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

Theory of the Hydrogen Atom
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    V(𝐫) = -e²/r

    For any hydrogenic ion with nuclear charge Z
        V(𝐫) = -Ze²/r

    Eigenfunctions in spherical coordinates:

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

    Derived from first principles the wave equation of an electron in the
    hydrogen atom.

        - Start with Dirac Notation
        - Replace with known general functions
        - transform to eigenvalue equation in position space
        - replace pieces with function F and find derivatives
            F = ∑ Cₖρᵏ

        pic 3 first take had factors that needed to be fixed

        - Led to the Laguerre Polynomials
            These are infinite, though, so
        - Force Laguerre Polynomial solutions to truncate

Derivation
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    The Hydrogen atom follows the central potential development from the
    previous lecture. 
    
    ❬r,θ,φ❙Ψ❭ = R(r)ₑ,ₗ yₗ﹐ₘ(θ,φ)

    Rₑ﹐ₗ(r) = Uₑ﹐ₗ(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) Fₑ﹐ₗ(ρ)

            d͟ U = (l+1)ρˡ exp(-ρ/2) Fₑ﹐ₗ(ρ)
            dρ
                        -½ ρˡ⁺¹ exp(-ρ/2) Fₑ﹐ₗ(ρ)

                            + ρˡ⁺¹ exp(-ρ/2) d͟ Fₑ﹐ₗ(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⎤Uₑ﹐ₗ(r) = E Uₑ﹐ₗ(r)
    ⎣2m  dr²  2mr²           r ⎦

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

    ⎛⎞
    ⎝⎠