## Time Dependent Perturbation Theory Homework Hotline

Physics 519 Homework Set #3 Spring 2013 Due in class 4/24/13 300 pts 1. (100 pts) Sakurai 5.35. Simpliﬁed photo-electric eﬀect. You do not need to repeat steps worked out in Sakurai (and also in lectures) to obtain a general formula for the transition rate. The point here is to calculate the resulting matrix elements and obtain a fully explicit expression for the rate and angular distribution. The ground state of a hydrogen atom ( n = 1, l = 0) is subjected to a time-dependent potential as follows: V ( ~x, t ) = V0 cos( kz-ωt ) . Using time-dependent perturbation theory, obtain an expression for the transi-tion rate at which the electron is emitted with momentum ~ p . Show, in particular, how you may compute the angular distribution of the ejected electron (in terms of θ and φ deﬁned with respect to the z-axis). Discuss brieﬂy the similarities and the diﬀerences between this problem and the (more realistic) photo-electric eﬀect. Note: The initial wave function is given by ψ n =1 ,l =0 ( ~x ) = 1 √ π ± 1 a0 ² 3 / 2 e-r/a0 . The ﬁnal wave function, at least in a very large box of size L , can be approxi-mated to be ψ f = ± 1 L 3 / 2 ² e i~ p · ~x/ ¯ h . incoming light is ignored by considering the perturbation V ( ~x, t ) = V0 2 ³ e i ( kz-ωt ) + e-i ( kz-ωt ) ´ . The e-iωt term leads to absorption of energy by the atom and thus (if the energy exceeds the ionization threshold) to ejection of a pho-toelectron. Following Sakurai’s development or our class notes, one ﬁnds, us-ing Fermi’s Golden rule (FGR), that the transition rate into electrons with momentum with solid angle d Ω of ~ p f is dw d Ω = 2 π ¯ h |{z} FGR mp f ¯ h 3 ± L 2 π ² 3 | {z } density of states µ µ h ~ p f | ( V0 / 2) e ikz | i i µ µ 2 | {z } matrix element . The required matrix element is h ~ p f | e ikz | i i = 1 L 3 / 2 Z d 3 xe-i~ p f · ~x/ ¯ h e ikz 1 √ π ± 1 a0 ² 3 / 2 e-r/a0 = 1 √ π ( a0 L ) 3 / 2 Z d 3 xe-i~ q · ~x e-r/a0 , 1

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