14.6 The hydrogen atom 405
are called spherical harmonics. They occur whenever a physical problem in three
dimensions is formulated in spherical polar coordinates.*Some of these functions
are listed in Table 14.1. The functions are complex whenm 1 ≠ 10 and (see equations
(14.63)) it is sometimes more convenient to use the corresponding real functions
(14.69)
Spherical harmonics multiplied by the factorr
lare called solid harmonics, and the
real forms of these are also listed in the table, with their conventional names in atomic
structure theory.
By virtue of the orthonormality relations (14.64) and (14.67), the spherical harmonics
form an orthonormal set over a complete solid angle (θ 1 = 101 → 1 π, φ 1 = 101 → 12 π):
(14.70)
Table 14.1
Spherical harmonics Solid harmonics (real)
Angular momentum
The angular equation (14.56) can be written as the eigenvalue equation (with
appropriate units)
− (14.71)
∂
∂
∂
∂
∂
∂
222211
sin
sin
sin
θθ
θ
θ
θ φ
=+
,,Yll Y
lm lm()1
2dxyd
xyxy2212
22
12
15
16
15
4
−
=
−, =
ππ
()
xy
Ye
i22
12
22
15
32
,±
±
=
π
sin θ
φdxzdyz
xz yz=
,=
12 12
15
4
15
ππ 4
Ye
i21
12
15
8
,±
±
=
π
sin cosθθ
φdzr
z212
22
5
8
= 3
−
π
()
Y
20
12
2
5
16
31
,
=
−
π
( cos θ )
pxpy
xy=
,=
12 12
3
4
3
ππ 4
Ye
i11
12
3
8
,±
±
=
π
sinθ
φpz
z=
12
3
4 π
Y
10
12
3
4
,
=
π
cosθ
s=
12
1
4 π
Y
00
12
1
4
,
=
π
ZZ
020ππYY dd
lm, l m′, ′ ll,′*( ) ( )sinθθθθ,,φφ φδδ=
mmm, ′1
2
1
2
()()YY 0
i
YY m
lm l m,,− lm l m,,−+, − , >
*The spherical harmonics are often defined with an additional ‘phase factor’(−1)
(m 1 + 1 |m|) 22that multiplies the
function by (−1)when mis odd and positive.