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 ˆˆ 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* 11 2 100 22 * 30 1 21 0 0 2 0 2 100 2 100 sin cos sin ,, s i n xy l m mi ll lm lm Q iQ d r i rd rR r R r d Y Y e φ ψ ψθ θφ θ ⋅= + = + ∫∫ Q ε rr r Of course, 0 0 14 Y π = , and comparison of the final factors with spherical harmonics makes it clear that it is simply ( ) 1 1 sin 8 3 , i eY πθ =− , so when we put this all together, we have () () 00 * 0 1 0 2 33 1 1 21 10 53 5 0 01 1 4 0 8 2 100 , , 3 24 2 1 3 324 2 128 2 4! 3 243 32 m ra r a lm lm r drR r R r d Y Y rdrR r R r rdr re e aa a a a δδ ∞∞ −− Ω ⎛⎞ ⎜⎟ ⎝⎠ Q ε Doing it by hand seemed easier than using Maple. We used orthogonality of the Y ’s to do the angular integral.
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This note was uploaded on 02/15/2011 for the course PHYS 742 taught by Professor Unknow during the Spring '04 term at Harvard.

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

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