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30 lines
735 B
Plaintext
30 lines
735 B
Plaintext
❙r❭=❙x,y,z❭
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with eigenvalue equations
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x̂❙r❭ = x❙r❭
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ŷ❙r❭ = y❙r❭
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ẑ❙r❭ = z❙r❭
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An arbitrary state
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❙Ψ❭ = ∫∫∫ dx dy dz ❙x,y,z❭❬x,y,z❙Ψ❭
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= ∫ d³r ❙r❭ ❬r❙Ψ❭
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Understanding a System:
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measure Ĥ, L̂², L̂𝓏 → Constitutes a complete set of commuting observables (except spin)
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I.E., There is a set of eigenstates that are eigenstates of all three operators.
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Ĥ❙E,l,mₗ❭ = E❙E,l,mₗ❭
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L̂²❙E,l,mₗ❭ = l(l+1)ħ²❙E,l,mₗ❭
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L̂𝓏❙E,l,mₗ❭ = mₗħ❙E,l,mₗ❭
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Ĥ must now include angular momentum
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L̂²= (r̂×p̂)(r̂×p̂) = (geometric identity) = r̂²p̂ - (r̂⋅p̂) + ιħr̂⋅p̂
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