852_2010hw7 - PHYS852 Quantum Mechanics II, Spring 2010...

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PHYS852 Quantum Mechanics II, Spring 2010 HOMEWORK ASSIGNMENT 7 Topics covered: addition of three angular momenta, degenerate perturbation theory 1. Consider a system of two spin-1/2 particles, described by ~ S 1 and ~ S 2 , and one spin-1 particle, described by ~ S 3 . Let ~ S 0 = ~ S 1 + ~ S 2 . What are the allowed values of s 0 ? For each allowed value, give the allowed m 0 values. Use Clebsch Gordan coefficients to express the | s 0 m 0 m 3 i states in terms of the | m 1 m 2 m 3 i states. Let ~ S = ~ S 1 + ~ S 2 + ~ S 3 . What are the allowed values of the quantum number s ? For each s value, list the alllowed m -values. Express the states | s 0 ,s,m i as linear superpositions of the | s 0 m 0 m 3 i states, and then as linear superpositions of the | m 1 m 2 m 3 i states. 2. Derive general expressions for the energy-shifts up to third-order and state vectors up to second-order in degenerate perturbation theory. 3. Consider a system described in the basis
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