Physical Chemistry Third Edition

800 19 The Electronic States of Atoms. III. Higher-Order Approximations

The ground state of the helium atom is nondegenerate. That is, there is only one

wave function corresponding to the ground-state energy. We now outline a perturba-

tion scheme to handle the nondegenerate case. The energy eigenvalues and energy

eigenfunctions are represented by power series inλ:

EnE(0)n +E(1)n λ+E(2)nλ^2 + ··· (19.3-5)

ΨnΨ(0)n +Ψ(1)nλ+Ψ(2)nλ^2 + ··· (19.3-6)

The idea of the perturbation method is to approximate these series by partial sums of

two or three terms. Figure 19.2 shows schematically a typical energy eigenvalue as a

function ofλand as represented by the first two partial sums of the series for values

ofλbetween zero and unity. To first order, the energy eigenvalue is given by letting

λ1 in the two-term partial sum:

En≈E(0)n +E(1)n (19.3-7)

Appendix G contains a derivation of the formula for the first-order correction to the

energy,E(1)n. The result is

E(1)n 

∫

Ψ(0)n∗̂H′Ψ(0)ndq (19.3-8)

EXAMPLE19.2

Apply first-order perturbation theory to the ground state of a system with the potential energy:

V(x)

{

∞ ifx<−aora<x

kx^2 /2if−a<x<a

Solution

We takeĤ(0)to be the particle-in-a-box Hamiltonian with−a<x<a, not 0<x<a.

Ĥ′kx

2

2

E(0) 1 E(1)

E(0) 1 λE(1)

E(correct)

E(0)

01

Correct energy

of real system

Approximate energy given by

first-order correction

E(λ)

λ

Figure 19.2 An Energy Eigenvalue as a Function ofλfor a Hypothetical System.