Darwin term in a hydrogen atom - evaluating expectation values

[astrocytosis]

Homework Statement

Homework Equations

V_D= -1/(8m²c²) [p_i,[p_i,V_c(r)]]

V_C(r) = -Ze²/r

Energy shift Δ = <nlm|V_D|nlm>

The Attempt at a Solution

I can't figure out how to evaluate the expectation values that result from the Δ equation. When I do out the commutator, I get p²V-2pVp+Vp². This results in expectation values such as <1/r² p> and <1/r p²>. I'm not sure how to calculate them when there is a momentum operator hanging off the end like that, since I don't know the exact form of the wavefunction (n,l,m not specified) and don't know how to do the integral. Also, most online sources write V_D in terms of the Laplacian of V_C. I know the Laplacian arises from the momentum operator squared, but I am confused as to how this can be equivalent the equation given here.

My question is really just how to do Δ = <nlm|-1/(8m²c²) [p_i,[p_i,V_c(r)]]|nlm>.

 

[DrClaude]

Remember that $\nabla (AB) = B (\nabla A) + A (\nabla B)$, so imagine that the operators are applied to an arbitrary wave function $\psi$.

What you have to be careful here is $\nabla^2 \frac{1}{r}$ (see eq. (24) in http://mathworld.wolfram.com/Laplacian.html).