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The Darwin Term

  1. Feb 27, 2013 #1
    I'm trying to do a HW involving the Darwin Term in the fine structure of hydrogen. I'm given that the perturbation from the Darwin term is equal to (times a constant factor which I'll ignore),

    [tex] V_D = [p_i,[p_i, \frac{e^2}{r}]] =
    e^2\vec{p}^2\frac{1}{r} -2e^2\vec{p}\frac{1}{r}\vec{p} +
    e^2\frac{1}{r}\vec{p}^2 [/tex]

    I know that the alterntive form of the Darwin term is,

    [tex] V_D = 4\pi\delta(r) [/tex]

    This comes from the first term in the first expression when you project the momentum operater onto real space, but there are two other terms which I can't seem to get to cancel... can anyone explain how these terms cancel? If they don't, then I have to do an intergal involving the laplacian of the wave function, and an integral involving the first derivite wrt to r of a wave function, for aribitrary n,l,m... and that is going to be a nightmare!
     
  2. jcsd
  3. Feb 28, 2013 #2

    TSny

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    Homework Helper
    Gold Member

    I think you've left out some terms. When simplifying operator expressions like this, I find it helpful to let the expressions act on an arbitrary function ##\psi##. So

    [itex] V_D \psi = [p_i,[p_i, \frac{e^2}{r}]] \psi [/itex]

    When expanding the commutators and then simplifying; make sure that you let the p operators act on everything to their right using the product rule.

    EDIT: You can also just use the well-known operator identity: ##[p_x, f(x)] = -i\hbar \frac{df(x)}{dx}##
     
    Last edited: Feb 28, 2013
  4. Feb 28, 2013 #3
    Doh! I totally forgot about that identity! Yes I do think maybe when I operate on some wave function i should get extra terms, but it's still messy to me, since I'm trying to do everything in spherical coordinates. :(

    I'll use the identity to finish the problem, then go back and try to work out how things should cancel. Thanks a bunch!
     
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