H.O. with anharmonic perturbation ( \bgroup\color{black}$ax^4$\egroup).

We add an anharmonic perturbation to the Harmonic Oscillator problem.

\begin{displaymath}\bgroup\color{black}H_1 = ax^4\egroup\end{displaymath}

Since this is a symmetric perturbation we expect that it will give a nonzero result in first order perturbation theory. First, write x in terms of \bgroup\color{black}$A$\egroup and \bgroup\color{black}$A^\dag $\egroup and compute the expectation value as we have done before.

\Delta E^{(1)}_n
&=& a\langle n \vert x^4\vert n\rangle = {a...
...n(n-1)\right] \\
&=& {3a\hbar^2\over{4m^2\omega^2}}(2n^2+2n+1)

Jim Branson 2013-04-22