Perturbation from a quantum harmonic oscillator potential

[Mayan Fung]

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?

 

[vela]

Try using the creation and annihilation operators.

 

[Mayan Fung]

Try using the creation and annihilation operators.

Thanks! It helps a lot.

I also tried to solve it analytically by substituting:
$$\hat{p'} = \hat{p} + \hat{q}$$
$$\hat{q'} = \hat{p} - \hat{q}$$
I finally got something like $\hat{H} = a\hat{p'} + b\hat{q'}$

However, $p',q'$ are composed of both p and q. How I can get the wavefunction in the q space?