11.7: Linear Stark Effect
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Apr 1, 2025
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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 ψ21−1 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|2,1,0⟩⟨2,1,0|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/√2−1/√2).
It follows that the simultaneous eigenstates of H0 and H1 take the form ψ1=ψ200+ψ210√2,ψ2=ψ200−ψ210√2.
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 ψ21−1 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.)
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