hw7solutions - HOMEWORK ASSIGNMENT 7 PHYS852 Quantum...

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Unformatted text preview: 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 ℓ 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 | nℓsjm j ) states onto | nℓsm ℓ m z ) states. Then use the selection rule m j = m ℓ + m s to show that ( nℓsjm j | L z + 2 S z | nℓ ′ sjm ′ j ) ∝ δ ℓ,ℓ ′ 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 a state with arbitrary n , ℓ , 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 = − α 2 Mc 2 2 bracketleftbigg 1 n 2 + α 2 j + 1 / 2 1 n 3 − 3 α 2 4 1 n 4 bracketrightbigg , (1) where α = 1 / 137, M = 9 . 11 × 10 − 31 kg, and c = 3 . 00 × 110 8 m/s. Clearly states with different ℓ 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 ℓ and m j . The matrix elements of V D = P D V P D are thus ( nℓsjm j | L z + 2 S z | nℓ ′ sjm ′ j ) . (2) If we can verify that they are proportional to δ ℓ,ℓ ′ δ m j ,m ′ j , then we will have proven that V D is diagonal, so that the ‘good states’ are just the | nℓsjm j ) states themselves. To prove the diagonality of V D , we insert the projector onto the | nℓsm ℓ m s ) basis into Eq. (1). This gives ( nℓsjm j | L z + 2 S z | nℓ ′ sjm ′ j ) = summationdisplay ℓ ′′ m ′′ ℓ m ′′ s ( nℓsjm j | ( L z + 2 S z ) | nℓ ′′ sm ′′ ℓ m ′′ s )( nℓ ′′ sm ′′ ℓ m ′′ s | nℓ ′ sjm ′ j ) = summationdisplay ℓ ′′ m ′′ ℓ m ′′ s ( nℓsjm j | ( m ′′ ℓ + 2 m ′′ s ) | nℓ ′′ sm ′′ ℓ m ′′ s ) δ ℓ ′′ ,ℓ ′ ( m ′′ ℓ m ′′ s | jm ′ j ) = δ ℓ,ℓ ′ summationdisplay m ′′ ℓ m ′′ s ( m ′′ ℓ + 2 m ′′ s ) δ ℓ,ℓ ′ ( jm j | m ′′ ℓ m ′′ s )( m ′′ ℓ...
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hw7solutions - HOMEWORK ASSIGNMENT 7 PHYS852 Quantum...

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