Time-Independent Perturbation Theory — David J. Griffiths
Physics — Quantum Mechanics · stark_effect
Problem
When an atom is placed in a uniform external electric field Eext, the energy levels are shifted—a phenomenon known as the Stark effect (it is the electrical analog to the Zeeman effect). In this problem we analyze the Stark effect for the n=1 and n=2 states of hydrogen. Let the field point in the z direction, so the potential energy of the electron is
HS′=eEextz=eEextrcosθ.
Treat this as a perturbation on the Bohr Hamiltonian (Equation 7.43). (Spin is irrelevant to this problem, so ignore it, and neglect the fine structure.)
(a) Show that the ground state energy is not affected by this perturbation, in first order.
(b) The first excited state is four-fold degenerate: ψ200, ψ211, ψ210, ψ21−1. Using degenerate perturbation theory, determine the first-order corrections to the energy. Into how many levels does E2 split?
(c) What are the "good" wave functions for part (b)? Find the expectation value of the electric dipole moment pe=−er, in each of these "good" states. Notice that the results are independent of the applied field—evidently hydrogen in its first excited state can carry a *permanent* electric dipole moment.
*Hint:* There are lots of integrals in this problem, but almost all of them are zero. So study each one carefully, before you do any calculations: If the ϕ integral vanishes, there's not much point in doing the r and θ integrals! You can avoid those integrals altogether if you use the selection rules of Sections 6.4.3 and 6.7.2. *Partial answer:* W13=W31=−3eaEext; all other elements are zero.