[QUE/QM-09010]

\(\newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}}\)
\(\newcommand{\dd}[2][]{\frac{d #1}{d #2}}\) \(\newcommand{\ket}[1]{\vert#1\rangle}\)
Consider the anharmonic oscillator problem with Hamiltonian \[H = \frac{p^2}{2m} + \frac{1}{2}k x^2 + \lambda x^4 \] Investigate if we can define  a new picture such that the operators evolve according to the equation \begin{eqnarray}   \dd[X_n]{t} &=&\pp[X_n]{t} + \frac{1}{i\hbar }\big[X_n,H_1]\\     H_1&=&\frac{p^2}{2m_1} + \frac{1}{2}k x^2 \end{eqnarray} and the states evolve according to the Schrodinger equation  \begin{eqnarray}   i\hbar \dd[\ket{\psi\,t}_n]{t} &=& H_2 \ket{\psi\,t}_n\\   H_2  &=& \frac{p^2}{2m_2} + \lambda x^4 \end{eqnarray}  where the suffix \(n\) means states and operators in the new picture. If such a picture exists, what should be the relation between \(m_1\) and \(m_2\)?