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Hermicity of radial equation

by geoduck
Tags: equation, hermicity, radial
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Jul8-12, 09:48 AM
P: 230
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?
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Jul8-12, 10:26 AM
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Bill_K's Avatar
P: 4,160
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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