phy-4600/lecture_notes/4-15/Overview
2016-04-28 16:51:34 -04:00

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Two Spin-½ Particles
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uncoupled basis
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❙s₁ s₂ m₁ m₂❭
❙+ +❭, ❙- -❭, ❙+ -❭, ❙- +❭
coupled basis
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❙S Mₛ❭
❙1 1❭, ❙1 -1❭, ❙1 0❭, ❙0 0❭
Given the state ❙1 0❭ = 1/√2(❙+ -❭ + ❙- +❭) :
P₁₂❙1 0❭ = ❙1 0❭ (symmetric)
❙0 0❭ = 1/√2(❙+ -❭ - ❙- +❭) :
P₁₂❙0 0❭ = -❙0 0❭ (antisymmetric)
Spatial Representation
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❙Ψ❭ = ❙Ψₛₚₐₜᵢₐₗ❭❙Ψₛₚᵢₙ❭
Symmetric:
❙Ψ❭ˢ = ❙Ψₛₚₐₜᵢₐₗ❭ˢ❙Ψₛₚᵢₙ❭ˢ OR ❙Ψₛₚₐₜᵢₐₗ❭ᴬ❙Ψₛₚᵢₙ❭ᴬ
Antisymmetric:
❙Ψ❭ᴬ = ❙Ψₛₚₐₜᵢₐₗ❭ᴬ❙Ψₛₚᵢₙ❭ˢ OR ❙Ψₛₚₐₜᵢₐₗ❭ˢ❙Ψₛₚᵢₙ❭ᴬ
Helium Atom
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Fermions, so overall must be antisymmetric
Configuration Term Energy
1s² 1s 0
❙Ψ❭ = 1/√2 (❙a❭₁ₛ❙b❭₁ₛ + ❙b❭₁ₛ❙a❭₁ₛ)❙0 0❭
1s²s 3s 1.46
❙Ψ❭ = 1/√2(❙a❭₁ₛ❙b❭₁ₛ - ❙b❭₁ₛ❙a❭₁ₛ)❙1 m❭, m=-1,0,1
1s2s 1s 1.52
❙Ψ❭ = 1/√2 (❙a❭₁ₛ❙b❭₁ₛ + ❙b❭₁ₛ❙a❭₁ₛ)❙0 0❭
1s2p 3pᵒ ~1.60
❙Ψ❭ = 1√2
1s²p 1pᵒ ~1.65