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lecture20_umn - Lecture 20 Antisymmetry Spin Operators Spin...

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Lecture 20 October 28, 2009 Antisymmetry Spin Operators Spin in Many-electron Systems
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Two Indistinguishable Particles System with two indistinguishable particles: two electrons. Probability of finding: electron 1 (e1) in volume r a r r b , θ a θ θ b , and φ a φ φ b , and at the same time electron 2 (e2) in volume r c r r d , θ c θ θ d , and φ c φ φ d (20-1) P V 1 1 ( ) , V 2 2 ( ) [ ] = " # 1 1 ( ) , # 2 2 ( ) [ ] $ c d % c d % r c r d % a b % a b % r a r b % 2 r 1 2 dr 1 sin 1 d 1 d 1 r 2 2 dr 2 sin 2 d 2 d 2
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Probability (20-1) e1 in volume 1 and e2 in volume 2. Ψ depends on two individual electron wave functions ψ , the first of which is occupied by e1 and the second by e2 . First set of integration coordinates corresponds to e1 and second set to e2 . Electrons are indistinguishable ! We can not label them . P V 1 1 ( ) , V 2 2 ( ) [ ] = " # 1 1 ( ) , # 2 2 ( ) [ ] $ c d % c d % r c r d % a b % a b % r a r b % 2 r 1 2 dr 1 sin 1 d 1 d 1 r 2 2 dr 2 sin 2 d 2 d 2
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Switching Electrons Probability of finding e2 in V 1 and e1 in V 2 must be the same as probability of finding e1 in V 1 and e2 in V 2 (20-1) (20-2) where we have swapped the coordinates for e2 and e1 . Single probability of finding an electron in V 1 and another electron in V 2 . P V 1 2 ( ) , V 2 1 ( ) [ ] = " # 2 1 ( ) , # 1 2 ( ) [ ] $ c d % c d % r c r d % a b % a b % r a r b % 2 r 1 2 dr 1 sin 1 d 1 d 1 r 2 2 dr 2 sin 2 d 2 d 2
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Equal Probabilities Probabilities in 20-1 and 20-2 are equal. Both integrations are over the identical volumes, then it must be the case that (20-3) If we work with real wave functions (20-4) Two solutions to this equation (20-5) " # 1 1 ( ) , # 2 2 ( ) [ ] 2 = " # 1 2 ( ) , 2 1 ( ) [ ] 2 " 2 1 1 ( ) , 2 2 ( ) [ ] = " 2 1 2 ( ) , 2 1 ( ) [ ] " # 1 1 ( ) , 2 2 ( ) [ ] = ± " # 1 2 ( ) , 2 1 ( ) [ ]
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Symmetry and Antisymmetry (20-5) All particles in the universe fall into one of these two classes. Particles whose wave functions are unchanged are called ‘bosons’ and their
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lecture20_umn - Lecture 20 Antisymmetry Spin Operators Spin...

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