The Chemistry Maths Book, Second Edition

356 Chapter 12Second-order differential equations. Constant coefficients

It follows therefore that

(12.58)

The functions are said to be orthogonal. For normalizedwave functions, when

we can write

(12.59)

The quantity δ

mn

, equal to 1 if m 1 = 1 n(normalization) and equal to 0 if m 1 ≠ 1 n

(orthogonality), is called the Kronecker delta. Functions that satisfy (12.59) are said

to be orthonormal(orthogonal and normalized).

0 Exercise 26

12.7 The particle in a ring

The Schrödinger equation for a particle of mass mmoving freely in

a circle of radius ris, Figure 12.7,

(12.60)

where Eis the (positive) kinetic energy of the particle andI 1 = 1 mr

2

is

its moment of inertia with respect to the centre of the circle. Setting

(12.61)

we have

(12.62)

with general solution, in exponential form

ψ(θ) 1 = 1 c

1

e

iωθ

1 + 1 c

2

e

−iωθ

(12.63)

For the wave function to be continuous around the circle, it must satisfy the periodic

boundary condition

ψ(θ 1 + 12 π) 1 = 1 ψ(θ) (12.64)

d

d

2

2

2

0

ψ

ωψ

θ

+=

ω

2

2

2

=

IE



Hψ

ψ

θ ψ

θ

θ

() θ

()

=− = ()



22

2

2 I

d

d

E

Z

0

1

0

l

mn mn

xxdx

mn

mn

ψψ() () ==δ

=

≠











if

if

Z

0

1

l

nn

ψψ() ()xxdx= ,

Z

0

0

l

mn

ψψ() ()xxdx=≠when mn

o

• 
  

r

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Figure 12.7