1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: [QM] Darwin term is not hermitian

  1. Jun 24, 2010 #1
    Hi there! I'm solving the dirac equation to get the fine structure hamiltonian of the hydrogen atom. In the hamiltonian there is this term:
    [tex]\frac{\hbar ^2 e}{4m^2c^2}\frac{dV}{dr}\frac{\partial}{\partial r}[/tex]

    This term gives rise to some difficulty because it is not hermitian. So Darwin proposed to use instead the symmetrical combination:
    [tex]\frac{1}{2}\left[\left(\frac{\hbar ^2 e}{4m^2c^2}\frac{dV}{dr}\frac{\partial}{\partial r}\right)+[\left(\frac{\hbar ^2 e}{4m^2c^2}\frac{dV}{dr}\frac{\partial}{\partial r}\right)^*\right]=\frac{\hbar^2}{8m^2c^2}\nabla^2V[/tex]

    but the adjoint of [tex]\frac{\hbar ^2 e}{4m^2c^2}\frac{dV}{dr}\frac{\partial}{\partial r}[/tex] is the adjoint of the spatial derivative which is anti-hermitian, so the symmetric combination should be 0... where am I wrong??
     
  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted