Problem
I am trying to calculate the Equations of Motion in the Heisenberg picture for $\hat{x}$ and $\hat{p}$ in a perturbed Hamiltonian, $$ \tag{1} \hat{H} = \hat{H}_0 + \hat{H}' $$
Assume the Heisenberg Equations of Motion are given by,
$$ \tag{2} \frac{\partial \hat{A}(t)}{\partial t} = -i\left[ \hat{A}(t),\hat{H} \right] $$ with $\hbar=1$ to simplify.
Issue
Can I use the operators $\hat{x}(t)$, $\hat{p}(t)$ for the unperturbed hamiltonian, $\hat{H}_0$, in Equation $(2)$ to get the perturbed equations of motion?
- Is this the correct approach?
- Do I need a correction in $\hat{x}(t)$, $\hat{p}(t)$ for the perturbation?
Specifics
In my case, $\hat{H}_0$ is an oscillator such that,
$$ \hat{H}_0 = \frac{1}{2m}\hat{p}^2 + \frac{k}{2}\hat{x}^2 $$ and the perturbation, $\hat{H}'$, is quartic in $\hat{x}$ like $\lambda\frac{1}{4}\hat{x}^4$ with a coupling constant of $\lambda$