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Homework Help: Time independent perturbation - potential with inner function

  1. Nov 17, 2013 #1
    1. The problem statement, all variables and given/known data
    I need to find the corrected ground state for perturbed harmonic oscillator (1D) with perturbation of the form V(x) = λ sin(κx), κ>0.

    My problem is I have no idea how to handle a potential that has its operator as an inner function.

    2. Relevant equations
    The perturbed Hamiltonian of the system:
    H = H[itex]_{0}[/itex] + λV = [itex]\frac{1}{2}[/itex]mω[itex]^{2}[/itex]x[itex]^{2}[/itex] + λsin(κx).

    At least the first correction to the ground state is defined
    [itex]\sum\frac{<m| λV |0>}{E_{m}-E_{n}}[/itex] where the sum is taken over m, m goes from 1 to infinity? (m and 0 are states, sorry I couldn't get the latex notation working there)

    3. The attempt at a solution
    I tried to use the ladder operators.
    In terms of the ladder operators
    x = [itex]\sqrt{\frac{h}{2mω}}{}(a+a^\dagger)[/itex]
    But the problem is I don't know how to operate the states when the operators are inside sines argument. I could use the taylor series of sin, but it would leave me with (a+a^\dagger)[itex]^{2n+1}[/itex] and that doesn't seem good either.

    Thank you in advance, I'm really stuck here.
  2. jcsd
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