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solo(1)

# solo(1) - Physics 742 Graduate Quantum Mechanics 2 Solution...

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Physics 742 – Graduate Quantum Mechanics 2 Solution Set O 1. [20] A hydrogen atom is in the 1s ground state while being bathed in light of sufficient frequency to excite it to the n = 2 states. The light is traveling in the + z direction and has circular polarization, ( ) 1 2 ˆ ˆ i = + ε i j . (a) [10] Calculate all relevant dipole moments FI r for final states 2 lm . The matrix elements we need are ( ) ( ) ( ) [ ] ( ) ( ) ( ) ( ) 3 * 1 1 2 100 2 2 * 3 0 1 2 10 0 2 0 2 100 2 100 sin cos sin , , sin x y lm m i l l lm lm Q iQ d r i r drR r R r d Y Y e φ ψ ψ θ φ φ θ φ θ φ θ = + = + = Ω Q ε r r r Of course, 0 0 1 4 Y π = , and comparison of the final factors with spherical harmonics makes it clear that it is simply ( ) 1 1 sin 8 3 , i e Y φ θ π θ φ = − , so when we put this all together, we have ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 0 * 3 0 2 10 1 0 2 3 3 1 1 1 1 21 10 5 3 0 0 0 0 5 1 1 0 0 1 1 4 0 1 1 8 2 100 , , 3 2 4 2 1 3 3 24 2 128 2 4! 3 243 3 2 m l l r a r a l m l m l m l m lm r drR r R r d Y Y r drR r R r r dr re e a a a a a π θ φ θ φ π δ δ δ δ δ δ δ δ = − Ω = − = −

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