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7 Spherical Symmetry and Spins
Table 7.1 Complex and cubic real forms of the spherical harmonics for ℓ = 0, 1, 2, 3. The constants N ℓ are the common normalizing factors over the θ and φ coordinates
ℓN ℓ
|LM| Γγ
s
1
4π
|00=1
|A 1g =1
p
3
4π r −1
|1 + 1=−
1
√
2
(x + iy)
|T 1u x=x
|1 − 1=
1
√
2
(x − iy)
|T 1u y=y
|10=z
|T 1u z=z
d
15
8π r −2
|2 + 2=
1
2 (x + iy) 2
|E g θ=
1
√
6
(3z 2 − r 2 )
|2 − 2=
1
2 (x − iy) 2
|E g ǫ=
1
√
2
(x 2 − y 2 )
|2 + 1=−(x + iy)z
|T 2g ξ =
√
2yz
|2 − 1=(x − iy)z
|T 2g η=
√
2xz
|20=
1
√
6
(3z 2 − r 2 )
|T 2g ζ =
√
2xy
f
35
8π r −3
|3 + 3=−
1
2
√
2
(x + iy) 3
|A 2u =
3
2 xyz
|3 − 3=
1
2
√
2
(x − iy) 3
|T 1u x=
1
√
10
x(5x 2 − 3r 2 )
|3 + 2=
√
3
2 z(x + iy) 2
|T 1u y=
1
√
10
y(5y 2 − 3r 2 )
|3 − 2=
√
3
2 z(x − iy) 2
|T 1u z=
1
√
10
z(5z 2 − 3r 2 )
|3 + 1=−
√
3
2
√
10
(x + iy)(5z 2 − 3r 2 )
|T 2u ξ =
3
2 x(z 2 − y 2 )
|3 − 1=
√
3
2
√
10
(x − iy)(5z 2 − 3r 2 )
|T 2u η=
3
2 y(x 2 − z 2 )
|30=
1
√
10
z(5z 2 − 3r 2 )
|T 2u ζ =
3
2 z(y 2 − x 2 )
spherical harmonics is given by
Φ m ℓ (φ) =
1
√
2π
exp(im ℓ φ)
(7.5)
A rotation ˆ
C α about the z-direction affects this function in the following way:
ˆ
C α Φ m ℓ (φ) = Φ m ℓ (φ − α) = exp(−im ℓ α)Φ m ℓ (φ)
(7.6)
The trace over the entire function space is then given by
χ
ℓ (C α ) =
ℓ
m ℓ =−ℓ
exp(−im ℓ α) =
sin(ℓ + 1/2)α
sin(α/2)
(7.7)