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Unformatted text preview: HOMEWORK ASSIGNMENT 7 PHYS852 Quantum Mechanics II, Spring 2008 New topics covered: Greens 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 | nsjm 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 2 M ( L z + 2 S z ). What are the good eigenstates? (hint: Use Clebsch Gordan coefficients to expand the | nsjm j ) states onto | nsm m z ) states. Then use the selection rule m j = m + m s to show that ( nsjm j | L z + 2 S z | n sjm j ) , 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 a state with arbitrary n , , and j . Answer : From Eq. (138) in the TIPT notes, we see that the bare-energies of the fine-structure eigenstates are given by E n,j = 2 Mc 2 2 bracketleftbigg 1 n 2 + 2 j + 1 / 2 1 n 3 3 2 4 1 n 4 bracketrightbigg , (1) where = 1 / 137, M = 9 . 11 10 31 kg, and c = 3 . 00 110 8 m/s. Clearly states with different and m j , but the same n and j are degenerate, since the bare energies depend on n and j only. To find the good eigenstates of V = eB 2 M ( L z + 2 S z ), we need to diagonalize V in the degenerate subspace. The degenerate subspace consists of all states with the same n , s , and j , but different and m j . The matrix elements of V D = P D V P D are thus ( nsjm j | L z + 2 S z | n sjm j ) . (2) If we can verify that they are proportional to , m j ,m j , then we will have proven that V D is diagonal, so that the good states are just the | nsjm j ) states themselves. To prove the diagonality of V D , we insert the projector onto the | nsm m s ) basis into Eq. (1). This gives ( nsjm j | L z + 2 S z | n sjm j ) = summationdisplay m m s ( nsjm j | ( L z + 2 S z ) | n sm m s )( n sm m s | n sjm j ) = summationdisplay m m s ( nsjm j | ( m + 2 m s ) | n sm m s ) , ( m m s | jm j ) = , summationdisplay m m s ( m + 2 m s ) , ( jm j | m m s )( m...
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