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 fi nal exam grade. 1. Let {| n x , n y , n z i} denote the eigenstates of an isotropic 3D oscillator of frequency ω centered at the origin, where n x denotes the number of vibrational quanta associated with Cartesian coordinate x . Determine, at least to fi rst order, the splitting of the threefold degenerate n = 1 states | 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 · ~ W is a scalar with respect to rotations, i.e., show that it obeys commutation relations
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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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