17hw1.pdf - Graduate QM(PHYS 519 Problem Set#1 Due Friday...

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Graduate QM (PHYS 519) Problem Set #1 Due Friday March 31 1. Simplified photoelectric effect The ground state of a hydrogen atom is subject to the time-dependent potential: V ( r , t ) = V 0 cos( kz - ωt ). Use TDPT to obtain an expression for the transition rate at which an electron is emitted with momentum p . Show how you may compute the angular distribution of the ejected electron. You may take the final electron wave function to be ψ f ( r ) = L - 3 / 2 e i p · r . 2. Sum rules This problem concerns sum rules, which are very useful in the analysis of radiation by atoms, molecules, nuclei and particles. Before proceeding with the problem, we give an example. Define the x oscillator strength of a transition between two atomic states n and k as f x kn 2 m e ω kn ¯ h |h k | x | n i| 2 and similarly for y and z . The single-electron oscillator strength sum rule is X k f x kn = X k f y kn = X k f z kn = 1 where the sum is taken over all states k including the continuum, and n is a given state. For example, the 1 s - 2 p x - oscillator strength for hydrogen is 0.4162 by explicit computation. Thus the sum of lthe
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