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132 lines
6.0 KiB
Plaintext
132 lines
6.0 KiB
Plaintext
Bound states of a central potential
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For any central potential V(r) = V(│r│) the eigenfunctions of H can be
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separated as
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❬r❙E,l,mₗ❭ = Rₑ﹐ₗ(r) Yₗ﹐ₘ(θ,φ)
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(pic) The radial S.E. is
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⎡−͟ħ͟²⎛ ∂͟²͟ + 2͟ ∂͟ ⎞ + l͟(l͟+͟1͟)ħ͟² + V(│r│) ⎤ Rₑ﹐ₗ(r) = E Rₑ﹐ₗ(r)
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⎣2m ⎝ ∂r² r ∂r ⎠ 2 m r² ⎦
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(pic) Developed radial schrodinger equation using U(r) replacement
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Rₑ﹐ₗ(r) = Uₑ﹐ₗ(r) /r
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⎡−͟ħ͟²⎛ ∂͟²͟ + 2͟ ∂͟ ⎞⎤ U͟ = E ∂͟²͟ U
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⎣2m ⎝ ∂r² r ∂r ⎠⎦ r ∂r²
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∂͟ U͟ = 1͟ ∂͟ U - U͟
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∂r r r ∂r r²
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∂͟²͟ U͟ = 1͟ ∂͟²͟ U + 2͟U͟ - 2͟ ∂͟ U
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∂r² r r ∂r² r³ r²∂r
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⎡−͟ħ͟²⎛ ∂͟²͟ U͟ + 2͟ ∂͟ U͟ ⎞⎤ = E ∂͟²͟ U
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⎣2m ⎝ ∂r² r r ∂r r ⎠⎦ ∂r²
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────────────────────────────────────────────────────────────────────────────────
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⎡−͟ħ͟²⎛ 1͟ ∂͟²͟ U + 2͟U͟ - 2͟ ∂͟ U + 2͟ 1͟ ∂͟ U - 2͟ U͟ ⎞⎤
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⎣2m ⎝ r ∂r² r³ r²∂r r r ∂r r r² ⎠⎦
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= E 1͟ ∂͟²͟ U + E 2͟U͟ - E 2͟ ∂͟U
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r ∂r² r³ r²∂r
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────────────────────────────────────────────────────────────────────────────────
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⎡−͟ħ͟²⎛ 1͟ ∂͟²͟ + 2͟ - 2͟ ∂͟ + 2͟ ∂͟ - 2͟ ⎞U⎤
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⎣2m ⎝ r ∂r² r³ r²∂r r² ∂r r³⎠ ⎦
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= E ⎛1͟ ∂͟²͟ + 2͟ - 2͟ ∂ ⎞U
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⎝r ∂r² r³ r²∂r⎠
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────────────────────────────────────────────────────────────────────────────────
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⎡⎛−͟ħ͟² 1͟ ∂͟²͟ + −͟ħ͟² 2͟ - −͟ħ͟² 2͟ ∂͟ + −͟ħ͟² 2͟ ∂͟ - −͟ħ͟² 2͟ ⎞U⎤
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⎣⎝2m r ∂r² 2m r³ 2m r²∂r 2m r² ∂r 2m r³⎠ ⎦
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= E ⎛1͟ ∂͟²͟ + 2͟ - 2͟ ∂ ⎞U
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⎝r ∂r² r³ r²∂r⎠
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────────────────────────────────────────────────────────────────────────────────
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⎡⎛−͟ħ͟² 1͟ ∂͟²͟ - −͟ħ͟² 2͟ ∂͟ + −͟ħ͟² 2͟ ∂͟ + −͟ħ͟² 2͟ - −͟ħ͟² 2͟ ⎞U⎤
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⎣⎝2m r ∂r² 2m r²∂r 2m r² ∂r 2m r³ 2m r³⎠ ⎦
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= E ⎛1͟ ∂͟²͟ + 2͟ - 2͟ ∂ ⎞U
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⎝r ∂r² r³ r²∂r⎠
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────────────────────────────────────────────────────────────────────────────────
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↓
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Not sure this makes sense, but the final result is
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↓
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⎡−͟ħ͟²∂͟²͟ - ħ͟² l(l+1) + V(│r│)⎤ Uₑ﹐ₗ(r) = E Uₑ﹐ₗ(r)
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⎣2m ∂r² 2mr² ⎦
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Normalization Condition
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∞ ∞
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∫ │Rₑ﹐ₗ│² r² dr = ∫ │Uₑ﹐ₗ│² dr
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0 0
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If V(r) is not more singular at the origin than 1/r^2 then the SE has power
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series solutions.
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Thus for small r we take U(r) → rˢ
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(pic) substitute U(r) = rˢ into S.E.
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Uₑ﹐ₗ(r) ≈ rˢ
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⎡−͟ħ͟² ∂͟²͟ + ħ͟² l(l+1) + V(│r│)⎤ rˢ = E rˢ
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⎣2m ∂r² 2mr² ⎦
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−͟ħ͟² s(s-1)∂͟ rˢ⁻² + l͟ (l+1)ħ²rˢ⁻² + V(│r│)rˢ = E rˢ
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2m ∂r 2m
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−͟ħ͟² ⎛s(s-1)∂͟ + l͟ (l+1)ħ²⎞rˢ⁻² + V(│r│)rˢ = E rˢ
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2m ⎝ ∂r 2m ⎠
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−͟ħ͟² ⎛s(s-1)∂͟ + l͟ (l+1)ħ²⎞rˢ⁻² + V(│r│)rˢ = E rˢ
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2m ⎝ ∂r 2m ⎠
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−͟ħ͟² ⎛s(s-1)∂͟ + l͟ (l+1)ħ²⎞ + V(│r│)r² = E r²
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2m ⎝ ∂r 2m ⎠
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For r → 0,
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r² → 0,
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V(r) r² → 0.
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⇓
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s(s-1) + l(l+1) = 0
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⇓
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s = l+1 or s = -l
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If s = -l, the normalization conditions
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∞ │∞
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∫ r⁻²ˡ dr = 1/(2l-1) 1/(r²ˡ⁻¹) │ → diverges
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0 │0
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So, for small r,
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Uₑ﹐ₗ(r) → (r→0) → rˡ⁺¹;
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Rₑ﹐ₗ(r) → (r→0) → rˡ.
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Eigenfunctions
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━━━━━━━━━━━━━━
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Ψₑ﹐ₗ﹐ₘ(r,θ,φ) = Rₑ﹐ₗ(r) Yₗ﹐ₘ(θ,φ) = Uₑ﹐ₗ /r Yₗ﹐ₘ(θ,φ)
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⇒ d͟²͟ U - 1͟ l(l+1) + ⎛λ͟ - 1͟⎞U = 0
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dρ² ρ² ⎝ρ 4⎠
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⎡−͟ħ͟² d² + l͟ (l+1)ħ² - Z͟e͟²⎤Uₑ﹐ₗ(r) = E Uₑ﹐ₗ(r)
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⎣2m dr² 2mr² r ⎦
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ρ = √⎛8͟m͟ │E│⎞r
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⎝ ħ² ⎠
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λ = Z͟e͟² √⎛_͟m͟_͟ ⎞
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ħ ⎝2│E│⎠
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