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Quantum harmonical oscillator with electric field

  1. Jan 21, 2006 #1
    Hi,

    I have a particle of mass m and charge q, which is located in the potential of an harmonic oscillator and also subject to a constant electric field. The Hamiltonian is given as:

    [tex]H = \frac{p^2}{2m} + \frac{1}{2}m \omega ^2 x^2 - q E' x[/tex]

    And I need to find a change of variables from x to u, so that the eigenvalue equation:

    [tex]H \phi (x) = E \phi (x)[/tex]

    Becomes:

    [tex][-\frac{h^2}{2m}\frac{d^2}{du^2}+\frac{1}{2}m \omega ^2u^2] \phi (u) = (E + \frac{q^2 E'^2}{2m \omega ^2}) \phi (u)[/tex]

    (It's an h-bar there, of course.) I don't even know where to start. I tried plugging u(x) into the original eigenvalue equation and find some constraint on u from there, to no avail.

    Thanks
     
    Last edited: Jan 21, 2006
  2. jcsd
  3. Jan 21, 2006 #2

    George Jones

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    Complete the square on the last 2 terms in the Hamiltonian, and the transformation might become a bit more obvious.

    Regards,
    George
     
  4. Jan 21, 2006 #3
    Doh... thanks! :smile:
     
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