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852_2010hw7_Solutions

# 852_2010hw7_Solutions - PHYS852 Quantum Mechanics II Spring...

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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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852_2010hw7_Solutions - PHYS852 Quantum Mechanics II Spring...

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