40
F. E. Harris
Expansion of the spherical harmonic product Y
m 1
l 1
Y
m 2
l 2
in the orthonormal set Y
M
L
,
carried out by taking scalar products with (Y
M
L
)
∗ , leads after use of Eqs. (33) and
(34) to
Y
m 1
l 1
(í µí»º)Y
m 2
l 2
(í µí»º) =
∑
LM
(−1)
M
[
l 1 l 2 L
m 1 m 2 −M
]
Y
M
L
(í µí»º).
(35)
Because harmonics with upper indices m 1 and m 2 form a product all of whose terms
have the same value of M, Eq. (35) can be simplified by dropping the M summation,
setting M = m 1 + m 2 .
The Gaunt coefficients can be written in terms of Wigner 3-j symbols [39]. Using
the standard notation for that symbol (an array of l and m values in ordinary parentheses), the Gaunt coefficients as defined here can be written
[
l 1 l 2 l 3
m 1 m 2 m 3
]
=
√ (2l 1 + 1)(2l 2 + 1)(2l 3 + 1)
4í µí¼
(
l 1 l 2 l 3
m 1 m 2 m 3
) (
l 1 l 2 l 3
0 0 0
)
. (36)
A pair of angular momenta in two independent variables can be coupled to form
a quantity of definite resultant angular momentum by forming a linear combination
of products of the individual angular momenta; the coefficients in that expansion
are called Clebsch-Gordan coefficients. Coupling of the angular-momentum wave
functions í µí¼
m 1
j 1
(1) and í µí¼
m 2
j 2
(2) with fixed values of j 1 and j 2 to form the combined
function í µí»¹
M
J
(1, 2) is described by
í µí»¹
M
J
(1, 2) =
∑
m 1 m 2
⟨
j 1 j 2 J
m 1 m 2 M
⟩
í µí¼
m 1
j 1
(1)í µí¼
m 2
j 2
(2),
(37)
where the array in angle brackets is our (nonstandard) notation for the ClebschGordan coefficient. Here all contributing terms must satisfy m 1 + m 2 = M, so we
can actually reduce Eq. (37) to a single sum over, say, m 2 , with m 1 set to M − m 2 .
The Clebsch-Gordan coefficients can also be written in terms of 3-j symbols:
⟨
j 1 j 2 j 3
m 1 m 2 m 3
⟩
= (−1)
j 1 −j 2 +m 3
√
2j 3 + 1
(
j 1 j 2 j 3
m 1 m 2 −m 3
)
.
(38)
Well-documented computer programs exist for the evaluation of the 3-j symbols,
making it straightforward to evaluate expressions involving Gaunt or Clebsch-Gordan
coefficients.
References
1. Hylleraas EA (1929) Z Phys 54:347
2. James HM, Coolidge AS (1936) Phys Rev 49:688
F. E. Harris
Expansion of the spherical harmonic product Y
m 1
l 1
Y
m 2
l 2
in the orthonormal set Y
M
L
,
carried out by taking scalar products with (Y
M
L
)
∗ , leads after use of Eqs. (33) and
(34) to
Y
m 1
l 1
(í µí»º)Y
m 2
l 2
(í µí»º) =
∑
LM
(−1)
M
[
l 1 l 2 L
m 1 m 2 −M
]
Y
M
L
(í µí»º).
(35)
Because harmonics with upper indices m 1 and m 2 form a product all of whose terms
have the same value of M, Eq. (35) can be simplified by dropping the M summation,
setting M = m 1 + m 2 .
The Gaunt coefficients can be written in terms of Wigner 3-j symbols [39]. Using
the standard notation for that symbol (an array of l and m values in ordinary parentheses), the Gaunt coefficients as defined here can be written
[
l 1 l 2 l 3
m 1 m 2 m 3
]
=
√ (2l 1 + 1)(2l 2 + 1)(2l 3 + 1)
4í µí¼
(
l 1 l 2 l 3
m 1 m 2 m 3
) (
l 1 l 2 l 3
0 0 0
)
. (36)
A pair of angular momenta in two independent variables can be coupled to form
a quantity of definite resultant angular momentum by forming a linear combination
of products of the individual angular momenta; the coefficients in that expansion
are called Clebsch-Gordan coefficients. Coupling of the angular-momentum wave
functions í µí¼
m 1
j 1
(1) and í µí¼
m 2
j 2
(2) with fixed values of j 1 and j 2 to form the combined
function í µí»¹
M
J
(1, 2) is described by
í µí»¹
M
J
(1, 2) =
∑
m 1 m 2
⟨
j 1 j 2 J
m 1 m 2 M
⟩
í µí¼
m 1
j 1
(1)í µí¼
m 2
j 2
(2),
(37)
where the array in angle brackets is our (nonstandard) notation for the ClebschGordan coefficient. Here all contributing terms must satisfy m 1 + m 2 = M, so we
can actually reduce Eq. (37) to a single sum over, say, m 2 , with m 1 set to M − m 2 .
The Clebsch-Gordan coefficients can also be written in terms of 3-j symbols:
⟨
j 1 j 2 j 3
m 1 m 2 m 3
⟩
= (−1)
j 1 −j 2 +m 3
√
2j 3 + 1
(
j 1 j 2 j 3
m 1 m 2 −m 3
)
.
(38)
Well-documented computer programs exist for the evaluation of the 3-j symbols,
making it straightforward to evaluate expressions involving Gaunt or Clebsch-Gordan
coefficients.
References
1. Hylleraas EA (1929) Z Phys 54:347
2. James HM, Coolidge AS (1936) Phys Rev 49:688