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