The Chemistry Maths Book, Second Edition

12.8 Inhomogeneous linear equations 359

The general case

If the end-points of the linear box discussed in Section 12.6 are joined to form a

simple closed loop, the Schrödinger equation is unchanged,

(12.72)

if the variable xis measured along the loop, but the two boundary conditions (12.45)

are replaced by the single periodic condition

ψ(x 1 + 1 l) 1 = 1 ψ(x) (12.73)

Equations (12.72) and (12.73) are the periodic boundary problem discussed in

Section 12.4 (with λreplaced by l). The allowed values of ωareω

n

1 = 12 πn 2 l, and the

solutions, in trigonometric form, are given by equation (12.26),

(12.74)

These results are valid for a simple closed loop with any shape. The results for the

circle of radius rare then obtained by replacing lby the circumference 2πr, and the

variable xbyrθ.

0 Exercise 29

12.8 Inhomogeneous linear equations

The general inhomogeneous second-order linear equation with constant coefficients is

(12.75)

where aand bare constants. Particular solutions of this equation can be found by

elementary methods for several important types of inhomogeneityr(x).

EXAMPLE 12.12Find a particular solution of the equationy′′ 1 + 13 y′ 1 + 12 y 1 = 12 x

2

.

The form of the function on the right of the equation suggests a solution of type

y 1 = 1 a

0

1 + 1 a

1

x 1 + 1 a

2

x

2

Then

y′ 1 = 1 a

1

1 + 12 a

2

x, y′′ 1 = 12 a

2

and

y′′ 1 + 13 y′ 1 + 12 y 1 = 1 (2a

2

1 + 13 a

1

1 + 12 a

0

) 1 + 1 (6a

2

1 + 12 a

1

)x 1 + 12 a

2

x

2

dy

dx

a

dy

dx

by r x

2

2

++=()

ψ

n

xd

nx

l

d

nx

l

() cos=+sin

12

22 ππ

d

dx

mE

2

2

22

2

0

ψ 2

+=,ωψ ω=