Perturbed Hamiltonian Matrix for Quantum Harmonic Oscillator

  • #1
14
0
Homework Statement

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?

Thanks
 

Answers and Replies

  • #2
DrClaude
Mentor
7,607
4,009
You need to calculate ##\langle i | \hat{H}' | j \rangle = -2 \langle i | \cos(\pi x) | j \rangle##.
 

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