The Chemistry Maths Book, Second Edition

14.6 The hydrogen atom 409

ψ

n,l,m

(r,θ,φ) 1 = 1 R

n,l

(r)Θ

l,m

(θ)Φ

m

(φ) (14.83)

and the numbers n,land mare called the principal quantum number, (n), the

angular momentum quantum number, (l), and the component of angular momentum

(or magnetic) quantum number (m). From the orthormality relations (14.64) forΦ

m

,

(14.67) forΘ

l,m

and (14.81) forR

n,l

it follows that the total wave functions form an

orthonormal set in three-dimensional space:

= 1 δ

n,n′

δ

l,l′

δ

m,m′

(14.84)

Some of the total wave functions of the hydrogen atom, the atomic orbitals, are

illustrated in Figure 14.5 by contour diagrams in an appropriate plane containing the

nucleus.*Solid contours represent positive values of the orbitals, dashed contours

represent negative values, dotted contours are nodes. We note that wave functions

ψ

n,l,m

with energyE

n

have a total of (n 1 − 11 ) nodal surfaces. It is these diagrams that

have led to the conventional pictorial representation of atomic orbitals.

0 Exercises 12–15

ZZψψ ZZψ θφ

nlm n l m nlm

dr

,, ′,′,′ ,,

• v=,( ,

000

2

∞

ππ

*))()sinψφθθθφ

′,′,′

,,

nlm

rr drdd

2

*The diagrams in Figure 14.5 are to scale within boxes of side 20 (a

0

). The contour values are±0.002 1 × 12

i

, i 1 = 1 0,

1, 2, 3, =

1 s 2 s 2 p

x

3 s 3 p

x

3 d

xy

Figure 14.5