Physical Chemistry Third Edition

726 17 The Electronic States of Atoms. I. The Hydrogen Atom

17.1 The Hydrogen Atom and the Central Force System

The hydrogen atom is one of the few systems for which the Schrödinger equation can

be solved. It consists of a single electron with massmeand charge−eand a nucleus

consisting of a single proton with massmnand chargee, as depicted in Figure 17.1.

We first assume that the atom is not confined in any container. Its potential energy

depends only on the distance between the particles and is given byCoulomb’s law, for

which the potential energy is given in Eq. (14.4-16):

V(r)−

e^2

4 πε 0 r

(17.1-1)

whereε 0 is the permittivity of the vacuum, and whereris the distance between the

nucleus and the electron. This formula corresponds to the choice that the potential

energy is equal to zero when the electron and nucleus are infinitely far apart. The

hydrogen atom is an example of acentral-force system, which is any two-particle

system with a potential energy that depends only on the distance between the particles.

z

zn

xn yn

y

x

Nucleus (proton)

Electron

xe

ze

ye

Figure 17.1 The Hydrogen Atom
System Consisting of a Nucleus and
an Electron.
We denote the Cartesian coordinates of the nucleus byxn,yn, andzn, and the
Cartesian coordinates of the electron byxe,ye, andze. The distance between the
particles is given by a three-dimensional version of the theorem of Pythagoras:

r

(

(xe−xn)^2 +(ye−yn)^2 +(ze−zn)^2

) 1 / 2

(17.1-2)

This is an inconvenient function of all six coordinates. To simplify the expression for the

potential energy, we will make two transformations of coordinates. We first transform

to relative coordinates and center-of-mass coordinates. Therelative coordinatesx,y,

andzare defined by

xxe−xn (17.1-3a)

yye−yn (17.1-3b)

zze−zn (17.1-3c)

Thecenter-of-mass coordinatesxc,yc, andzcare given by:

xc

mexe+mnxn

M

(17.1-4a)

yc

meye+mnyn

M

(17.1-4b)

zc

meze+mnzn

M

(17.1-4c)

where the sum of the masses is denoted byM:

Mme+mn (17.1-5)

The kinetic energy can be expressed in terms of the velocity of the center of mass

and the relative velocity, as shown in Appendix E:

K 

M

2

(v^2 cx+v^2 cy+v^2 cz)+

μ

2

(v^2 x+v^2 y+v^2 z) (17.1-6)