lecture22_umn

# lecture22_umn - Lecture 22 More Spin Algebra Triplet State...

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Lecture 22 November 2, 2009 More Spin Algebra: Triplet State Open-Shell Singlet State

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Triplet States 2 is a proper eigenvalue of S 2 corresponding to S = 1 S ( S +1). α (1) α (2) is an eigenfunction of S 2 S = 1; M S = 1, 0, or 1 Evaluation of M S : add one-half for each α electron and subtract one-half for each β electron. For α (1) α (2): M S = 1 Cases 1 and 0 , degenerate in energy in the absence of a magnetic field . State is triply degenerate: Triplet states: 3 Ψ III Now " 1 ( ) 2 ( ) S 2 1 ( ) 2 ( ) = 2 " IV S 2 " IV 1 ( ) 2 ( ) S 2 1 ( ) 2 ( ) = 2 (22-1) This is the M S = 1 component of the triplet: 3 Ψ IV
Evaluation of (22-2) No integer value of S for which S ( S + 1) = 1 : Ψ V is not an eigenfunction of S 2 . Same for Ψ VI " V S 2 " V = 1 2 a 1 ( ) # 1 ( ) b 2 ( ) \$ 2 ( ) S 2 a 1 ( ) 1 ( ) b 2 ( ) 2 ( ) % a 1 ( ) 1 ( ) b 2 ( ) 2 ( ) S 2 a 2 ( ) 2 ( ) b 1 ( ) 1 ( ) % a 2 ( ) 2 ( ) b 1 ( ) 1 ( ) S 2 a 1 ( ) 1 ( ) b 2 ( ) 2 ( ) + a 2 ( ) 2 ( ) b 1 ( ) 1 ( ) S 2 a 2 ( ) 2 ( ) b 1 ( ) 1 ( ) ( ( ( ( ( ( ( ) * + + + + + + + = 1 2 (1 % 0 % 0 + 1) = 1 " V S 2 " V

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Linear Combination Ψ V + Ψ VI < S 2 > =1: average of a singlet (< S 2 > = 0) and a triplet (< S 2 > = 2) Wave function that is a 50:50 mixture of singlet and triplet If both Ψ V and Ψ VI are equal mixtures of singlet and triplet taking linear combinations of the two (and renormalizing) might give pure spin states. (22-3) " V+VI = 1 2 " V + " VI ( ) = 1 2 1 2 a 1 ( ) # 1 ( ) b 2 ( ) \$ 2 ( ) % a 2 ( ) 2 ( ) b 1 ( ) 1 ( ) [ ] + 1 2 a 1 ( ) 1 ( ) b 2 ( ) 2 ( ) % a 2 ( ) 2 ( ) b 1 ( ) 1 ( ) [ ] ( ( ) ( ( * + ( ( , ( ( = 1 2 a 1 ( ) b 2 ( ) % a 2 ( ) b 1 ( ) [ ] 1 ( ) 2 ( ) + 2 ( ) 1 ( ) [ ]
Expectation Value Normalized sum of the two wave functions is an eigenfunction of S 2 : a triplet The only component of the triplet not found so far is M S = 0 Eq. 22-3 is that wave function: one up and one down electron Expectation value of S z is zero The two electrons,

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lecture22_umn - Lecture 22 More Spin Algebra Triplet State...

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