hw3 - order. Compare with the Taylor’s expansions of the...

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HOMEWORK ASSIGNMENT 3 PHYS852 Quantum Mechanics I, Spring 2008 Topics covered: Time-independent perturbation theory up to 2nd order and the degenerate case 1. Orthogonality : Start from equation (33) in the lecture notes, and prove that in 2nd order non- degenearte perturbation theory the perturbed states are normalized to second order, by showing explicitly a n | n A = 1 + O ( λ 3 ) . Similarly, prove that orthogonality is only guaranteed up to Frst order, by proving a n | ν A = λ 2 s k n = n,ν V nk V E nk E , where n n = ν . 2. Two-level system : ±or the two-level system governed by H = δS z + Ω S x with δ > 0, use pertur- bation theory to compute E 1 and E 2 including terms up to fourth-order in Ω. Expand the exact solution in Taylor series around Ω = 0 and compare the two results. 3. Consider the shifted harmonic oscillator H = P 2 2 M + 2 2 X 2 + aX. Use perturbation theory to compute the eigenvalues to second order in a and the eigenstates to Frst
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Unformatted text preview: order. Compare with the Taylor’s expansions of the exact results. 4. ±ind the Frst non-vanishing corrections to the ground-state energy and wavefunction of the har-monic oscillator when an anharmonic term is added tot he potential. ±irst consider the asymmetric anharmonic oscillator H = P 2 2 M + Mω 2 2 X 2 + λX 3 , then do the same for symmetric anharmonic oscillator H = P 2 2 M + Mω 2 2 X 2 + λX 4 . 5. Consider a 3-dimensional Hilbert space spanned by the states | 1 A , | 2 A , and | 3 A . Let the unperturbed Hamiltonian be H = 16 4 28 4 1 7 28 7 49 . Let the perturbation operator be V = 3 − 5 3 3 − 5 3 . ±ind the eigenvalues of H = H + λV to second order in λ , and Fnd the eigenstates of H to Frst-order....
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This note was uploaded on 11/26/2010 for the course PHYSICS PHYS 852 taught by Professor Michaelmoore during the Spring '10 term at Michigan State University.

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