hw5ajj - PHYS 507 Answers to Homework V 1. The assumption...

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Unformatted text preview: PHYS 507 Answers to Homework V 1. The assumption ~μ = k ~ S is not necessary. We only need to use the commutation relation [ S i ,μ j ] = i ¯ h X k ² ijk μ k , which expresses the fact that ~μ is a vector operator. d dt h S i i t = i ¯ h h [ H,S i ] i t =- i ¯ h X j B j h [ μ j ,S i ] i t = i ¯ h X j B j h [ S i ,μ j ] i t =- X jk B j ² ijk h μ k i t =- ( ~ B × h ~μ i t ) i = +( h ~μ i t × ~ B ) i . 2. We have calculated the rotation operator for a single spin 1/2 particle before. It is D ( θ, ˆ x ) = exp- i 2 θσ x ¶ = cos θ 2- i sin θ 2 σ x = • cos θ 2- i sin θ 2- i sin θ 2 cos θ 2 ‚ As a result we have the following D ( θ, ˆ x ) | ↑i = cos θ 2 | ↑i - i sin θ 2 | ↓i , D ( θ, ˆ x ) | ↓i =- i sin θ 2 | ↑i + cos θ 2 | ↓i . (a) We will rotate the two-particle state | , i by applying individual rotations to each particle using the relations above. Let us start with | ↑↓i . We have D ( θ, ˆ x ) | ↑↓i = ( D 1 ( θ, ˆ x ) | ↑i ) ⊗ ( D 2 ( θ, ˆ x ) | ↓i ) = cos θ 2 | ↑i - i sin θ 2 | ↓i ¶ ⊗- i sin θ 2 | ↑i + cos θ 2 | ↓i ¶ = cos 2 θ 2 | ↑↓i - i sin θ 2 cos θ 2 ( | ↑↑i + | ↓↓i )- sin 2 θ 2 | ↓↑i . Continue with | ↓↑i , D ( θ, ˆ x ) | ↓↑i = ( D 1 ( θ, ˆ x ) | ↓i ) ⊗ ( D 2 ( θ, ˆ x ) | ↑i ) =- i sin θ 2 | ↑i + cos θ 2 | ↓i ¶ ⊗ cos θ 2 | ↑i - i sin θ 2 | ↓i ¶ = cos 2 θ 2 | ↓↑i - i sin θ 2 cos θ 2 ( | ↑↑i + | ↓↓i )- sin 2 θ 2 | ↑↓i . 1 For the state | , i we have D ( θ, ˆ x ) | , i = 1 √ 2 cos 2 θ 2 + sin 2 θ 2 ¶ ( | ↑↓i - | ↓↑i ) = | , i ....
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hw5ajj - PHYS 507 Answers to Homework V 1. The assumption...

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