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Physics LibreTexts

11.7: Linear Stark Effect

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Returning to the Stark effect, let us examine the effect of an external electric field on the energy levels of the n=2 states of a hydrogen atom. There are four such states: an l=0 state, usually referred to as 2S, and three l=1 states (with m=1,0,1), usually referred to as 2P. All of these states possess the same unperturbed energy, E200=e2/(32πϵ0a0). As before, the perturbing Hamiltonian is H1=e|E|z.

According to the previously determined selection rules (i.e., m=m, and l=l±1), this Hamiltonian couples ψ200 and ψ210. 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 ψ211 and ψ211 states, because these are not coupled to any other n=2 states by the perturbing Hamiltonian.

In order to apply perturbation theory to the ψ200 and ψ210 states, we have to solve the matrix eigenvalue equation Ux=λx,

where U is the matrix of the matrix elements of H1 between these states. Thus, U=e|E|(0,2,0,0|z|2,1,02,1,0|z|2,0,0,0),
where the rows and columns correspond to ψ200 and ψ210, respectively. Here, we have again made use of the selection rules, which tell us that the matrix element of z between two hydrogen atom states is zero unless the states possess l quantum numbers that differ by unity. It is easily demonstrated, from the exact forms of the 2S and 2P wavefunctions, that 2,0,0|z|2,1,0=2,1,0|z|2,0,0=3a0.

It can be seen, by inspection, that the eigenvalues of U are λ1=3ea0|E| and λ2=3ea0|E|. The corresponding normalized eigenvectors are x1=(1/21/2),x2=(1/21/2).

It follows that the simultaneous eigenstates of H0 and H1 take the form ψ1=ψ200+ψ2102,ψ2=ψ200ψ2102.
In the absence of an external electric field, both of these states possess the same energy, E200. The first-order energy shifts induced by an external electric field are given by ΔE1=+3ea0|E|,ΔE2=3ea0|E|.
Thus, in the presence of an electric field, the energies of states 1 and 2 are shifted upwards and downwards, respectively, by an amount 3ea0|E|. These states are orthogonal linear combinations of the original ψ200 and ψ210 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 Section 1.5. Note, also, that the energies of the ψ211 and ψ211 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 Section 1.5.)

Contributors and Attributions

  • Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)


This page titled 11.7: Linear Stark Effect is shared under a not declared license and was authored, remixed, and/or curated by Richard Fitzpatrick.

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