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Midterm 1 sol - Suggested Solutions to Midterm 1 Problem...

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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 λ : H = H + λH , ψ n = ψ (0) n + λψ (1) n + ... , E n = E (0) n + λE (1) n + ... . Thus, keeping terms up to linear order in λ , we have ( H + λH ) ψ (0) n + λψ (1) n = E (0) n + λE (1) n ψ (0) n + λψ (1) n (1) The terms linear in λ can be collected to yield H ψ (1) n + H ψ (0) n = E (1) n ψ (0) n + E (0) n ψ (1) n (2) Taking the inner product with h ψ (0) n | , we have E (1) n = h ψ (0) n | H | ψ (0) n i (3) (b)Assuming completeness, we can write the trial wavefunction ≡ ψ t as a sum of the energy eigenstates of the Hamiltonian. Thus, ψ t = ∑ n c n ψ n , and since it is normalized, we also have ∑ n | c n | 2 = 1. Taking the expectation value of the Hamiltonian, we have: h ψ t | H | ψ t i = X n,m c * n c m h ψ n | H | ψ 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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Midterm 1 sol - Suggested Solutions to Midterm 1 Problem...

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