The Chemistry Maths Book, Second Edition

12.7 The particle in a ring 357

Equations (12.62) and (12.64) are the periodic boundary value problem discussed in

Section 12.4, with xreplaced by θand λby 2π. The allowed values of ωare therefore

ω

n

1 = 1 n, and the corresponding solutions are the eigenfunctions

ψ

n

(θ) 1 = 1 c

1

e

inθ

1 + 1 c

2

e

−inθ

, n 1 = 1 0, ±1, ±2,= (12.65)

with eigenvalues

(12.66)

We note that the states of the system with quantum numbern 1 ≠ 10 occur in degenerate

pairs,ψ

n

andψ

−n

,

Hψ

n

1 = 1 E

n

ψ

n

, Hψ

−n

1 = 1 E

n

ψ

−n

By the principle of superposition (Section 12.2), every linear combination of a pair of

degenerate eigenfunctions is itself an eigenfunction with the same eigenvalue,

H(aψ

n

1 + 1 bψ

−n

) 1 = 1 E

n

(aψ

n

1 + 1 bψ

−n

)

where aand bare arbitrary. It is physically possible to distinguish degenerate states of

a quantum-mechanical system only by the application of an external force to break

the degeneracy. In the absence of such a force, therefore, every choice of coefficients

c

1

and c

2

is equally good. It is conventional to choosec

2

1 = 10 in (12.65), to give the set

of eigenfunctions

ψ

n

(θ) 1 = 1 c

1

e

inθ

, n 1 = 1 0, ±1, ±2,= (12.67)

and to choose c

1

to normalize these functions. The normalization condition for the

complexfunctions is

(12.68)

whereψ*

n

(θ) 1 = 1 c*

1

e

−inθ

is the complex conjugate function ofψ

n

(θ). Then

and this is unity if. The normalized eigenfunctions are therefore

(12.69)

These functions form an orthonormal setin the interval 01 ≤ 1 θ 1 ≤ 12 π; they are

orthogonal as well as normalized:

(12.70)

Z

0

2

0

π

ψψθθθ

nm

*( ) ( )dnm=≠if

ψ θ

θ

n

in

()=,=,±,±,en

1

2

012

π

...

c

1

= 12 π

ZZZ

0

2

1

2

0

2

1

2

0

ππ 2

ψψθθθ θ

θθ

nn

in in

*( ) ( )dc eedc==

−

ππ

dcθ= 2 π

1

2

Z

0

2

1

π

ψψθθθ

nn

*( ) ( )d =

E

n

I

n

=



22

2