PauliEx

# PauliEx - Pauli Exclusion Principle Ground State of He...

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Colby College Pauli Exclusion Principle Ground State of He: Ψ He (1,2) = Ψ 1s (1) Ψ 1s (2) * spin part Ψ He (1,2) = Ψ 1s (1) Ψ 1s (2) [ α(1)β(2) ] 12 1 2 Indistinguishable from: Ψ He (1,2) = Ψ 1s (1) Ψ 1s (2) [ β(1)α(2) ] Take linear combinations with equal weight: Ψ s = 1 / 2 Ψ 1s (1) Ψ 1s (2) [ α(1)β(2) + β(1)α(2) ] Ψ a = 1 / 2 Ψ 1s (1) Ψ 1s (2) [ α(1)β(2) − β(1)α(2) ] Exchange labels -> Ψ 2 stays the same, but: Ψ s --> Ψ s Ψ a --> 1 / 2 Ψ 1s (1) Ψ 1s (2) [ α(2)β(1) − β(2)α(1) ] =– 1 / 2 Ψ 1s (1) Ψ 1s (2) [ α(1)β(2) − β(1)α(2) ] Ψ a --> – Ψ a Ψ s = 1 / 2 Ψ 1s (1) Ψ 1s (2) [ α(1)β(2) + β(1)α(2) ] Ψ a = 1 / 2 Ψ 1s (1) Ψ 1s (2) [ α(1)β(2) − β(1)α(2) ] SPATIAL PART* SPIN PART symmetric antisymmetric Pauli Exclusion Principle: The wavefunction is overall antisymmetric with respect to exchange of spin labels. Guaranteeing Antisymmetry: Determinantal Wavefunctions: Ψ a = 1 / 2 Ψ 1s (1) α(1) Ψ 1s (1) β(1) Ψ 1s (2) α(2) Ψ 1s (2) β(2) rows: different electrons columns: different wavefunctions He Excited State: 1s 1 2s 1 1s 2s or 1s 2s Ψ

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PauliEx - Pauli Exclusion Principle Ground State of He...

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