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59 lines
1.8 KiB
Plaintext
59 lines
1.8 KiB
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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❬r❙r̂²p̂²❙Ψ❭ = r² ❬r❙p̂²❙Ψ❭
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r̂² p̂² = L̂² + (r̂⋅p̂)² - ιħ r̂⋅p̂
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❬r❙p̂²❙Ψ❭ = 1/r² ❬r❙L̂² + (r̂⋅p̂)² - ιħ r̂⋅p̂❙Ψ❭
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❬r❙L̂²❙Ψ❭
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❬r❙r̂²⋅p̂²❙Ψ❭ = r ⋅ ħ/ι ∇ ❬r❙Ψ❭ = ħ/ι r ∂/∂r ❬r❙Ψ❭
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❬r❙(r̂⋅p̂)²❙Ψ❭ = ❬r❙(r̂⋅p̂)(r̂⋅p̂)❙Ψ❭
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= r ħ/ι ∂/∂r ❬r❙r̂⋅p̂❙Ψ❭
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= -ħ² r ∂/∂r (r ∂/∂r) ❬r❙Ψ❭
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1/2m ❬r❙p̂²❙Ψ❭ = -ħ/2m r/r² ∂/∂r r ∂/∂r ❬r❙Ψ❭ + 1/(2mr²) ❬r❙L²❙Ψ❭
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= -ħ²/2m (∂²/∂r² + 2/r ∂/∂r) ❬r❙Ψ❭ + 1/(2mr²) ❬r❙L̂²❙Ψ❭
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| ↓ | | ↓ |
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linear energy rotational energy
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Hamiltonian can now be written
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Ĥ = -ħ²/2m (∂²/∂r² + 2/r ∂/∂r) + L̂²/(2mr²) + V(│r│)
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With the eigenvalue equation
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❬r❙E,l,mₗ❭ = E❬r❙E,l,mₗ❭
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