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Unformatted text preview: Suggested Solutions to Midterm 1 Problem 1 (a)Expand the wavefunctions and energy eigenvalues in terms of the perturbation parameter : H = H + H , n = (0) n + (1) n + ... , E n = E (0) n + E (1) n + ... . Thus, keeping terms up to linear order in , we have ( H + H ) (0) n + (1) n = E (0) n + E (1) n (0) n + (1) n (1) The terms linear in can be collected to yield H (1) n + H (0) n = E (1) n (0) n + E (0) n (1) n (2) Taking the inner product with h (0) n | , we have E (1) n = h (0) n | H | (0) n i (3) (b)Assuming completeness, we can write the trial wavefunction t as a sum of the energy eigenstates of the Hamiltonian. Thus, t = n c n n , and since it is normalized, we also have n | c n | 2 = 1. Taking the expectation value of the Hamiltonian, we have: h t | H | t i = X n,m c * n c m h n | H | m i = X n | c n | 2 E n X n | c n | 2 E = E (4) (c)A sufficient condition is that: the basis of states used to perform the perturbation scheme is a...
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This note was uploaded on 11/21/2010 for the course PHYSICS 137B taught by Professor Moore during the Spring '07 term at University of California, Berkeley.
- Spring '07