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Oct20-10, 01:47 PM
Sci Advisor
HW Helper
PF Gold
P: 2,603
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

[tex]\hat{H} = \hat{H}_0 + \lambda \hat{H}^{(1)}, [/tex]

where [tex]\lambda[/tex] 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 [tex]\lambda[/tex] is conveniently written in terms of a ratio of a scale corresponding to the average momentum over the mass. [tex]\hat{H}_1[/tex] can then be written in terms of [tex]\hat{V}[/tex].

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 [tex]\lambda[/tex] of the ground state wavefunction and energy eigenvalue are

[tex] E_0 = E_0^{(0)} + \lambda E_0^{(1)} + \cdots , [/tex]

[tex] \psi_0 = \psi_0^{(0)} + \lambda \psi_0^{(1)} + \cdots , [/tex]

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

[tex] (\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) ).[/tex]