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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
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