- #1
dipole
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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!
[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!