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Perturbation Theory - Shift of Ground State

I can guarantee that there's a section on time-independent perturbation theory in your textbook. You want to rewrite your Hamiltonian in the form

$$\hat{H} = \hat{H}_0 + \lambda \hat{H}^{(1)},$$

where $$\lambda$$ is a small, dimensionless constant. To do this, it might be convenient to note that the average momentum of the electron is small compared to its mass (times c). Therefore $$\lambda$$ is conveniently written in terms of a ratio of a scale corresponding to the average momentum over the mass. $$\hat{H}_1$$ can then be written in terms of $$\hat{V}$$.

You really should find the relevant discussion of PT in your text, but the basics of the first order correction are that the expansions in $$\lambda$$ of the ground state wavefunction and energy eigenvalue are

$$E_0 = E_0^{(0)} + \lambda E_0^{(1)} + \cdots ,$$

$$\psi_0 = \psi_0^{(0)} + \lambda \psi_0^{(1)} + \cdots ,$$

The superscripts correspond to the order in the expansion in powers of $$\lambda$$ and the question is asking you to compute $$E_0^{(1)}$$. You should try to find the relevant term in the $$\lambda$$ expansion of

$$(\psi_0 , \hat{H} \psi_0) = ( \psi_0^{(0)} + \lambda \psi_0^{(1)} + \cdots, (\hat{H}_0 + \lambda \hat{H}^{(1)}) ( \psi_0^{(0)} + \lambda \psi_0^{(1)} + \cdots) ).$$