852HW3 - Then use Frst-order perturbation theory to compute...

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PHYS852 Quantum Mechanics II, Spring 2009 HOMEWORK ASSIGNMENT 3: 1. Cohen-Tannoudji problem 10.1, page 1086 2. Cohen-Tannoudji problem 10.5, page 1087 3. The exact normalized eigenvalues and eigenstates of the hamiltonian H = δS z + Ω S x are ω ± = ± 1 2 r δ 2 + Ω 2 | ω ± a = p ± δ 2 + Ω 2 + δ P | ↑a + Ω | ↓a R p ± δ 2 + Ω 2 + δ P 2 + Ω 2 expand these eigenvalues and eigenvectors to second order in Ω. (I recommend you use the command “Series” in Mathematics for this). Use perturbation theory to compute the energy eigenvalues to third-order and the eigenvectors to second order. You will know you have done it correctly when your answer agrees with the series expansion of the exact result. 4. According to special relativity, the energy-momentum relation is E ( P ) = M 2 c 4 + c 2 P 2 . To leading order in P , this gives E = Mc 2 + P 2 2 M , which is the origin of the kinetic energy term in non-relativistic quantum mechanics. Using T = E Mc 2 , as the deFnition of the kinetic energy, compute the next term in the series expansion of the energy to Fnd the Frst relativistic correction to the kinetic energy.
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Unformatted text preview: Then use Frst-order perturbation theory to compute the energy shift to the ground state of a harmonic oscillator with frequency ω . Here you should use H = P 2 2 M + Mω 2 2 X 2 , and for the perturbation V use your relativistic correction to the kinetic energy. Compute the ratio E (1) /E (0) , and evaluate numerically for an electron, taking the harmonic oscillator length to be the Bohr radius. Do you expect this to be a good estimate of the relativistic correction to the hydrogen atom ground-state energy? 5. A three-state quantum system is described by the Hamiltonian matrix H = 0 0 0 0 1 0 0 0 2 . The system is perturbed by the operator V = 0 1 2 1 0 1 2 1 0 . ±or the Hamiltonian H = H + λV, compute the eigenvalues to third order and the eigenvectors to second-order. 1...
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This note was uploaded on 10/25/2010 for the course PHYSICS PHYS 851 taught by Professor Michaelmoore during the Fall '08 term at Michigan State University.

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