The Chemistry Maths Book, Second Edition

21.8 The Gaussian distribution 615

If the variables are transformed to spherical polar coordinates in the space then

is the probability that the point lies in the section of spherical shell between radii

r

1

andr

2

, anglesθ

1

andθ

2

, and anglesφ

1

andφ

2

(see the discussion of Figure 10.6).

The probability that the point lies anywherein the shell betweenr

1

andr

2

is then

P(r

1

1 < 1 r 1 < 1 r

2

) 1 = 1 P(r

1

1 < 1 r 1 < 1 r

2

; 0 1 < 1 θ 1 < 1 π; 0 1 < 1 φ 1 < 12 π)

The quantity in square brackets,

is called a radial density function, and p(r)dris the probability that the variable

has value in the (infinitesimal) interval rto r 1 + 1 drthat is, that

(x, y, z)lies in a spherical shell of radius rand thicknessdr.

For example, the modulus square of the wave function of an electron,ρ(r, θ, φ)

= 1 |ψ(r,θ,φ)|

2

is an electron probability density function. For the electron in the

hydrogen atom, the wave function has the form (see equation 14.83)

ψ

n,l,m

(r, θ, φ) 1 = 1 R

n,l

(r)Θ

l,m

(θ)Φ

m

(φ),

and the corresponding radialdensity function is

p(r) 1 = 1 r

2

R

2

n,l

(r)

0 Exercises 24, 25

21.8 The Gaussian distribution

The Gaussian distribution, also called the normal distribution, is the continuous

distribution with mean μ and standard deviation σwhose probability density

function is

(21.39)

for all values of x. The graphs for three values of σare shown in Figure 21.7.

ρ

σ

μσ

()

()

xe

x

=

−−

1

2

22

2

π

rxyz=++

222

pr()=,,ZZ( ) sinr r d d

0

2

0

2

π π

ρφθθθφ

=,,

















ZZZ

r

r

rr dddr

1

2

0

2

0

2

π π

ρφ()sinθθθφ

Pr r r

r

r

()

121 21 2

1

2

1

2

1

2

<<; < < ; << =θθθφφφ ρ

φ

φ

θ

θ

ZZZ(()sinrr drdd,,θθθφ φ

2