Perturbation Theory for a Degenerate Energy Level

As A 0, the eigenfunctions satisfying (9.72) approach eigenfunctions satisfying (9.70). Does this mean that lim i o Not necessarily. If is nondegener- 

Our first task is thus to determine the correct zeroth-order wave functions (9.75) for the perturbation H. Calling these correct zeroth-order functions l f we have 

Each different function has a different set of coefficients in (9.76). The correct set of zeroth-order functions depends on what the perturbation H is. 

The treatment of the d-fold degenerate level proceeds like the nondegenerate treatment of Section 9.2, except that instead of we use Instead of Eqs. (9.13) and (9.14), we have 

We now multiply (9.79) by and integrate over all space, where m is one of the states corresponding to the d-fold degenerate unperturbed level under consideration that is, 1 m d. We get 

---

Section 9.5 Perturbation Theory for a Degenerate Energy Level 259... 

At this point we may introduce the spin of the electrons into the wave function (in the same manner as for helium) by multiplying each single-electron orbital function by either < ( ) or /3(w). For convenience we shall include these spin factors in the functions u (l), etc., so that hereafter a, j3, y, represent four quantum numbers n, l, mi, and m, for each electron and 1, 2, represent four coordinates n, d,-, electron case, treatment of this degenerate energy level by perturbation theory (the electron interactions being the perturbation) leads to certain combinations... 

---