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45. Clebsch-Gordan coefficients
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45. Clebsch-Gordan Coefficients, Spherical Harmonics, and d Functions
p
Note: A square-root sign is to be understood over every coefficient, e.g., for −8/15 read − 8/15.
Y11
Y20
Y21
Y22
r
3
cos θ
4π
r
3
=−
sin θ eiφ
8π
r 5 3
1
cos2 θ −
=
4π 2
2
r
15
sin θ cos θ eiφ
=−
8π
r
1 15
sin2 θ e2iφ
=
4 2π
Y10 =
Yℓ−m = (−1)m Yℓm∗
d ℓm,0 =
j
′
j
r
hj1 j2 m1 m2 |j1 j2 JM i
4π
Y m e−imφ
2ℓ + 1 ℓ
j
d m′ ,m = (−1)m−m d m,m′ = d −m,−m′
= (−1)J−j1 −j2 hj2 j1 m2 m1 |j2 j1 JM i
d 10,0 = cos θ
1/2
d 1/2,1/2 = cos
θ
2
1/2
d 1/2,−1/2 = − sin
1 + cos θ
θ
cos
2
2
√ 1 + cos θ
θ
3/2
sin
d 3/2,1/2 = − 3
2
2
√ 1 − cos θ
θ
3/2
d 3/2,−1/2 = 3
cos
2
2
1 − cos θ
θ
3/2
d 3/2,−3/2 = −
sin
2
2
3 cos θ − 1
θ
3/2
d 1/2,1/2 =
cos
2
2
θ
3 cos θ + 1
3/2
sin
d 1/2,−1/2 = −
2
2
1 + cos θ
2
sin θ
1
d 1,0 = − √
2
1 − cos θ
1
d 1,−1 =
2
d 11,1 =
θ
2
3/2
d 3/2,3/2 =
d 22,2 =
1 + cos θ 2
2
1 + cos θ
sin θ
=−
2
√
6
d 22,0 =
sin2 θ
4
1 − cos θ
d 22,−1 = −
sin θ
2
1 − cos θ 2
d 22,−2 =
2
d 22,1
1 + cos θ
(2 cos θ − 1)
2
r
3
d 21,0 = −
sin θ cos θ
2
1 − cos θ
(2 cos θ + 1)
d 21,−1 =
2
d 21,1 =
d 20,0 =
3
2
cos2 θ −
1
2
Figure 45.1: The sign convention is that of Wigner (Group Theory, Academic Press, New York, 1959), also used by Condon and Shortley (The
Theory of Atomic Spectra, Cambridge Univ. Press, New York, 1953), Rose (Elementary Theory of Angular Momentum, Wiley, New York, 1957),
and Cohen (Tables of the Clebsch-Gordan Coefficients, North American Rockwell Science Center, Thousand Oaks, Calif., 1974).