H.O. with anharmonic perturbation ( $ax^4$).

Anharmonic Oscillator

$$\bgroup\color{black}H_1 = ax^4\egroup$$

We will just write x in terms of $A$ and $A^\dag$ and compute.

$$\bgroup\color{black}\Delta E^{(1)}_n = a\langle\phi_n \vert x^4\vert \phi_n\rangle = a{3\hbar^2\over{4m^2\omega^2}}\egroup$$