The Chemistry Maths Book, Second Edition

14.4 The particle in a rectangular box 397

andC

x

1 = 1 −p

2

π

2

2 a

2

. Similarly, (14.21b) describes the motion of the particle along the

y-direction, and has normalized solutions

(14.22b)

and C

y

1 = 1 −q

2

π

2

2 b

2

. The total solutions (eigenfunctions) of the two-dimensional

Schrödinger equation (14.13) are the products

ψ

p,q

(x, 1 y) 1 = 1 X

p

(x) 1 × 1 Y

q

(y)

(14.23)

and the corresponding total energies (eigenvalues) are

(14.24)

We note that the quantities

(14.25)

are to be identified with the kinetic energies of motion along the xand ydirections,

respectively.

A square box. Degeneracy

When the sides of the box are not equal and neither is an integer multiple of the other,

the eigenvalues (14.24) are all distinct; the states of the system are then said to be

nondegenerate. For a square box however, witha 1 = 1 b,

(14.26)

and the eigenvalues forp 1 ≠ 1 qoccur in pairs withE

p,q

1 = 1 E

q,p

; for example,E

1,2

1 = 1 E

2,1

1 =

5 h

2

28 ma

2

. States of the system with equal energies are called degenerate states.

The occurence of degeneracy for the square box is a consequence of the symmetry

of the system. The eigenfunctions (14.23) are

(14.27)

and an interchange of the xand ycoordinate axes gives

(14.28)

ψψ

pq q p

yx

a

py

a

qx

a

,,

() sin sin,=

























=

2 ππ

(()xy,

ψ

pq

xy

a

px

a

qy

a

,

() sin sin,=

























2 ππ

E

h

ma

pq

pq,

=+

2

2

22

8

()

E

hp

ma

E

hq

mb

pq

==

22

2

22

2

88

and

E

h

m

p

a

q

b

pq,

=+













22

2

2

2

8

=













×













,,=,,,

22

123

a

px

ab

qy

b

sin sin pq

ππ

......

Yy

b

qy

b

q

q

() sin=













,=,,,

2

123

π

...