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Perturbation Theory with Atoms

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When an atom is placed in a uniform external electric field Eext, the energy levels are shifted - a phenomenon known as the stark 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's = -e Eext z = -e Eext r cos θ.

Treat this as a perturbation on the Bohn Hamiltonian:

Spin is irrelevant to this problem, so ignore it.

a. Show that the ground state energy is not affected by this perturbation, in first order
b. The first excited state is fourfold 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 in 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 a lot of integrals in this problem, but most of them are zero. 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. Partial answer: W13 = W31 = 3eaEext; all other elements are zero.

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This solution discusses the perturbation theory and applies it to atoms and determines the ground state energy, first-order corrections in energy, how many levels E2 splits into, good wave functions, and the expectation value of the electric dipole moment. All calculations steps are shown with brief explanations.

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