the
    particle in the classically excluded region?

    2. Relevant equations

    gif.latex?%5Cint%5E%7B%5Cinfty%7D_%7B%5Csqrt%7B3%7D%7Dx%5E2e%5E%7B-x%5E2%7Ddx%3D0.0495.gif

    7B%5Cpartial%20x%5E2%7D%20%3D%20%5Cpsi(%5Cfrac%7Bm%5E2%5Comega%5E2x%5E2-2mE%7D%7B%5Chbar%5E2%7D).gif

    3. The attempt at a solution

    I sub in
    ga%5E2%7D%7B%5Chbar%5E2%7D%0A%5C%5C%0A%5C%5C%0A%5Cbeta%20%3D%20%5Cfrac%7B2mE%7D%7B%5Chbar%5E2%7D.gif

    and get a wave function:

    7B%5Cfrac%7B1%7D%7B4%7D%7D%5Csqrt%7B2%5Calpha%7Dxe%5E%7B%5Cfrac%7B-%5Calpha%20x%5E2%7D%7B2%7D%7D.gif

    But I don't know how to set my bounds for the normalization integral.

    I've been advised that the classical limits are:
    gif.gif

    But I'm still stuck.
     
  2. jcsd
  3. Aug 1, 2014 #2
    theWapiti - the classically excluded region is where the potential [itex]V(x)[/itex] exceeds the total energy of the system, which in this case is [itex]\frac{3}{2}\hbar\omega[/itex]. You need to find out how much of your wavefunction lies in this region. The integral given will probably come in useful for doing that.

    [I suggest the powers that be move this thread to "Advanced Physics Homework"]
     
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