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Perturbed Hamiltonian Matrix for Quantum Harmonic Oscillator

  1. Nov 1, 2016 #1
    The problem statement, all variables and given/known data

    How to calculate the matrix elements of the quantum harmonic oscillator Hamiltonian with perturbation to potential of [itex]-2cos(\pi x)[/itex]
    The attempt at a solution
    [itex] H=H_o +H' [/itex] so [itex] H=\frac{p^2}{2m}+\frac{1}{2} m \omega x^2-2cos(\pi x) [/itex]

    I know how to find the matrix of the normal Hamiltonian as [itex]H \psi_j =E_j \psi_j[/itex] then [itex] H_{ij}=<i|H|j>=E_j\delta_{ij}=(j+1/2)\hbar \omega \delta_{ij}[/itex] therefore we get 1/2, 3/2,5/2 etc on the diagonal. However i am not sure how to apply this to this situation. Can I obtain the matrix just from here or do I need to do perturbation theory first?

  2. jcsd
  3. Nov 2, 2016 #2


    User Avatar

    Staff: Mentor

    You need to calculate ##\langle i | \hat{H}' | j \rangle = -2 \langle i | \cos(\pi x) | j \rangle##.
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