Perturbed Hamiltonian Matrix for Quantum Harmonic Oscillator

[Luke1121]

Homework Statement

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

I know how to find the matrix of the normal Hamiltonian as H \psi_j =E_j \psi_j then H_{ij}=<i｜H｜j>=E_j\delta_{ij}=(j+1/2)\hbar \omega \delta_{ij} 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

 

[DrClaude]

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