The Chemistry Maths Book, Second Edition

10.4 Volume integrals 303

Average values

We shall see in Chapter 21 that if xis a continuous variable in the intervala 1 ≤ 1 x 1 ≤ 1 b,

and ifp(x) 1 dxis the probability that the variable have value between xandx 1 + 1 dx,

then the quantity

(10.14)

is the average value of the functionf(x)in the interval. The generalization to functions

of more than one variable involves the corresponding multiple integral. For example,

letf(x, 1 y, 1 z)be a function of position in three dimensions, and letp(x, 1 y, 1 z)dx dy dz

be the probability that the x-coordinate have value between xandx 1 + 1 dx, that the

y-coordinate have value between yandy 1 + 1 dy, and that the z-coordinate have value

between zandz 1 + 1 dz. The average value of the function is then the volume integral

over all space

(10.15)

The corresponding expression in spherical polar coordinates is

(10.16)

If the probability density is the modulus square of a wave function in quantum

mechanics,p 1 = 1 |ψ|

2

, the average value

(10.17)

is usually called the expectation value of fin the state ψ.

EXAMPLE 10.9Find the average distance of the electron from the nucleus in (i) the

1 sorbital and (ii) the 2 p

z

orbital of the hydrogen atom (see Table 10.1 in Example 10.4).

In these cases,f(r, 1 θ, 1 φ) 1 = 1 rin equation (10.17), and

(i)

r

a

erdrdd

s

ra

1

0

3

0

2

00

2

3

1

=

−

π

ππ

ZZZ

∞

2

0

sinθθφ

ψ

1

2

0

3

2

1

s

ra

a

=,e

−

π

2

0

rr d=||Z ψ

2

v

ffd=Z ||ψ

2

v

ff=,ZZZ rp,,rrd,rdd

0

2

00

2

ππ

∞

()()sinθθ θθφφ φ

f=,fxyzpxyz dxdydz,,,

−

+

−

+

−

+

ZZZ

∞

∞

∞

∞

∞

∞

()()

ffxpxdx

a

b

=Z ()()