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)

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           ⎦


    Rₑ﹐ₗ(r) = U͟ₑ͟﹐͟ₗ͟(r)
            r

            (pic) ...

(pic) Developed radial schrodinger equation  using U(r) replacement

    - Developed normalization condition

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.

    -ħ²/2m [(s(s-1) + l(l+1)] + V r² = E r²
    
    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


    Uₑ﹐ₗ(r) → (r→0) → rˡ⁺¹

    Rₑ﹐ₗ(r) → (r→0) → rˡ


The Hydrogen Atom
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    V(r) = -e²/r

    For a hydrogenic ion with nuclear charge Z

    V(r) = -Ze²/r

    Eigenfunctions:

        Ψₑ﹐ₗ﹐ₘ(r,θ,φ) = Rₑ﹐ₗ(r) Yₗ﹐ₘ(θ,φ) = Uₑ﹐ₗ/r Yₗ﹐ₘ(θ,φ)