ph135cS4 - Ph135c. Solution set #4, 4/29/10 1. This problem...

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Unformatted text preview: 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 ( W + BP σ + MP x + HP σ P x ) ψ 1 s =- V ( 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     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 ( ~σ 1 + ~σ 2 ) · ( ~σ 1 + ~σ 2 ) = 1 4 (3 + 2 ~σ 1 · ~σ 2 + 3) = 3 2 + 1 2 ~σ 1 · ~σ 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 · ~σ 1 )( ~ r · ~σ 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 · ~σ 1 )( ~ r · ~σ 2 ) + 1 2 r 2 1 Using the definition of...
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This note was uploaded on 12/04/2011 for the course PHY 7070 taught by Professor Smith during the Spring '11 term at University of Wisconsin Colleges Online.

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ph135cS4 - Ph135c. Solution set #4, 4/29/10 1. This problem...

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