- #1
astrocytosis
- 51
- 2
Homework Statement
Homework Equations
VD= -1/(8m2c2) [pi,[pi,Vc(r)]]
VC(r) = -Ze2/r
Energy shift Δ = <nlm|VD|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 VD in terms of the Laplacian of VC. 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) [pi,[pi,Vc(r)]]|nlm>.