Really sorry for this simple question, but I think it will be useful/interesting in general.
Consider a quantum simple harmonic oscillator. Add a perturbation $H_I = -\lambda \hat{x}$
Calculate $\langle \psi| \hat{x} |\psi \rangle$ in the new ground state, to first order in perturbation theory in $\lambda$, and relate it to $\langle 0| x^2 |0 \rangle $ where $| 0 \rangle$ is the unperturbed ground state.
Can someone help me on this
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