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# exam - highest(M=1 are separated by an amount ∆ E 1,0 =...

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Physics 829 Spring 2001 Dr. Herbst MIDTERM EXAMINATION (In Class; Closed Book) 11 MAY (100 POINTS) (1) (30 points) Estimate the energy through first-order of perturbation theory for a hydrogen atom in its ground state with a finite spherical nucleus of radius R<<a 0 . You may assume that the nuclear charge resides on the surface of the nucleus. 100 = 1 a 0 3 exp( - r / a 0 ) E 100 0 = - e 2 2 a 0 W = e 2 ( 1 r - 1 R ); r R What is the ratio of the first-order to the zeroth-order term if R = 10 -4 a 0 ? (2) (30 points) Use the WKB method to determine the energies of the states of a one-dimensional harmonic oscillator of charge q placed in a static electric field. (The answer is the same as the exact one.) (3) (40 points) A system with angular momentum quantum number J=1 is maintained in a magnetic field such that the energy levels from lowest (M=-1) to
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Unformatted text preview: highest (M=1) are separated by an amount ∆ E 1,0 = ∆ E 0,-1 = h . A second, rotating, magnetic field (with angular frequency ϖ ) is applied at t=0 and turned off at τ , leading to the perturbation W ( t ) = 1 [ J + exp(-i t ) + J-exp( i t )] where J + and J-are the raising and lowering operators. At t=0, the system is in the M=0 state. a) (30 points) Determine at t= τ to first order (i) the state vector of the system and (ii) the probability that the system is in each of its three M states, as a function of ϖ . You must start with perturbation theory; do not use the formula for P fi given on the equation page. b) (10 points) Determine the second-order contribution to the population of the M=1 state....
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