Operators for a Perturbed Hamiltonian: Heisenberg Picture ($\hat{x}$, $\hat{p}$) 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 $\lambda$
 A: Basically, you do need to treat it as a perturbation and no correction is necessary...
Calculating the commutator,
        \begin{equation}
   \left[\hat{x}, \hat{H}\right]
   =
   \left[
    \hat{x},
    \frac{\hat{p}^2}{2m} 
    - \frac{k}{2}\hat{x}^2 
    + \frac{\lambda}{4}\hat{x}^4
   \right]
  \end{equation}
        but as $\left[\hat{x},\hat{x}\right]=0$,
        \begin{equation}
   \left[\hat{x}, \hat{H}\right]
   =
   \left[
    \hat{x},
    \frac{\hat{p}^2}{2m}
   \right]
   = \frac{1}{2m}\left[\hat{x},\hat{p}\hat{p}\right]
   =
   \frac{1}{2m}
   \left[\hat{x},\hat{p}\right]\hat{p}
   +
   \frac{1}{2m}
   \hat{p}\left[\hat{x},\hat{p}\right]
  \end{equation}
        rmembering I'm setting $\hbar=1$ so we get,
        \begin{equation}
   \tag{1}
   \left[\hat{x}, \hat{H}\right]
   =
   \frac{i}{2m}
   \hat{p}
   +
   \frac{i}{2m}
   \hat{p}
   =
   \frac{i}{m}
   \hat{p}
  \end{equation}
        Using Equation $(1)$ with the Heisenberg equations of motion,
        \begin{equation}
   \frac{\partial}{\partial t} \hat{x}
   = 
   -i\left[\hat{x}, \hat{H}\right]
   =
   \frac{\hat{p}}{m}
  \end{equation}
The same can be done for $\hat{p}$. The two can then be related to get two pairs of $1^\text{st}$ and $2^\text{nd}$ order coupled ODEs
A: It seems like your answer sidestepped the whole question.
When you do the same for $\hat p$ you'll find that its derivative depends on $\hat x$, and on $\lambda$.  
But these coupled equations can then be solved as a second order equation for the terms individually, which should be what you are looking for.
