Is the radial part of the Laplacian in spherical coordinates Hermitian?

[geoduck]

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

\frac{d^2}{dr^2}\psi+\frac{2}{r}\frac{d}{dr}\psi

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?

 

[Bill_K]

∇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