36. Clebsch-Gordan coefficients 1

36. Clebsch-Gordan coefficients 1

36. CLEBSCH-GORDAN COEFFICIENTS, SPHERICAL HARMONICS, AND d FUNCTIONS

Note: A square-root sign is to be understood over every coefficient, e.g., for − 8 / 15 read − 8 / 15.

Y

₁⁰

=

3

4 π cos θ Y

₁¹

= −

3

8 π sin θ e

^iφ

Y

₂⁰

=

5

4 π 3

2 cos

²

θ − 1 2 Y

₂¹

= −

15

8 π sin θ cos θ e

^iφ

Y

₂²

= 1

4

15

2 π sin

²

θ e

^2iφ

Y

^−m

= ( − 1)

^m

Y

^m∗

j

1

j

2

m

1

m

2

|j

1

j

2

JM

= ( − 1)

^J−j1−j₂

j

₂

j

₁

m

₂

m

₁

|j

₂

j

₁

JM d

_m,0

=

4 π

2 + 1 Y

^m

e

^−imφ

d

^j_m,m

= ( − 1)

^m−m

d

^j_m,m

= d

^j_−m,−m

d

¹_0,0

= cos θ d

^1/2_1/2,1/2

= cos θ 2 d

^1/2_1/2,−1/2

= − sin θ

2

d

¹_1,1

= 1 + cos θ 2 d

¹_1,0

= − sin √ θ

2 d

¹_1,−1

= 1 − cos θ

2

d

^3/2_3/2,3/2

= 1 + cos θ 2 cos θ

2 d

^3/2_3/2,1/2

= − √

3 1 + cos θ 2 sin θ

2 d

^3/2_3/2,−1/2

= √

3 1 − cos θ 2 cos θ

2 d

^3/2_3/2,−3/2

= − 1 − cos θ

2 sin θ 2 d

^3/2_1/2,1/2

= 3 cos θ − 1

2 cos θ 2 d

^3/2_1/2,−1/2

= − 3 cos θ + 1

2 sin θ 2

d

²_2,2

= 1 + cos θ 2

₂

d

²_2,1

= − 1 + cos θ

2 sin θ d

²_2,0

=

√ 6 4 sin

²

θ d

²_2,−1

= − 1 − cos θ

2 sin θ d

²_2,−2

=

1 − cos θ 2

₂

d

²_1,1

= 1 + cos θ

2 (2 cos θ − 1) d

²_1,0

= −

3

2 sin θ cos θ d

²_1,−1

= 1 − cos θ

2 (2 cos θ + 1) d

²_0,0

= 3

2 cos

²

θ − 1 2

+1

5/2 5/2 +3/2 3/2

+3/2 1/54/5 4/5

−1/5 5/2

5/2

−1/2 3/52/5

−1−2 3/2

−1/2

2/5 5/2 3/2

−3/2

−3/2 4/51/5−4/51/5

−1/2

−2 1

−5/25/2

−3/5

−1/2+1/2 +1−1/2 2/5 3/5

−2/5

−1/2 +22

+3/2 +3/2

+5/2 5/25/2

5/2 3/2 1/2

−1/31/2

−1 +10 1/6 +1/2

+1/2−1/2

−3/2 +1/2 1/152/5

−8/15 +1/2 1/10

3/103/5 5/2 3/2 1/2

−1/2

−1/3 5/21/6

−5/25/2 1

−3/23/2

−3/52/5

−3/2

−3/2 3/52/5 1/2

−1

−1 0

−1/2

−1/158/15

−2/5

−1/2−3/2

−1/2 3/103/5 1/10 +3/2

+3/2+1/2

−1/2 +3/2+1/2

+2+1

+2+1 0

+1 2/5 3/5

3/2 3/5

−2/5

−1 +10 +3/2 +1 1

+3 +1 1 0

3 1/3 +2 2/3

2 3/2 3/2

1/32/3 +1/2

−10 1/2 +1/2

−1/32/3

−1/2+1/2 1

+1 1 0 1/21/2

−1/2 0 0 1/2

−1/2 1 1

−1

−1/2 1 1

−1/2 +1/2 +1/2+1/2

+1/2

−1/2

−1/2+1/2 −1/2

−1 3/2

2/3 3/2

−3/2 1 1/3

−1/2

−1/2 1/2

−2/31/3 +1+1/2

+10 +3/2

2/3 3

3

3

3 3

−1 1

−2

−3 2/3 1/3

−2 2 1/3

−2/3

−2 0

−1

−2

−10 +1

−1 8/152/5 1/15

2

−1

−1−2−1 0 1/2

−1/6

−1/3 1

−1 1/10

−3/10 3/5

0 2

0 1

0 3/10−2/5 3/10 1/20

−1/2 1/5 1/5 3/5 +1

+1

−10 0−1 +1 1/15

8/152/5 2

+2 2 +1 1/21/2

1 1/2 2

0 1/6 1/6 2/3

1 1/2

