The Chemistry Maths Book, Second Edition

8.6 Periodicity 243

system is called a ‘rigid rotor’ and the equation of motion in quantum mechanics (the

Schrödinger equation) for a rigid rotor in a plane is

(8.49)

where the wave functionψ 1 = 1 ψ(θ)is a function of the orientation variable θ, and Eis

the (positive) kinetic energy of rotation. The rigid rotor is used in chemistry to model

the rotational motion of a molecule.

Equation (8.49) can be written

(8.50)

wherea

2

1 = 12 IE2A

2

1 > 10 , and it is readily verified that a solution of this equation is

ψ(θ) 1 = 1 Ce

iaθ

(8.51)

where Cis an arbitrary constant (see Section 12.7 for a more complete discussion).

Thus,

For the solution (8.51) to be physically significant, and represent rotation, it is neces-

sary that the wave function be unchanged when θis replaced byθ 1 + 12 π; it must satisfy

the periodicity condition

ψ(θ 1 + 12 π) 1 = 1 ψ(θ) (8.52)

For the function (8.51),

ψ(θ 1 + 12 π) 1 = 1 Ce

ia(θ+ 2 π)

1 = 1 Ce

iaθ

1 × 1 e

2 πai

1 = 1 ψ(θ) 1 × 1 e

2 πai

and the periodicity condition is satisfied if 2 πais a multiple of 2 π; that is, ifa 1 = 1 nfor

n 1 = 1 0, ±1, ±2,1=The physically significant solutions of the Schrödinger equation are

therefore

ψ

n

(θ) 1 = 1 Ce

inθ

, n 1 = 1 0, ±1, ±2,1= (8.53)

where the ‘quantum number’ nhas been used to label the solutions. The corresponding

values of the energyE 1 =1A

2

a

2

22 Iare

(8.54)

E

n

I

n

=



22

2

d

d

iaCe

d

d

ia Ce a Ce a

ia ia ia

ψψ

θ

θ

θθθ

=, = =− =−

2

2

222

() ψψ

d

d

a

2

2

2

ψ

ψ

θ

=−

−=



22

2

2 I

d

d

E

ψ

ψ

θ