phy-4600/lecture_notes/3-25/central potential states

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