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Showing f is a solution to quantum oscillator SWE

  1. Feb 16, 2016 #1
    1. The problem statement, all variables and given/known data
    For a 1-dimensional simple harmonic oscillator, the Hamiltonian operator is of the form:
    H = -ħ2/2m ∂xx + 1/2 mω2x2
    n = Enψn = (n+1/2)ħωψn

    where ψn is the wave function of the nth state.

    defining a new function fn to be:

    fn = xψn + ħ/mω ∂xψn

    show that fn is a solution to the low energy SWE. I.e. that:

    Hfn = (En - ħω)fn

    3. The attempt at a solution
    I know that:
    ψn = Cne-mωx2/2Hn(x)
    where Hn(x) is an nth order Hermite Polynomial.
    I was wondering what the procedure for determining this would be. I'm assuming I could plug this expression for ψn into the expression for fn, and then apply the Hamiltonian operator. But I'm not sure how that would simplify, and it also seems very complicated to plug in. Any help would be appreciated
  2. jcsd
  3. Feb 16, 2016 #2


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    I believe you are meant to let the operator H act on the expression for fn and show that Hfn reduces to (En - ħω)fn.

    It will require a fair amount of manipulation and use of the fact that Hψn = Enψn. But nothing else is needed.
    Last edited: Feb 16, 2016
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