In quantum mechanics, the Wigner's 3-j symbols, also called 3-jm symbols, are an alternative to Clebsch–Gordan coefficients for the purpose of adding angular momenta. While the two approaches address exactly the same physical problem, the 3-j symbols do so more symmetrically.
Mathematical relation to Clebsch–Gordan coefficients The 3-j symbols are given in terms of the Clebsch–Gordan coefficients by
( j 1 j 2 j 3 m 1 m 2 m 3 ) ≡ ( − 1 ) j 1 − j 2 − m 3 2 j 3 + 1 ⟨ j 1 m 1 j 2 m 2 | j 3 ( − m 3 ) ⟩ . {\displaystyle {\begin{pmatrix}j_{1}&j_{2}&j_{3}\\m_{1}&m_{2}&m_{3}\end{pmatrix}}\equiv {\frac {(-1)^{j_{1}-j_{2}-m_{3}}}{\sqrt {2j_{3}+1}}}\langle j_{1}\,m_{1}\,j_{2}\,m_{2}|j_{3}\,(-m_{3})\rangle .}
The j and m components are angular-momentum quantum numbers, i.e., every j (and every corresponding m) is either a nonnegative integer or half-odd-integer. The exponent of the sign factor is always an integer, so it remains the same when transposed to the left, and the inverse relation follows upon making the substitution m3 → −m3:
⟨ j 1 m 1 j 2 m 2 | j 3 m 3 ⟩ = ( − 1 ) − j 1 + j 2 − m 3 2 j 3 + 1 ( j 1 j 2 j 3 m 1 m 2 − m 3 ) . {\displaystyle \langle j_{1}\,m_{1}\,j_{2}\,m_{2}|j_{3}\,m_{3}\rangle =(-1)^{-j_{1}+j_{2}-m_{3}}{\sqrt {2j_{3}+1}}{\begin{pmatrix}j_{1}&j_{2}&j_{3}\\m_{1}&m_{2}&-m_{3}\end{pmatrix}}.}
Explicit expression
( j 1 j 2 j 3 m 1 m 2 m 3 ) ≡ δ ( m 1 + m 2 + m 3 , 0 ) ( − 1 ) j 1 − j 2 − m 3
( j 1 + j 2 − j 3 ) ! ( j 1 − j 2 + j 3 ) ! ( − j 1 + j 2 + j 3 ) ! ( j 1 + j 2 + j 3 + 1 ) ! ×
× ( j 1 − m 1 ) ! ( j 1 + m 1 ) ! ( j 2 − m 2 ) ! ( j 2 + m 2 ) ! ( j 3 − m 3 ) ! ( j 3 + m 3 ) ! ×
× ∑ k = K N ( − 1 ) k k ! ( j 1 + j 2 − j 3 − k ) ! ( j 1 − m 1 − k ) ! ( j 2 + m 2 − k ) ! ( j 3 − j 2 + m 1 + k ) ! ( j 3 − j 1 − m 2 + k ) ! , {\displaystyle {\begin{aligned}{\begin{pmatrix}j_{1}&j_{2}&j_{3}\\m_{1}&m_{2}&m_{3}\end{pmatrix}}&\equiv \delta (m_{1}+m_{2}+m_{3},0)(-1)^{j_{1}-j_{2}-m_{3}}{}{\sqrt {\frac {(j_{1}+j_{2}-j_{3})!(j_{1}-j_{2}+j_{3})!(-j_{1}+j_{2}+j_{3})!}{(j_{1}+j_{2}+j_{3}+1)!}}}\ \times {}\\[6pt]&\times {\sqrt {(j_{1}-m_{1})!(j_{1}+m_{1})!(j_{2}-m_{2})!(j_{2}+m_{2})!(j_{3}-m_{3})!(j_{3}+m_{3})!}}\ \times {}\\[6pt]&\times \sum _{k=K}^{N}{\frac {(-1)^{k}}{k!(j_{1}+j_{2}-j_{3}-k)!(j_{1}-m_{1}-k)!(j_{2}+m_{2}-k)!(j_{3}-j_{2}+m_{1}+k)!(j_{3}-j_{1}-m_{2}+k)!}},\end{aligned}}}
where δ ( i , j ) {\displaystyle \delta (i,j)} is the Kronecker delta. The summation is performed over those integer values k for which the argument of each factorial in the denominator is non-negative, i.e. summation limits K and N are taken equal: the lower one K = max ( 0 , j 2 − j 3 − m 1 , j 1 − j 3 + m 2 ) , {\displaystyle K=\max(0,j_{2}-j_{3}-m_{1},j_{1}-j_{3}+m_{2}),} the upper one N = min ( j 1 + j 2 − j 3 , j 1 − m 1 , j 2 + m 2 ) . {\displaystyle N=\min(j_{1}+j_{2}-j_{3},j_{1}-m_{1},j_{2}+m_{2}).} Factorials of negative numbers are conventionally taken equal to zero, so that the values of the 3j symbol at, for example, j 3 > j 1 + j 2 {\displaystyle j_{3}>j_{1}+j_{2}} or j 1 < m 1 {\displaystyle j_{1}<m_{1}} are automatically set to zero.
