Consider the Stark effect (Problem 7.45) for the n = 3 states of hydrogen. There are initially nine degenerate states, 3m (neglecting spin, as before),

Consider the Stark effect (Problem 7.45) for the n = 3 states of hydrogen. There are initially nine degenerate states, Ψ3ℓm (neglecting spin, as before), and we turn on an electric field in the z direction.

(a) Construct the 9x9 matrix representing the perturbing Hamiltonian.

(b) Find the eigenvalues, and their degeneracies.

Problem 7.45 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 H = e Eextz = e Eextr cos 9. 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: 200, 211, 210, 21-1- 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 (p = -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 integral vanishes, there's not much point in doing the r and 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 = -3ea Eext; all other elements are zero.