bei48482_FM

values the electron can have are specified by the formula EnE 1     n^2. The quantiza-

tion of electron energy in the hydrogen atom is therefore described by the principal

quantum number n.

The theory of planetary motion can also be worked out from Schrödinger’s equa-

tion, and it yields a similar energy restriction. However, the total quantum number n

for any of the planets turns out to be so immense (see Exercise 11 of Chap. 4) that

the separation of permitted levels is far too small to be observable. For this reason clas-

sical physics provides an adequate description of planetary motion but fails within the

atom.

6.5 ORBITAL QUANTUM NUMBER

Quantization of angular-momentum magnitude

The interpretation of the orbital quantum number lis less obvious. Let us look at the

differential equation for the radial part R(r) of the wave function :

r

2

 E R^0 (6.14)

This equation is solely concerned with the radial aspect of the electron’s motion, that

is, its motion toward or away from the nucleus. However, we notice the presence of

E, the total electron energy, in the equation. The total energy Eincludes the electron’s

kinetic energy of orbital motion, which should have nothing to do with its radial motion.

This contradiction may be removed by the following argument. The kinetic energy

KE of the electron has two parts, KEradialdue to its motion toward or away from the

nucleus, and KEorbitaldue to its motion around the nucleus. The potential energy Uof

the electron is the electric energy

U (6.2)

Hence the total energy of the electron is

EKEradialKEorbitalUKEradialKEorbital

Inserting this expression for Ein Eq. (6.14) we obtain, after a slight rearrangement,

r

2

 KEradialKEorbital R^0 (6.19)

If the last two terms in the square brackets of this equation cancel each other out, we

shall have what we want: a differential equation for R(r) that involves functions of the

radius vector rexclusively.

We therefore require that

KEorbital (6.20)

^2 l(l1)



2 mr^2

^2 l(l1)



2 mr^2

2 m



^2

dR



dr

d



dr

1



r^2

e^2



4  0 r

e^2



4  0 r

l(l1)



r^2

e^2



4  0 r

2 m



^2

dR



dr

d



dr

1



r^2

208 Chapter Six

bei48482_ch06 1/23/02 8:16 AM Page 208