hw7 - HOMEWORK ASSIGNMENT 7 PHYS852 Quantum Mechanics II...

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HOMEWORK ASSIGNMENT 7 PHYS852 Quantum Mechanics II, Spring 2008 New topics covered: Green’s functions, T-matrix . 1. ‘Weak-field Zeeman Effect’ : Consider a hydrogen atom in a uniform magnetic field. Assume that the Zeeman shift is large compared to the hyperfine splitting, but smaller than the fine-structure shift. Thus we can ignore nuclear spin (i.e. the hyperfine structure) and consider the magnetic field as a perturbation on the fine structure. The fine-structure lifts the degeneracy of states with different j , so the bare eigenstates are | nℓsjm j ) states. Use your previous work, and/or reference materials, to find the bare energy eigenvalues, which should depend on n and j only. Show that the states with the same n , s and j , but different and m j are degenerate. Now add the magnetic field interaction V = eB 0 2 M ( L z + 2 S z ). What are the good eigenstates? (hint: Use Clebsch Gordan coefficients to expand the | nℓsjm j ) states onto | nℓsm m z ) states. Then use the selection rule m j = m + m s to show that ( nℓsjm j | L z + 2 S z | nℓ sjm j ) ∝ δ ℓℓ δ m j ,m j . This result should allow you to deduce the good basis). Now use first-order perturbation theory to compute the Zeeman sublevels of
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