Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Hermicity of radial equation

  1. Jul 8, 2012 #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. Jul 8, 2012 #2

    Bill_K

    User Avatar
    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
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook




Loading...
Similar Threads for Hermicity radial equation Date
I Probability vs radial density-confusion Oct 25, 2017
I Derivation of the radial equation Aug 16, 2017
Does time reversal symmetry imply hermicity of s-matrix? Apr 21, 2014
Problem with Hermicity of Density matrix May 1, 2012
Testing Hermicity Nov 16, 2007