HẳH0ỵlV ð9-24ị
HereHandHoare known as the perturbed and unperturbed Hamiltonian operators, respectively, and the differencelV is called the perturbation. The latter contains a scaling parameterl that is assumed to be small.
PERTURBATION THEORY FOR A NONDEGENERATE STATE 113
In the case analyzed by Schro¨dinger,H0 represented the hydrogen atom and its eigenvalues and eigenfunctions are known exactly. It should be noted that in most subsequent applications of perturbation theory to more complex atoms and mole- cules, the exact eigenvalues and eigenfunction of the unperturbed system are not known. Nevertheless, the formal derivation of perturbation theory is based on the assumption that the exact solutions ofH0 are known. Quantitative results are then obtained by substituting approximate eigenvalues and eigenfunctions into the per- turbation expressions.
In order to derive formal expressions for the various perturbation terms, we define the eigenfunctions and eigenvalues of the Hamiltonian operatorH0 as
H0ckẳekck ð9-25ị and those of the operatorHas
HfkẳEkfk ð9-26ị
We now consider the effect of the perturbationlV on a nondegenerate eigenvalue ofH0 that we denote ase0 and on its eigenfunctionc0. The corresponding eigen- value E0 and eigenfunctionf0 of H may then be represented as power series in terms of the scaling parameterl:
E0ẳe0ỵlE00ỵl2E000ỵ
f0ẳc0ỵl f00ỵl2f000ỵ ð9-27ị Substitution of these power series expansions into the Schro¨dinger equation (9-26) gives
ðH0e0ỵlVlE00l2E000 ịðc0ỵl f00ỵl2f000ỵ ị ẳ0 ð9-28ị The various perturbation equations are now derived by expanding Eq. (9-28) as a power series in terms of the scaling parameter l and by setting each successive coefficient of this power series expansion equal to zero:
ðH0e0ịc0ẳ0 ðH0e0ịf00ỵ ðVE00ịc0ẳ0 ðH0e0ịf000ỵ ðVE00ịf00E000c0ẳ0; etc:
ð9-29ị
The first of the above perturbation equations is, of course, automatically satis- fied. The other two equations may be simplified by multipliying them on the left by c0 and by subsequent integration:
hc0jH0e0jf00i ỵ hc0jVE00jc0i ẳ0
hc0 jH0e0jf000i ỵ hc0jVE00jf00i ẳE000 ð9-30ị
SinceH0 is Hermition, the first terms are zero and we find that E00ẳ hc0jVjc0i
E000 ẳ hc0jVE00jf00i ð9-31ị The first-order energy perturbationE00 is obtained by a straightforward integration, but the second-order termE000 depends on the first-order wave function perturbation f00. The determination of the latter function requires solution of the inhomogeneous differential equation
ðH0e0ịf00ẳ ðVE00ịc0 ð9-32ị We discuss three different approaches to the solution of this differential equation.
We note first that the solution of an inhomogeneous differential equation is not unique since it is always permissible to add an arbitrary amount of the solution of the homogeneous equation (in this casec0) to any solution. However, we define a unique solution by imposing the condition
hf00jc0i ẳ0 ð9-33ị The best-known method for solving the perturbation equation (9-32) consists of expanding the unknown functionf00in terms of the complete set of eigenfunctions ckof the operatorH0:
f00ẳX1
nẳ1
ancn ð9-34ị
We note that the expansion coefficienta0is equal to zero because of the condition (9-33).
Substitution of the expansion (9-34) into the differential equation (9-32) gives X
n
anðH0e0ịcnẳ ðVE00ịc0 ð9-35ị
The coefficientsan are then obtained by multiplying the equation by one of the eigenfunctionsck and subsequent integration:
X
n
anðene0ịhckjcni ẳ hckjV jc0i ð9-36ị
or
akẳ hckjVjcoi ekeo
ð9-37ị
PERTURBATION THEORY FOR A NONDEGENERATE STATE 115
The second-order energy perturbationE000 is then derived by substituting this result into Eq. (9-31):
E000ẳ X
k
hcojV jcki hckjVjcoi ekeo
ð9-38ị It may be shown that the third-order energy perturbation may also be derived from the first-order wave function perturbationf00 and that theð2nỵ1ị-th energy perturbation may be obtained by simple integration from the nth eigenfunction perturbation. The higher-order perturbation terms have become of interest lately because of experimental advances in nonlinear optics, but their derivation is beyond the scope of this book.
The perturbation expression (9-38) was used extensively as a basis for the discus- sion of electric and magnetic properties of molecules, but it offers only a qualitative representation of the various effects and it is not well suited for quantitative evalua- tions. Even in cases where approximate ground state wave functions are available, there is much less information about the excited state eigenfunctions, so numerical evaluations of the perturbation expression (9-38) present awkward problems.
As an interesting alternative, we will show how the perturbation equation (9-32) may also be solved by making use of the variational theorem of Eq. (9-10). We first note that by substituting Eq. (9-32) into Eq. (9-31), we may also writeE000 as
E000ẳ hf00jH0e0jf00i ð9-39ị In the special case wheree0 is the lowest eigenvalue ofH0 we have
hgf00jHe0jgf00i 0 ð9-40ị for any functiong. We may rewrite Eq. (9-41) as
hgjH0e0 jgi ỵ hgjVE00jc0i ỵ hc0jVE00jgi E000 ð9-41ị by making use of Eqs. (9-32) and (9-39). This inequality presents a variational approach to the derivation of the second-order energy perturbation, especially in situations where we can make an education guess about the nature of the perturba- tion functionf00.
The third approach to the perturbation problem is to solve the differential equa- tion analytically. This is, of course, the best approach, but unfortunately it is feasi- ble in only a few cases. One of these is the perturbation of the hydrogen atom by a homogeneous electric field. We discuss this problem in the next section.