Introduction to Quantum Mechanics · Problem 7.42

Time-Independent Perturbation Theory — David J. Griffiths

Physics — Quantum Mechanics · feynman_hellmann_hydrogen

Problem

The Feynman–Hellmann theorem (Problem 7.38) can be used to determine the expectation values of

1/r

and

1/r^2

for hydrogen. The effective Hamiltonian for the radial wave functions is (Equation 4.53)

H = -\frac{\hbar^2}{2m}\frac{d^2}{dr^2} + \frac{\hbar^2}{2m}\frac{\ell(\ell+1)}{r^2} - \frac{e^2}{4\pi\epsilon_0}\frac{1}{r},

and the eigenvalues (expressed in terms of

\ell

) are (Equation 4.70)

E_n = -\frac{me^4}{32\pi^2\epsilon_0^2\hbar^2(N+\ell)^2}.

(a) Use

\lambda = e

in the Feynman–Hellmann theorem to obtain

\langle 1/r \rangle

. Check your result against Equation 7.56. (b) Use

\lambda = \ell

to obtain

\langle 1/r^2 \rangle

. Check your answer with Equation 7.57.