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ph135cS4 - Ph135c Solution set#4 1 This problem amounts to...

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Ph135c. Solution set #4, 4/29/10 1. This problem amounts to determining the action of P σ and P x on the various angular momentum states. The spin exchange operator acts on the singlet as - 1 and on the triplet as +1, since these states are anti-symmetric and symmetric respectively. Similarly, the spatial coordinate exchange operator acts as ( - 1) . To summarize: P σ P x 1 S - 1 +1 3 S +1 +1 1 P - 1 - 1 3 P +1 - 1 Thus we have, for example: V 1 s ψ 1 s = - V 0 ( W + BP σ + MP x + HP σ P x ) ψ 1 s = - V 0 ( W - B + M - H ) ψ 1 s along with three similar equations. These can be summarized as: V 1 s V 3 s V 1 p V 3 p = - V 0 1 - 1 1 - 1 1 1 1 1 1 - 1 - 1 1 1 1 - 1 - 1 W B M H 2. (a) S · S = 1 4 ( 1 + 2 ) · ( 1 + 2 ) = 1 4 (3 + 2 1 · 2 + 3) = 3 2 + 1 2 1 · 2 (b) ( ~ r · ~ S ) 2 = 1 4 r i r j ( σ 1 i + σ 2 i )( σ 1 j + σ 2 j ) = 1 2 ( ~ r · 1 )( ~ r · 2 ) + 1 4 r i r j ( σ 1 i σ 1 j + σ 2 i σ 2 j ) Since r i r j is symmetric under i j , we can replace σ i σ j with 1 2 { σ i , σ j } = δ ij : = 1 2 ( ~ r · 1 )( ~ r · 2 ) + 1 2 r 2 1
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Using the definition of S 12 , we can rewrite the first term, leaving: ( ~ r · ~ S ) 2 = r 2 2 (1 + 1 3 ( S 12 + 1 ·
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