The Chemistry Maths Book, Second Edition

406 Chapter 14Partial differential equations

or

L

2

Y

l,m

1 = 1 l(l 1 + 1 1)A

2

Y

l,m

(14.72)

in whichL

2

is the quantum-mechanical operator for the square of angular momentum.

The spherical harmonics are therefore the eigenfunctions of L

2

. They describe the

possible states of angular momentum of the system, and the eigenvaluesl(l 1 + 1 1)A

2

are

the allowed values of the square of angular momentum. In addition, it follows from

(14.62) that

and, therefore, that

(14.73)

or

L

z

Y

l,m

1 = 1 mAY

l,m

(14.74)

L

z

is the quantum-mechanical operator representing the component of angular

momentum in the z-direction, and the eigenvaluesmAofL

z

are the allowed values

of this component. The number lis called the angular momentum quantum number

and mthe component of angular momentum (or magnetic) quantum number.

The radial equation

(14.55)

The equation has two sets of solutions for the hydrogen atom. In the bound states

of the atom the electron is effectively confined to the vicinity of the nucleus by

application of the boundary conditionR(r) 1 → 10 asr 1 → 1 ∞. The energies of these states

are negative (the zero of energy is for the two charges at rest and at infinite separation)

and energy must be supplied to the system in order to ionize the atom. States with

positive energy are unbound states (continuum states) in which the electron moves

freely in the presence of the nucleus but is not bound to it. We consider here only the

bound states, withE 1 < 10. We set

α

2

1 = 1 − 2 E, (14.75)

and introduce the new variable

ρ 1 = 12 αr (14.76)

λ

α

=

Z

112

2

2

2

2

r

d

dr

r

dR

dr

ll

r

Z

r

ER













+−

• 
  

++













=

()

00

−

∂

∂

=

,

,

i

Y

mY

lm

lm



φ

−=i

d

d

m

m

m

Φ

Φ

φ