The expectation value refers to only a single

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Unformatted text preview: - ÅÅÅÅÅe ÅÅ ƒ 1 sÆ 1 s 2 s_ ÅÅÅ ƒ ÅÅÅ ƒ ÅÅÅ ƒ ƒ ƒ ƒ r1 ƒ r2 ƒ r3 ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ In the first term, the second and third electrons are not affected by the operator and ƒØ2 ƒ ƒp ƒ ƒ Z2ƒ ƒ 2 ÅÅÅ [1 sÆ 1 s 2 s ƒ ÅÅÅÅ1ÅmÅ - ÅÅÅÅÅe ÅÅ ƒ 1 sÆ 1 s 2 s_ ÅÅÅ ƒ ƒ r1 ƒ ƒ ƒ ƒ ƒ 2 ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ i y iØ p ƒ ƒ ƒ ƒ Z2y j z ƒ ƒ ƒ ƒ j 2ÅÅÅÅ = j[1 sÆ ƒ1 ⊗[1 s ƒ2 ⊗X2 s »3 L j ÅÅÅÅ1mÅ - ÅÅÅÅÅe ÅÅ z H » 1 sÆ \1 ⊗ ƒ 1 s _ ⊗ ƒ 2 s_ z ÅÅÅ z j z ƒ ƒ ƒ ƒ r1 ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ k { 2 3{ k ƒ Ø2 ƒ ƒp ƒ ƒ Z2ƒ ƒ 2ÅÅÅÅ = [1 sÆ ƒ1 ÅÅÅÅ1mÅ - ÅÅÅÅÅe ÅÅ ƒ 1 sÆ _ X1 s »2 » 1 s \2 X2 s »3 » 2 s\3 ÅÅÅ ƒ ƒ r1 ƒ ƒ ƒ ƒ ƒ 1 p Z = Z1 sÆ À ÅÅÅÅÅmÅ - ÅÅÅÅÅre ÅÅ À 1 sÆ ^ ÅÅÅ ÅÅÅ 2 Ø2 2 In the last line, I used the fact that the single-particle states are properly normalized. The expectation value refers to only a single-particle state, and I dropped the particle index. Therefore, p p p Z Z Z X1 sÆ 1 s 2 s » H0 » 1 sÆ 1 s 2 s\ = Z1 sÆ À ÅÅÅÅÅmÅ - ÅÅÅÅÅre ÅÅ À 1 sÆ ^ + Z1 s À ÅÅÅÅÅmÅ - ÅÅÅÅÅre ÅÅ À 1 s ^ + Z2 s À ÅÅÅÅÅmÅ - ÅÅÅÅÅre ÅÅ À 2 s^ ÅÅÅ ÅÅÅ ÅÅÅ 2 ÅÅÅ 2 ÅÅÅ 2 ÅÅÅ Ø2 2 Ø2 2 Ø2 2 = E1 s + E1 s + E2 s and hence the expectation value is simply the sum of single-particle energies. The same applies to all the diagonal pieces in the expectation value. For the off-diagonal (the ket and the bra are different) pieces, we find, for example, ƒØ2 ƒ ƒp ƒ ƒ Z2ƒ ƒ 2 ÅÅÅ [1 sÆ 1 s 2 s ƒ ÅÅÅÅ1ÅmÅ - ÅÅÅÅÅe ÅÅ 1 s 2 s 1 sÆ _ ÅÅÅ ƒ ƒ r1 ƒ ƒ ƒ ƒ ƒ 2 ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ i y iØ p ƒ ƒ ƒ ƒ Z2y j z ƒ ƒ ƒ ƒ j 2ÅÅÅÅ = j[1 sÆ ƒ1 ⊗[1 s ƒ2 ⊗X2 s »3 L j ÅÅÅÅ1mÅ - ÅÅÅÅÅe ÅÅ z H » 1 s \1 ⊗ ƒ 2 s_ ⊗ ƒ 1 sÆ _ z ÅÅÅ z j z ƒ ƒ ƒ ƒ r1 ƒ ƒ ƒ ƒ k { ƒ ƒ ƒ ƒ 2 3{ k ƒ Ø2 ƒ ƒp ƒ ƒ Z2ƒ ƒ 2ÅÅÅÅ = [1 sÆ ƒ1 ÅÅÅÅ1mÅ - ÅÅÅÅÅe ÅÅ ƒ 1 s _ X1 s »2 » 2 s\2 X2 s »3 »...
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