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

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PHYS 507 Answers to Homework V 1. The assumption = 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 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 i t ) i = +( h 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 | 0 , 0 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

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For the state | 0 , 0 i we have D ( θ, ˆ x ) | 0 , 0 i = 1 2 cos 2 θ 2 + sin 2 θ 2 ( | ↑↓i - | ↓↑i ) = | 0 , 0 i . (b) Here 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 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 2 (1 + cos θ ) | 1 , 1 i - i 2 sin θ · 2 | 1 , 0 i - 1 2 (1 - cos θ ) | 1 , - 1 i .
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hw5ajj - PHYS 507 Answers to Homework V 1 The assumption =...

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