Fall 2006 6 Qm

A one dimensional harmonic oscillator has momentum p, mass m, and angular frequency $\omega$. It is subject to a perturbation with a potential energy $U=\lambda x^4$ where $\lambda$ is suitably small so that perturbation theory is applicable.

(a) Derive the expressions for $a$ and $a^{\dagger}$ in terms of x and p using the fact that they satisfy $\left [ a,a^{\dagger} \right ]=1,\;H=\hbar \omega (a^{\dagger} a + 1/2)$.

(b) Calculate the energy shift $\Delta E_n$ of the state $|n \rangle$ due to the perturbation to the first order in $\lambda$, using creation and annihilation operators.

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