Physical Chemistry Third Edition

728 17 The Electronic States of Atoms. I. The Hydrogen Atom

In previous chapters we used a capitalΨto represent a time-dependent wave function.

In the next few chapters we will useΨfor a time-independent wave function of two or

more particles.

Equation (17.1-16) can be solved by the separation of variables. We assume the trial

function:

Ψψc(xc,yc,zc)ψ(x,y,z) (17.1-17)

When the product function of Eq. (17.1-17) is substituted into Eq. (17.1-16) and the

variables are separated, we obtain the two equations

̂HcψcEcψc (17.1-18)

ĤrelψErelψ (17.1-19)

whereEcis the center-of-mass energy andErelis the relative energy and where

EEc+Erel (17.1-20)

Exercise 17.1

Carry out the steps to obtain Eqs. (17.1-18) to (17.1-20).

Equation (17.1-18) is the same as the Schrödinger equation for a free particle in three

dimensions, which we discussed in Chapter 15. Equation (17.1-19) is mathematically

equivalent to the Schrödinger equation for the motion of a particle of massμrelative

to a fixed origin (see Appendix E). Figure 17.2 depicts this equivalence. The vector

from the nucleus (labeledn) to the electron (labelede) in Figure 17.2a is equal to the

vector from the fixed origin to the fictitious particle of massμin Figure 17.2b. Since

the nucleus is much more massive than the electron the motion of the electron is nearly

the same as though the nucleus were stationary, and we will often refer to the relative

motion as electronic motion.

Particle of mass m

Vector of

lengthr

fromn to

Center of mass

Particle of mass mn

(a)(b)

Vector of

lengthr

from fixed

origin to

fictitious

particle of

mass

Fictitious particle of

mass

Fixed origin

e

e

meemn

memn

Figure 17.2 Figure to Illustrate the Equivalence between the Motion of a Particle of

Massμaround a Fixed Center and the Relative Motion of Two Particles.(a) The actual

two-particle system. (b) The fictitious particle of massμ.