Question

1. De Generalization of De Generacy In class, we argued that the first-order corrections to the energies of d degenerate states are given by the eigenvalues of the matrix H., and the eigenvectors give us the "correct" set of states in the degenerate sub-space. These claims were based on working out the d=2 case explicitly and then generalizing the results in an obvious' way. For this problem, prove that these claims are true by considering a set of d degenerate states, {9}, with j = 1, 2, 3, ..., d, that obey Hº 4= E° 45° with (3743) = dij In analogy with Eq. 7.17 in the text, consider the linear combinations d =l Now follow the steps used in section 7.2.1 in the text to arrive at the generalized form of the eigenequation (Eq. 7.27): HQ = Eta; where Hj = (?|H'|6:9) i= Note that this is a more concise and general expression than that given in section 7.2.3 in the text.

Answer 1

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