final - Physics 829 Dr Herbst Spring 2001 FINAL...

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Physics 829 Spring 2001 Dr. Herbst FINAL EXAMINATION (In Class; Closed Book) 4 JUNE (150 POINTS) 1. (40 pts) In the theory of the interaction of radiation with matter, consider the W = 1 2 m q 2 A z 2 interaction term we neglected in class. Using the semiclassical theory of radiation and first-order time-dependent perturbation theory, determine the absorption probability P fi as a function of ϖ for the hydrogen atom transition 1s -> 2p if only W' is considered. You need not compute any spatial integrals. Use the term in W' that yields the largest non-zero answer. What is unusual about your expression? A 0 = B / 2 ik 2. (30 pts) Use the variational method to determine an approximate value for the ground state energy of a particle trapped by the one-dimensional potential: V ( x ) = k | x | . Choose as your trial function a Gaussian: = exp( - x 2 ). How does your result compare with the exact value of E 0 = 0.8086( k 2 h 2 / m ) 1/3 ? 3. (30 pts) Consider a tritium (heavy hydrogen) atom (Z=1) with an electron in its 1s state. The nucleus suddenly undergoes beta decay such that Z=2.
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Unformatted text preview: a) (15 points) What is the probability that the electron remains in its 1s state? b) (15 points) Can you think of a general selection rule for the sudden process? 1 s = Z 3 a 3 exp(-Zr / a ) 4. (50 pts) The energy levels of a nucleus can often be expressed in terms of the solutions to the three-dimensional isotropic harmonic oscillator. For one particle, the lowest three energy levels are as follows: 0s ( E = 1.5 h ), 1p ( E = 2.5 h ), and 2s and 2d, both with E = 3.5 h . a) (40 pts) Consider two non-interacting identical spin-1/2 fermions in a nucleus. Determine the energies, degeneracies, term symbols, and suitable kets for the three lowest configurations. Your kets can be determined by use of memory, symmetrizers/antisymmetrizers, or by angular momentum considerations. b) (10 pts) Do the same problem for the lowest configuration of three non-interacting spin-1/2 fermions....
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