test1b - with ~ J characteristic of a scalar observable(ii...

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Physics 463 - Final Exam Your grade will be based upon your answers to 4 of the following 6 problems. If you turn in more than 4 solutions, your 4 highest scores will be used to compute your f nal exam grade. 1. Let {| n x ,n y ,n z i} denote the eigenstates of an isotrop ic3Dosc i l latoro ffrequency ω centered at the origin, where n x denotes the number of vibrational quanta associated with Cartesian coordinate x . Determine, at least to f rst order, the splitting of the threefold degenerate n =1states | x i = | 1 , 0 , 0 i| y i = | 0 , 1 , 0 i| z i = | 0 , 0 , 1 i due to a weak perturbation V = α m ω 2 xz. (Recall, in 1D, q = x p m ω / ¯ h .) 2. Let ~ V and ~ W be two vector operators of a certain quantum mechanical system with angular momentum ~ J , so [ J i ,V j ]= i X k ε ijk V k [ J i ,W j ]= i X k ε ijk W k . (i) Show that ~ V ·
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Unformatted text preview: with ~ J characteristic of a scalar observable. (ii) Show that ~ U = ~ V × ~ W is a vector under rotations. 3. Consider an oscillator with a particle of mass m in a quadratic potential with spring constant k . The particle is initially in the ground state when, at t = 0 , the spring constant is suddenly reduced to 1 / 3 of its initial value. Find the probability to f nd the particle in the new ground state immediately after the change in spring constant. 4. Use the variational principle to f nd a lower bound for the ground state energy of a particle con f ned to a 1D potential V ( x ) = V exp ( − | x | /x )....
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This note was uploaded on 12/19/2010 for the course PHYSICS ph 463 taught by Professor Paule. during the Fall '10 term at Missouri S&T.

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