Stark Effect

An electric field partly lifts the degeneracies of atomic energy levels. This splitting was observed by Stark [1] and explained by Schrödinger [2]. We compute the Stark effect on atomic hydrogen using perturbation theory by diagonalizing the perturbation term in the N-fold degenerate multiplet of states with principal quantum number N . We exploit the symmetries of this problem to simplify the numerical computations. In particular, after assuming the N -N matrix elements of the hamiltonian 〈N LM ′|H|NLM〉 are not important, we use symmetry to show that these matrix elements: (i.) vanish unless M ′ = M ; (ii.) vanish unless L = L ± 1; (iii.) are the same for M and −M ; and (iv.) factor into a product of two simpler functions which are simple look-ups. The Stark effect partly breaks the N-fold degeneracy of the states in the principal quantum level N into one N -fold degenerate multiplet and two multiplets with degeneracies k, where k = 0, 1, 2, · · · , N − 1. The splitting is indicated in Fig. 1. The perturbing hamiltonian is introduced in Sect. 2. In Sect. 3 we construct the matrix elements of this hamiltonian. The effects of symmetry on this computation are described in Sect. 4 and applied to the 16-fold degenerate multiplet with N = 4 in Sect. 5. The nature of the splitting for arbitrary N is described in Sect. 6. In a sense, the splitting is similar to that encountered in the case of an external magnetic field B (Zeeman effect). In Sect. 7 we apply nondegenerate state perturbation theory to study the effect of the electric field on the ground state of the hydrogen atom. In this case first order perturbation theory provides a null result, and we must go to second order perturbation theory to estimate the atomic polarizability of atomic hydrogen in the ground state. I n Sects. 8 and 9 we return to the question of how good is the first approximation that was made: that the N -N matrix elements could be neglected. It turns out to be a good approximation in one way, but bad in an unexpected way. We summarize our results in the closing Section.