−1/2 0

0 2

2

−2

−1 1

−1 1

−1 1/2

−1/2

−1 1/2 1/2 00

0

−1 1/3

−1/31/3

−1/2 +1

−1

−1 0 +10 0

+1

−1 2 1 0+10 +1 +1

+1 1/3

−1/21/6 1 +1 3/5

−3/10 1/10

−1/3

−1 +1 0

0 +2

+1 +2 3

+3/2

+1/2 +1

1/4 2

2

−1 1

2

−2 1

−1 1/4

−1/2 1/2

1/2

−1/2−1/2 +1/2

−3/2

−3/2 1/2

10 0 3/4 +1/2

−1/2 −1/2

2 +1 3/4

3/4

−3/4 1/4

−1/2 +1/2

−1/4 1

+1/2−1/2 +1/2 1

+1/2 3/5

−10 +1/2 0

+1/2 3/2 +1/2 +5/2

+2 −1/2 +2+1/2

+1 +1/2 1

2× 1/2

3/2 × 1/2 3/2 × 1

2 × 1 1 × 1/2 1/2 × 1/2

1× 1

Notation: ^{J J} M M

. . . . . .

. . . . . .

m₁ m₂

m₁ m₂ Coefficients

−1/5 2

2/7

−3/72/7 3 1/2

−1/2

−1−2 −2

−1

0 4

1/21/2

−3 3

−1/21/2

−2 1

−44

−2 1/5

−27/70 +1/2 +7/2 7/27/2

+5/2 3/74/7 +2+1 0 1 +2+1

+4 1 4

4 +2 3/14 3/144/7

+2 1/2

−1/20 +2

−10 +1+2 +2+1

−10

3 2

4 1/14 1/14 3/73/7

+1 3

−1/51/5 3/10

−3/10

+1 2

+2+1

−10

−2

−2−1 +10 +2 3/7 3/7

−1/14

−1/14

+1 1

4 3 2

2/7

2/7

−2/71/14

1/14 4

1/14 1/143/73/7

3 3/10

−3/10

−1/51/5

−1

−2

−2−1 0

−10

−2

−10 +1 +10

−1−2

−1 2

4 3/14 3/144/7

−2 −2 −2 3/7

3/7

−1/14

−1/14

−1 1

−3/101/5 3/10

−1

1 0

0 1/70

1/70 18/358/35 8/35

0 1/10

−1/10 2/5

−2/50

0 0

0 2/5

−2/5

−1/10 1/10

0 1/5

−1/51/5

−1/5 1/5

−1/5

−3/10 3/10 +1 2/7

−3/72/7 +3

1/2 +2+1

0 1/2 +2+2

+2

+1+2+1 +3 1/2

−1/2 +10 +2 4 3

+1/2+3/2 +3/2

+2 +5/2

4/7 7/2 +3/2 1/74/7 2/7

+3/25/2

+2+1

−10 16/35

−18/351/35

12/351/35 18/35 4/35 3/2

+3/2 +3/2

−3/2−1/2 +1/2

−2/52/5 7/2

7/2 18/354/35 12/35 1/35

−1/2 5/2 27/70

−5/143/35

−6/35

−1/2 3/2

7/2

7/2

−5/2 4/73/7

5/2

−5/2

−4/73/7

−3/2

−2 2/74/7

1/7

−3/25/2

−1−2 18/35

−1/35

−16/35

−3/2

−2/51/5 2/5

−3/2−1/2

−3/2 3/2

7/2 1

−7/2

−1/2

−1/52/5 0

−10

−2 2/5

−1/21/2 1/10 3/10−1/5

−2/5

−3/2−1/2 +1/2 5/2 3/2 1/2

+1/2 2/5 1/5

−3/2−1/2 +1/2+3/2

−1/10

−3/10 +1/2

2/5 2/5 +10

−1−2 0 +33

+23 2 +2 +3/2 1

+3/2+1/2+1/2 1/2

−1/2

−1/2+1/2 +3/2

1/2 3 2

3 0 1/20

1/20 9/209/20

2 1

3

−1 1/5 1/53/5

2

3 3 1

−3

−2 1/21/2

−3/2 2

−1/21/2

−3/2

−2

−1 1/2

−1/2

−1/2−3/2 0

1

−1 3/10 3/10−2/5

−3/2−1/2 0

0 1/41/4

−1/4

−1/4 0 9/20

9/20 +1/2−1/2

−3/2

−1/20

−1/20 0 1/4

−1/41/4

−1/4

−3/2−1/2 +1/2 1/2

−1/20 1 3/10 3/10

−3/2−1/2 +1/2+3/2 +3/2+1/2

−1/2−3/2

−2/5 +1 +1 +1 1/53/5 1/5 1/2 +3/2+1/2

−1/2 +3/2 +3/2

−1/5 +1/2 6/355/14

−3/35 1/5

−3/7

−1/2+1/2 +3/2

2 × 3/2

5/2

2× 2

3/2 × 3/2

−3

Figure 36.1: 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).