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External field applied to Harmonic Oscillator

  1. Jun 14, 2014 #1
    1. The problem statement, all variables and given/known data

    For a particle of charge ##q## in a potential ##\frac{1}{2}m\omega^2x^2##, the wavefunction of ground state is given as ##\phi_0 = \left( \frac{m\omega }{\pi \hbar} \right)^{\frac{1}{4}} exp \left( -\frac{m\omega}{2\hbar} x^2 \right)##.

    Now an external electric field ##E## is applied.

    Part (a): Find the new energies and wavefunction of the ground state.

    Part (b): Find the probability that the particle will be in the ground state of the new potential.

    2. Relevant equations



    3. The attempt at a solution

    The Hamiltonian now becomes:

    [tex]H = \frac{p^2}{2m} + \frac{1}{2} m\omega^2 \left(x - \frac{qE}{m\omega^2} \right)^2 - \frac{q^2E^2}{2m\omega^2} [/tex]

    Thus the shift in energy is ## \frac{q^2E^2}{2m\omega^2} ##. New energies are given by: ##E_n = (n+1)\hbar \omega - \frac{q^2E^2}{2m\omega^2} ##.

    This represents a displaced harmonic oscillator, with eigenfunction:

    [tex]\phi_0 \space ' = \left( \frac{m\omega}{\hbar \pi} \right)^{\frac{1}{4}} e^{-\alpha (x-\frac{qE}{m\omega^2})^2 } [/tex]

    Part(b)

    Do I overlap this state with the old one and integrate?
     
    Last edited: Jun 14, 2014
  2. jcsd
  3. Jun 14, 2014 #2
    Edit: yes
    Edit take two: may I suggest to edit for typos not to change completely the questions?
     
    Last edited: Jun 14, 2014
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