- #1

- 131

- 14

- Homework Statement
- Given the hamiltonian ##\hat{H}##:

$$\hat{H} = \frac{\hat{p}^2}{2} + \frac{\hat{q}^2}{2} + \alpha (\hat{p}\hat{q}+\hat{q}\hat{p})$$

Approximate the eigenvectors and eigenvalues numerically using the known eigenvectors with ##\alpha = 0## (That means I am asked to truncate the infinitely large matrix and find the eigenvectors and eigenvalues of the truncated matrix)

- Relevant Equations
- $$ E = \hbar \omega (n+\frac{1}{2}) $$

For the off-diagonal term, it is obvious that (p^2+q^2) returns 0 in the integration (##<m|p^2+q^2|n> = E<m|n> = 0##). However, (pq+qp) seems to give a complicated expression because of the complicated wavefunctions of a quantum harmonic oscillator. I wonder whether there is a good method to compute the Hmn terms?