11.5: Quadratic Stark Effect
( \newcommand{\kernel}{\mathrm{null}\,}\)
Suppose that a hydrogen atom is subject to a uniform external electric field, of magnitude |E|, directed along the z-axis. The Hamiltonian of the system can be split into two parts. Namely, the unperturbed Hamiltonian, H0=p22me−e24πϵ0r,
Note that the electron spin is irrelevant to this problem (because the spin operators all commute with H1), so we can ignore the spin degrees of freedom of the system. Hence, the energy eigenstates of the unperturbed Hamiltonian are characterized by three quantum numbers—the radial quantum number n, and the two angular quantum numbers l and m. (See Chapter [scent].) Let us denote these states as the ψnlm, and let their corresponding energy eigenvalues be the Enlm. According to the analysis in the previous section, the change in energy of the eigenstate characterized by the quantum numbers n,l,m in the presence of a small electric field is given by ΔEnlm=e|E|⟨n,l,m|z|n,l,m⟩=+e2|E|2∑n′,l′,m′≠n,l,m|⟨n,l,m|z|n′,l′,m′⟩|2Enlm−En′l′m′.
The sum on the right-hand side of the previous equation seems very complicated. However, it turns out that most of the terms in this sum are zero. This follows because the matrix elements ⟨n,l,m|z|n′,l′,m′⟩ are zero for virtually all choices of the two sets of quantum number, n,l,m and n′,l′,m′. Let us try to find a set of rules that determine when these matrix elements are non-zero. These rules are usually referred to as the selection rules for the problem in hand.
Now, because [see Equation ([e8.3])] Lz=xpy−ypx,
Let us now determine the selection rule for l. We have =[L2x,z]+[L2y,z]=Lx[Lx,z]+[Lx,z]Lx+Ly[Ly,z]+[Ly,z]Ly=iℏ(−Lxy−yLx+Lyx+xLy)=2iℏ(Lyx−Lxy+iℏz)=2iℏ(Lyx−yLx)=2iℏ(xLy−Lxy),
L4z−2L2zL2+zL4−2ℏ2(L2z+zL2)=0.
Equation ([e12.69]) implies that ⟨n,l,m|L4z−2L2zL2+zL4−2ℏ2(L2z+zL2)|n′,l′,m⟩=0.
Application of the selection rules ([e12.63]) and ([e12.73]) to Equation ([e12.59]) yields ΔEnlm=e2|E|2∑n′,l′=l±1|⟨n,l,m|z|n′,l′,m⟩|2Enlm−En′l′m.
Unfortunately, there is one fairly obvious problem with Equation ([e12.74]). Namely, it predicts an infinite energy-shift if there exists some non-zero matrix element ⟨n,l,m|z|n′,l′,m⟩ that couples two degenerate unperturbed energy eigenstates: that is, if ⟨n,l,m|z|n′,l′,m⟩≠0 and Enlm=En′l′m. Clearly, our perturbation method breaks down completely in this situation. Hence, we conclude that Equations ([e12.74]) and ([e12.76]) are only applicable to cases where the coupled eigenstates are non-degenerate. For this reason, the type of perturbation theory employed here is known as non-degenerate perturbation theory. The unperturbed eigenstates of a hydrogen atom have energies that only depend on the radial quantum number n. (See Chapter [scent].) It follows that we can only apply the previous results to the n=1 eigenstate (because for n>1 there will be coupling to degenerate eigenstates with the same value of n but different values of l).
Thus, according to non-degenerate perturbation theory, the polarizability of the ground-state (i.e., n=1) of a hydrogen atom is given by α=2e2∑n>1|⟨1,0,0|z|n,1,0⟩|2En00−E100.
Contributors and Attributions
Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)