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

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│⎠