852_2010hw7_Solutions - PHYS852 Quantum Mechanics II,...

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Unformatted text preview: PHYS852 Quantum Mechanics II, Spring 2010 HOMEWORK ASSIGNMENT 7: Solutions 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 = ~ S 1 + ~ S 2 . What are the allowed values of s ? For each allowed value, give the allowed m values. Use Clebsch Gordan coefficients to express the | s m 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 ,s,m i as linear superpositions of the | s m m 3 i states, and then as linear superpositions of the | m 1 m 2 m 3 i states. Notation : For s = 2, we use | i for m = 2, | i for m = 1, | i for m = 0, | i for m =- 1, and | i for m =- 2; For s = 1, we use | i for m = 1, | i for m = 0, and | i for m =- 1; For s = 1 / 2, we use | i for m = 1 / 2, and | i for m =- 1 / 2. The allowed values of s are 0 , 1 For s = 0, the only allowed value for m is 0. For s = 1, the allowed values for m are- 1 , , 1. The | s m m 3 i states follow the standard singlet/triplet forms: | s m m 3 i = | m 1 m 2 m 3 i (1) | 1 m 3 i = | m 3 i (2) | 10 m 3 i = 1 2 ( | m 3 i + | m 3 i ) (3) | 1 m 3 i = | m 3 i (4) | 00 m 3 i = 1 2 ( | m 3 i - | m 3 i ) (5) For s = 0, the allowed value of s is 1. For s = 1, the allowed values of s are 0 , 1 , 2. For s = 0, the allowed value of m is 0. For s = 1, the allowed values of m are- 1 , , 1. For s = 2, the allowed values of m are- 2 ,- 1 , , 1 , 2. 1 | s sm i = | s m m 3 i = | m 1 m 2 m 3 i (6) | 12 i = | 1 i = | i (7) | 12 i = 1 2 ( | 1 i + | 10 i ) = 1 2 ( 2 | i + | i + | i ) (8) | 120 i = 1 6 ( | 1 i +2 | 100 i + | 1 i ) = 1 6 ( | i + 2 | i + 2 | i + | i ) (9) | 12 i = 1 2 ( | 10 i + | 1 i ) = 1 2 ( | i + | i + 2 | i ) (10) | 12 i = | 1 i = | i (11) | 11 i = 1 2 ( | 1 i - | 10 i ) = 1 2 ( 2 | i - | i - | i ) (12) | 110 i = 1 2 ( | 1 i - | 1 i ) = 1 2 ( | i - | i ) (13) | 11 i = 1 2 ( | 10 i - | 1 i ) = 1 2 ( | i + | i - 2 | i ) (14) | 100 i = 1 3 ( | 1 i-| 100 i + | 1 i ) = 1 6 ( 2 | i - | i - | i + 2 | i ) (15) | 01 i = | 00 i = 1 2 ( | i - | i ) (16) | 010 i = | 000 i = 1 2 ( | i - | i...
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This note was uploaded on 12/21/2011 for the course PHYS 852 taught by Professor Moore during the Spring '11 term at Michigan State University.

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852_2010hw7_Solutions - PHYS852 Quantum Mechanics II,...

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