Definitional relation to Clebsch–Gordan coefficients The CG coefficients are defined so as to express the addition of two angular momenta in terms of a third:
| j 3 m 3 ⟩ = ∑ m 1 = − j 1 j 1 ∑ m 2 = − j 2 j 2 ⟨ j 1 m 1 j 2 m 2 | j 3 m 3 ⟩ | j 1 m 1 j 2 m 2 ⟩ . {\displaystyle |j_{3}\,m_{3}\rangle =\sum _{m_{1}=-j_{1}}^{j_{1}}\sum _{m_{2}=-j_{2}}^{j_{2}}\langle j_{1}\,m_{1}\,j_{2}\,m_{2}|j_{3}\,m_{3}\rangle |j_{1}\,m_{1}\,j_{2}\,m_{2}\rangle .}
The 3-j symbols, on the other hand, are the coefficients with which three angular momenta must be added so that the resultant is zero:
∑ m 1 = − j 1 j 1 ∑ m 2 = − j 2 j 2 ∑ m 3 = − j 3 j 3 | j 1 m 1 ⟩ | j 2 m 2 ⟩ | j 3 m 3 ⟩ ( j 1 j 2 j 3 m 1 m 2 m 3 ) = | 0 0 ⟩ . {\displaystyle \sum _{m_{1}=-j_{1}}^{j_{1}}\sum _{m_{2}=-j_{2}}^{j_{2}}\sum _{m_{3}=-j_{3}}^{j_{3}}|j_{1}m_{1}\rangle |j_{2}m_{2}\rangle |j_{3}m_{3}\rangle {\begin{pmatrix}j_{1}&j_{2}&j_{3}\\m_{1}&m_{2}&m_{3}\end{pmatrix}}=|0\,0\rangle .}
Here | 0 0 ⟩ {\displaystyle |0\,0\rangle } is the zero-angular-momentum state ( j = m = 0 {\displaystyle j=m=0} ). It is apparent that the 3-j symbol treats all three angular momenta involved in the addition problem on an equal footing and is therefore more symmetrical than the CG coefficient. Since the state | 0 0 ⟩ {\displaystyle |0\,0\rangle } is unchanged by rotation, one also says that the contraction of the product of three rotational states with a 3-j symbol is invariant under rotations.
Selection rules The Wigner 3-j symbol is zero unless all these conditions are satisfied:
m i ∈ { − j i , − j i + 1 , − j i + 2 , … , j i } ( i = 1 , 2 , 3 ) , m 1 + m 2 + m 3 = 0 , | j 1 − j 2 | ≤ j 3 ≤ j 1 + j 2 , ( j 1 + j 2 + j 3 ) is an integer (and, moreover, an even integer if m 1 = m 2 = m 3 = 0 ) . {\displaystyle {\begin{aligned}&m_{i}\in \{-j_{i},-j_{i}+1,-j_{i}+2,\ldots ,j_{i}\}\quad (i=1,2,3),\\&m_{1}+m_{2}+m_{3}=0,\\&|j_{1}-j_{2}|\leq j_{3}\leq j_{1}+j_{2},\\&(j_{1}+j_{2}+j_{3}){\text{ is an integer (and, moreover, an even integer if }}m_{1}=m_{2}=m_{3}=0{\text{)}}.\\\end{aligned}}}
Symmetry properties A 3-j symbol is invariant under an even permutation of its columns:
( j 1 j 2 j 3 m 1 m 2 m 3 ) = ( j 2 j 3 j 1 m 2 m 3 m 1 ) = ( j 3 j 1 j 2 m 3 m 1 m 2 ) . {\displaystyle {\begin{pmatrix}j_{1}&j_{2}&j_{3}\\m_{1}&m_{2}&m_{3}\end{pmatrix}}={\begin{pmatrix}j_{2}&j_{3}&j_{1}\\m_{2}&m_{3}&m_{1}\end{pmatrix}}={\begin{pmatrix}j_{3}&j_{1}&j_{2}\\m_{3}&m_{1}&m_{2}\end{pmatrix}}.}
An odd permutation of the columns gives a phase factor:
( j 1 j 2 j 3 m 1 m 2 m 3 ) = ( − 1 ) j 1 + j 2 + j 3 ( j 2 j 1 j 3 m 2 m 1 m 3 ) {\displaystyle {\begin{pmatrix}j_{1}&j_{2}&j_{3}\\m_{1}&m_{2}&m_{3}\end{pmatrix}}=(-1)^{j_{1}+j_{2}+j_{3}}{\begin{pmatrix}j_{2}&j_{1}&j_{3}\\m_{2}&m_{1}&m_{3}\end{pmatrix}}}
= ( − 1 ) j 1 + j 2 + j 3 ( j 1 j 3 j 2 m 1 m 3 m 2 ) = ( − 1 ) j 1 + j 2 + j 3 ( j 3 j 2 j 1 m 3 m 2