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Unformatted text preview: Chem 120A Degenerate Perturbation Theory & the Variational Princi- ple 04/05/06 Spring 2006 Lecture 29 READING: Engel (CS): Continue with Ch. 10-11 (we will return to these later) The course notes! Additional optional reading: McQuarrie and Simon, Physical Chemistry , Ch. 7-8 See website for additional supplemental reading. Degenerate perturbation theory In the last class, we considered time-independent perturbation theory in the case where the eigenstates of the Hamiltonian, H , are not degenerate. If we were to try to use the expressions that we derived in Lecture 28 for a system where there are degeneracies there would be problems. One problem is that the second order correction to the energy and the first order correction to the eigenfunction would both blow up since p n would equal zero when the states n and p are degenerate. This means that the theory developed in Lecture 28 breaks down in the case where there are degeneracies in the eigenstates of H . Our perturbed Hamiltonian is given by H = H + V . (1) Suppose that there is an energy level for H , n , which has a g-fold degeneracy (i.e. there are g states with energy n .) We can construct any superposition of these states and it will also be an eigenstate of the Hamiltonian with energy n , n = g s = 1 d ns ns , (2) H n = n n . (3) where n is the energy label, s is a label for the degeneracy, and...
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