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Charged harmonic oscillator in an electric field

  1. Oct 28, 2009 #1
    1. The problem statement, all variables and given/known data

    A charged harmonic oscillator is placed in an external electric field [tex]\epsilon[/tex] i.e. its hamiltonian is [tex] H = \frac{p^2}{2m} + \frac{1}{2}m \omega ^2 x^2 - q \epsilon x [/tex] Find the eigenvalues and eigenstates of energy

    2. Relevant equations



    3. The attempt at a solution

    By completing the square i get
    [tex] [-\frac{\hbar^2}{2m}\frac{d^2}{du^2}+\frac{1}{2}m \omega ^2u^2] \phi (u) = (E + \frac{q^2 \epsilon^2}{2m \omega ^2}) \phi (u) [/tex]
    where
    [tex]u=x-\frac{q^2\epsilon^2}{2m\omega^2}[/tex].

    Then usually for Hamiltonians of this kind the energy eigenvalues are
    [tex]E_n=\hbar\omega(n+\frac{1}{2})[/tex]
    but how do I obtain them in this case? Or is this the right way to go?
    Do i call
    [tex]E + \frac{q^2 \epsilon^2}{2m \omega ^2}=E'[/tex]
    which would give me
    [tex]E'_n=\hbar\omega(n+\frac{1}{2})[/tex]
    And how do I swich back to x?
     
  2. jcsd
  3. Oct 28, 2009 #2

    gabbagabbahey

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    That doesn't look quite right...
     
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