Introduction to Quantum Mechanics · Problem 7.46

Time-Independent Perturbation Theory — David J. Griffiths

Physics — Quantum Mechanics · stark_effect_n3

Problem

Consider the Stark effect (Problem 7.45) for the

n = 3

states of hydrogen. There are initially nine degenerate states,

\psi_{3\ell m}

(neglecting spin, as before), and we turn on an electric field in the

z

direction. (a) Construct the

9 \times 9

matrix representing the perturbing Hamiltonian. *Partial answer:*

\langle 3\,0\,0|z|3\,1\,0\rangle = -3\sqrt{6}\,a

\langle 3\,1\,0|z|3\,2\,0\rangle = -3\sqrt{3}\,a

\langle 3\,1\,\pm 1|z|3\,2\,\pm 1\rangle = -(9/2)\,a

. (b) Find the eigenvalues, and their degeneracies.