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Unformatted text preview: Physics 137B, Fall 2007, Moore Problem Set 7 Solutions 1. (a) The applied perturbation is resonant when the driving frequency matches the natural frequency of the system, which in this case is the Bohr frequency for the 1 2 transition: = ba = 3 4 (13 . 6 eV) ~ = 3 4 13 . 6 eV 6 . 582 10- 16 eV s = 1 . 55 10 16 s- 1 . (b) Let a denote the 1 s state and b the 2 s state, with wavefunctions a ( r ) = ( a 3 )- 1 / 2 e- r/a , b ( r ) = (32 a 3 )- 1 / 2 2- r a e- r/ 2 a , Consider a perturbation H ( t ) =- eEx sin( t ). In spherical coordinates x = r sin cos , so H ba ( t ) = h b | - eEx sin( t ) | a i =- eE sin( t )( a 3 )- 1 / 2 (32 a 3 )- 1 / 2 Z d 3 r r sin cos 2- r a e- r/ 2 a e- r/a =- eE 4 2 a 3 sin( t ) Z 2 d cos Z d sin 2 Z r dr r 3 2- r a e- 3 r/ 2 a = 0 , 1 since the integral vanishes. The first-order transition amplitude from 1 s to 2 s is proportional to H ba ( t ), so (to first order) this perturbation will not create 1 s 2 s transitions. (This conclusion is immediate from the electric dipole selection rules, which de- mand that only l = 1 transitions are allowed.) 2. For a periodic perturbation H ( t ) = H sin t we define A = 1 2 i H , so that H ( t ) = H sin t = H 1 2 i ( e it- e- it ) = Ae it + A e- it . By Eq. (9.10), the first-order transition amplitudes c k ( t ) satisfy i ~ c k ( t ) = X l H kl ( t ) e i kl t c l ( t ) = X l A kl e it + A kl e- it e i kl t c l ( t ) ....
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