The Chemistry Maths Book, Second Edition

242 Chapter 8Complex numbers

A function that has the same circular periodicity as the figure with nequidistant

points on a circle is

f(θ) 1 = 1 e

inθ

(8.45)

This function is periodic in θwith period 2 π 2 n. Thus

f(θ 1 + 12 π 2 n) 1 = 1 e

in(θ+ 2 π 2 n)

1 = 1 e

inθ

1 × 1 e

2 πi

1 = 1 e

inθ

1 = 1 f(θ)

Such functions are important for the description of systems with circular periodicity.

Periodicity on a line

Figure 8.10 shows a simple linear array of equidistant points representing, for example,

a linear lattice. A function of xthat has the same periodicity as the lattice must satisfy

the periodicity condition

f(x 1 + 1 a) 1 = 1 f(x) (8.46)

The simplest such function is

f(x) 1 = 1 e

2 πxi 2 a

(8.47)

Thus,

f(x 1 + 1 a) 1 = 1 e

2 π(x+a)i 2 a

1 = 1 e

2 πxi 2 a

1 × 1 e

2 πi

1 = 1 f(x)e

2 πi

1 = 1 f(x)

Functions like (8.47) are important for the description of the properties of periodic

systems such as crystals. The functions are readily generalized for three-dimensional

periodic systems: the function

f(x, y, z) 1 = 1 e

2 πxi 2 a

e

2 πyi 2 b

e

2 πzi 2 c

(8.48)

has period ain the x-direction, bin the y-direction, and cin the z-direction.

Rotation in quantum mechanics

Figure 8.11 shows a system of two masses, m

1

and m

2

,

joined by a rigid rod (of negligible mass) rotating about

the centre of mass at O. As discussed in Section 5.6

(Example 5.13) the system has moment of inertiaI 1 = 1 μr

2

where μ 1 = 1 m

1

m

2

2 (m

1

1 + 1 m

2

) is the reduced mass and

r 1 = 1 r

1

1 + 1 r

2

is the distance between the masses. Such a

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Figure 8.10

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Figure 8.11