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