QFT for Gifted Amateur - Problem 2.2

[Daniel_C]

Homework Statement

For the Hamiltonian $$\hat{H} = \frac{\hat{p}^2}{2m} + \frac{1}{2} m \omega^2 \hat{x}^2 + \lambda \hat{x}^4$$ where lambda is small, use the creation annihilation operators with perturbation theory to find the energy eigenvalues of all the levels.

Relevant Equations

$$\hat{a} = \sqrt{\frac{m\omega}{2\hbar}} \left( \hat{x} + \frac{i}{mw} \hat{p} \right)$$
$$\hat{a}^\dagger = \sqrt{\frac{m\omega}{2\hbar}} \left( \hat{x} - \frac{i}{mw} \hat{p} \right)$$

I'm getting confused by the perturbation theory aspect of problem 2.2 in this book. We have to show that the energy eigenvalues are given by

$$E_n = \left(n + \frac{1}{2}\right) \hbar \omega + \frac{3\lambda}{4} \left(\frac{\hbar}{m\omega}\right)^2 (2n^2 + 2n + 1)$$

For the Hamiltonian

$$\hat{H} = \frac{\hat{p}^2}{2m} + \frac{1}{2} m \omega^2 \hat{x}^2 + \lambda \hat{x}^4$$

I'm going to try and apply first-order time-indepedent non-degenerate perturbation theory to this problem and use the equation
$$E^'_n = < n^0 | H' | n^0>$$
where H' is the perturbation of the Hamiltonian.

I've identified the perturbation of the Hamiltonian and have arrived at

$$E^'_n = < n^0 | \lambda \hat{x}^4 | n^0>$$

I'm getting confused as to start actually calculating from this point I've set myself up at with all the new creation and annihilation operators they've introduced. Could someone help put me on the right track?

My best guess would just be to explicitly evluate the integral by plugging in the quantum harmonic oscillator wave functions.

 

[kuruman]

What is the perturbation Hamiltonian in terms of $a$ and $a^{\dagger}$? Use that to calculate the first order perturbation energies. If you run into more problems, please post your work, not just the bottom line.