The Feynman–Hellmann theorem (Problem 7.38) can be used to determine the expectation values of 1/r and 1/r² for hydrogen. The effective Hamiltonian for the radial wave functions is (Equation 4.53)

and the eigenvalues (expressed in terms of ℓ) are (Equation 4.70)

(a) Use λ = e  in the Feynman–Hellmann theorem to obtain (1/r). Check your result against Equation 7.56.
(b) Use λ = ℓ to obtain (1/r²). Check your answer with Equation 7.57.

H h d 2m dr2 + h? 2m ( + 1) 12 e 41 for