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} Ψ = d^{2}/dr^{2} Ψ + 2/r dΨ/dr = (1/r^{2}) d/dr (r^{2} dΨ/dr) The inner product is ∫Φ*Ψ r^{2} dr dΩ = 4π ∫Φ*Ψ r^{2} dr Integrate twice by parts: 4π ∫Φ*∇^{2}Ψ r^{2} dr = - 4π ∫ (dΦ*/dr) (r^{2} dΨ/dr) dr = 4π ∫ d/dr (r^{2} dΦ*/dr) Ψ dr = 4π ∫ (1/r^{2}) d/dr (r^{2} dΦ*/dr) Ψ r^{2} dr = 4π ∫∇^{2}Φ* Ψ r^{2} dr