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Harmonic oscillator

  1. Aug 21, 2014 #1
    1. The problem statement, all variables and given/known data

    A harmonic oscillator of mass m and angular frequency ω experiences the potential:

    V(x) = 1/2m[itex]ω^{2}x^{2}[/itex] between -infinity < x < +infinity

    and solving the schrodinger equation for this potential yields the energy levels

    E_n = (n + 1/2) h_bar ω

    Determine the energy levels for the half oscillator for which

    V(x) = 1/2m[itex]ω^{2}x^{2}[/itex] between -infinity < x < 0

    = infinity otherwise

    3. The attempt at a solution

    -h_bar^2/2m *d^2ψ(x)/dx^2 + 1/2mω^2x^2 = Eψ(x)

    so d^2ψ(x)/dx^2 = -(E - 1/2mω^2x^2)*2m/h_bar^2 ψ(x) ==> d^2ψ(x)/dx^2 = k^2ψ(x)

    So the general solution is ψ(x) = Ae^kx + Be^-kx
    1. The problem statement, all variables and given/known data

    2. Relevant equations

    3. The attempt at a solution
  2. jcsd
  3. Aug 21, 2014 #2


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    Your k is not constant and generally depends on x, which means your differential equation is more difficult than that to solve.

    The fact that you have been given the energy levels for the full oscillator should be a hint. Can you think of a way to relate the problem of the half-oscillator to the full oscillator?
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