Physical Chemistry Third Edition

802 19 The Electronic States of Atoms. III. Higher-Order Approximations

The zero-order ground-state energy E

(0)

1 s 1 sis given by Eq. (18.3-13) andΨ

(0)

1 s 1 sis given

by Eq. (18.3-2). The first order correction to the energy is

E 1 (1)s 1 s

∫

ψ 100 (1)∗ψ 100 (2)∗

(

e^2

4 πε 0 r 12

)

ψ 100 (1)ψ 100 (2)d^3 r 1 d^3 r 2 (19.3-11)

This integral is the same as the integral in Eq. (19.1-6), so that our perturbation method

result to first order is

E(0) 1 s 1 s+E(1) 1 s 1 s− 108 .8eV+ 34 .0eV− 74 .8 eV (19.3-12)

This is the same as the result of the variation method using the zero-order wave function.

The first-order correction to the wave function and the second-order correction to

the energy eigenvalue are more complicated than the first-order correction to the energy

eigenvalue. Appendix F contains the formulas for these quantities. No exact calculation

of the second-order correction to the energy of the helium atom has been made, but a

calculation made by a combination of the perturbation and variation methods gives an

accurate upper bound:^10

E

(2)

1 s 1 s−^4 .3 eV (19.3-13)

so that the value of the energy through second order is− 79 .1 eV, within 0.1 eV of

the experimental value,− 79 .0 eV. Since we are not using the variation method, the

approximate energy can be lower than the correct energy. Approximate calculations

through thirteenth order have been made, and have given values that agree with exper-

iment nearly as well as the best results of the variation method.^11

PROBLEMS

Section 19.3: The Perturbation Method and Its

Application to the Ground State of the Helium Atom

19.18 a. Using first-order perturbation, find a formula for the
energy of a particle in its ground state in a box of
lengthawith an additional linear potential
term:

V(x)

{

∞ ifx<0ora<x

bx if 0<x<a

b.An electron in a one-dimensional box of length

10.00 Å is transported to a planet on which the

acceleration due to gravity is 6. 00 × 1018 ms−^2. The

box is placed in a vertical position. Find the energy of

the electron in first order. Do you think that your

answer is a reasonable approximation to the correct

energy?

c.Repeat the calculation for the surface of the earth. Do

you think that your answer is a reasonable

approximation to the correct energy?

19.19 a.An electron is confined in a one-dimensional box of

length 10.00 Å (1. 000 × 10 −^9 m). The left end of the

box corresponds tox0 and the right end of the box

corresponds tox 1. 00 × 10 −^9 m. An electric field

Eis imposed on the electron so that its potential

energy is given by

V−eEx

Assume thatE 1. 00 × 109 Vm−^1 (1.00 volt

potential difference between the ends of the box).

Using first-order perturbation theory, calculate the

ground-state energy of the electron in the box and

compare it to the result that you get in the absence of

(^10) C. W. Scherr and R. E. Knight,Rev. Mod. Phys., 35 , 436 (1963).
(^11) C. W. Scherr and R. E. Knight,loc. cit.(note 10).