What is the Darwin Term's Role in Hydrogen's Fine Structure?

[dipole]

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),

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

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

$$V_D = 4\pi\delta(r)$$

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!

 

[TSny]

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

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

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: $[p_x, f(x)] = -i\hbar \frac{df(x)}{dx}$

 

[dipole]

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

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

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: $[p_x, f(x)] = -i\hbar \frac{df(x)}{dx}$

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!