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# Linear Stark Effect

Returning to the Stark effect, let us examine the effect of an external electric field on the energy levels of the states of a hydrogen atom. There are four such states: an state, usually referred to as , and three states (with ), usually referred to as 2P. All of these states possess the same unperturbed energy, . As before, the perturbing Hamiltonian is (956)

According to the previously determined selection rules (i.e., , and ), this Hamiltonian couples and . Hence, non-degenerate perturbation theory breaks down when applied to these two states. On the other hand, non-degenerate perturbation theory works fine for the and states, since these are not coupled to any other states by the perturbing Hamiltonian.

In order to apply perturbation theory to the and states, we have to solve the matrix eigenvalue equation (957)

where is the matrix of the matrix elements of between these states. Thus, (958)

where the rows and columns correspond to and , respectively. Here, we have again made use of the selection rules, which tell us that the matrix element of between two hydrogen atom states is zero unless the states possess quantum numbers which differ by unity. It is easily demonstrated, from the exact forms of the 2S and 2P wavefunctions, that (959)

It can be seen, by inspection, that the eigenvalues of are and . The corresponding normalized eigenvectors are   (960)   (961)

It follows that the simultaneous eigenstates of and take the form   (962)   (963)

In the absence of an external electric field, both of these states possess the same energy, . The first-order energy shifts induced by an external electric field are given by   (964)   (965)

Thus, in the presence of an electric field, the energies of states 1 and 2 are shifted upwards and downwards, respectively, by an amount . These states are orthogonal linear combinations of the original and states. Note that the energy shifts are linear in the electric field-strength, so this effect--which is known as the linear Stark effect--is much larger than the quadratic effect described in Sect. 12.5. Note, also, that the energies of the and states are not affected by the electric field to first-order. Of course, to second-order the energies of these states are shifted by an amount which depends on the square of the electric field-strength (see Sect. 12.5).   Next: Fine Structure of Hydrogen Up: Time-Independent Perturbation Theory Previous: Degenerate Perturbation Theory
Richard Fitzpatrick 2010-07-20