phy-4600/lecture_notes/3-28/hydrogen atom

Derivation of Electron States in Hydrogen Atom
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    The Hydrogen atom follows the central potential development from the
    previous lecture. 
    
    ❬r,θ,φ❙Ψ❭ = R(r)ₑ,ₗ yₗₘ(θ,φ)

        Rₑₗ(r) = Uₑₗ(r) /r
                    
        ρ ≡ √⎛8͟m͟ │E│⎞r
             ⎝ ħ²   ⎠ 
    
        λ = Z͟e͟² √⎛_͟m͟_͟ ⎞
             ħ   ⎝2│E│⎠

    d͟²͟ U - 1͟ l(l+1) + ⎛λ͟ - 1͟⎞U = 0
    dρ²    ρ²         ⎝ρ   4⎠
    
    
        as ρ→0, U→ρˡ⁺¹

        for ρ→∞,

            d͟²͟ U - 1͟U = 0
            dρ²    4     

            d͟²͟ U - 1͟U = 0
            dρ²    4

            U(ρ) = A exp(-ρ/2) + B exp(ρ/2) = A exp(-ρ/2)
                                (ρ→∞, B=0)

            U(ρ) = ρˡ⁺¹ exp(-ρ/2) Fₑₗ(ρ)

            d͟ U =  (l+1)ρˡ exp(-ρ/2) Fₑₗ(ρ)
            dρ

                   -½ ρˡ⁺¹ exp(-ρ/2) Fₑₗ(ρ)


                   + ρˡ⁺¹ exp(-ρ/2) d͟ Fₑₗ(ρ)
                                    dρ

            d͟ U = ⎛l͟+͟1͟ - 1͟ ⎞ U + ρˡ⁺¹ exp(-ρ/2) d͟ F(ρ)
            dρ    ⎝ ρ    2 ⎠                    dρ



            d͟²͟ U =  -(l͟+͟1͟) U + ⎛l͟+͟1͟ - 1͟⎞² U 
            dρ²        ρ²      ⎝ ρ    2⎠

                    + 2⎛l͟+͟1͟ - 1͟⎞ρˡ⁺¹ exp(-ρ/2) d͟F͟
                       ⎝ ρ    2⎠               dρ

                    + ρˡ⁺¹ exp(-ρ/2) d͟²͟ F(ρ)
                                     dρ²




    ⎡−͟ħ͟² d² +  l͟  (l+1)ħ² - Z͟e͟²⎤Uₑₗ(r) = E Uₑₗ(r)
    ⎣2m  dr²  2mr²           r ⎦

    ρ = √⎛8͟m͟ │E│⎞r
         ⎝ ħ²   ⎠ 

    ⎛⎞
    ⎝⎠