Hermicity of radial equation

  1. The radial part of the Laplacian in spherical coordinates goes as:

    [tex]\frac{d^2}{dr^2}\psi+\frac{2}{r}\frac{d}{dr}\psi [/tex]

    How can this be Hermitian? The first term can be Hermitian, but the second term, with its 2/r factor, seems like it's not Hermitian?
     
  2. jcsd
  3. Bill_K

    Bill_K 4,157
    Science Advisor

    2 Ψ = d2/dr2 Ψ + 2/r dΨ/dr = (1/r2) d/dr (r2 dΨ/dr)

    The inner product is ∫Φ*Ψ r2 dr dΩ = 4π ∫Φ*Ψ r2 dr

    Integrate twice by parts:
    4π ∫Φ*∇2Ψ r2 dr = - 4π ∫ (dΦ*/dr) (r2 dΨ/dr) dr = 4π ∫ d/dr (r2 dΦ*/dr) Ψ dr = 4π ∫ (1/r2) d/dr (r2 dΦ*/dr) Ψ r2 dr
    = 4π ∫∇2Φ* Ψ r2 dr
     
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