3-j symbol

In quantum mechanics, the Wigner 3-j symbols, also called 3-jm symbols, are an alternative to Clebsch–Gordan coefficients for the purpose of adding angular momenta.^[1] 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

${\displaystyle {\begin{pmatrix}j_{1}&j_{2}&j_{3}\\m_{1}&m_{2}&m_{3}\end{pmatrix}}\equiv {\frac {(-1)^{j_{1}-j_{2}-m_{3}}}{\sqrt {2j_{3}+1}}}\langle j_{1}\,m_{1}\,j_{2}\,m_{2}|j_{3}\,(-m_{3})\rangle .}$

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:

${\displaystyle \langle j_{1}\,m_{1}\,j_{2}\,m_{2}|j_{3}\,m_{3}\rangle =(-1)^{-j_{1}+j_{2}-m_{3}}{\sqrt {2j_{3}+1}}{\begin{pmatrix}j_{1}&j_{2}&j_{3}\\m_{1}&m_{2}&-m_{3}\end{pmatrix}}.}$

Explicit expression

${\displaystyle {\begin{aligned}{\begin{pmatrix}j_{1}&j_{2}&j_{3}\\m_{1}&m_{2}&m_{3}\end{pmatrix}}&\equiv \delta (m_{1}+m_{2}+m_{3},0)(-1)^{j_{1}-j_{2}-m_{3}}{}{\sqrt {\frac {(j_{1}+j_{2}-j_{3})!(j_{1}-j_{2}+j_{3})!(-j_{1}+j_{2}+j_{3})!}{(j_{1}+j_{2}+j_{3}+1)!}}}\ \times {}\\[6pt]&\times {\sqrt {(j_{1}-m_{1})!(j_{1}+m_{1})!(j_{2}-m_{2})!(j_{2}+m_{2})!(j_{3}-m_{3})!(j_{3}+m_{3})!}}\ \times {}\\[6pt]&\times \sum _{k=K}^{N}{\frac {(-1)^{k}}{k!(j_{1}+j_{2}-j_{3}-k)!(j_{1}-m_{1}-k)!(j_{2}+m_{2}-k)!(j_{3}-j_{2}+m_{1}+k)!(j_{3}-j_{1}-m_{2}+k)!}},\end{aligned}}}$

where ${\displaystyle \delta (i,j)}$ 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 ${\displaystyle K=\max(0,j_{2}-j_{3}-m_{1},j_{1}-j_{3}+m_{2}),}$ the upper one ${\displaystyle N=\min(j_{1}+j_{2}-j_{3},j_{1}-m_{1},j_{2}+m_{2}).}$ Factorials of negative numbers are conventionally taken equal to zero, so that the values of the 3j symbol at, for example, ${\displaystyle j_{3}>j_{1}+j_{2}}$ or ${\displaystyle j_{1}<m_{1}}$ 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:

${\displaystyle |j_{3}\,m_{3}\rangle =\sum _{m_{1}=-j_{1}}^{j_{1}}\sum _{m_{2}=-j_{2}}^{j_{2}}\langle j_{1}\,m_{1}\,j_{2}\,m_{2}|j_{3}\,m_{3}\rangle |j_{1}\,m_{1}\,j_{2}\,m_{2}\rangle .}$

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:

${\displaystyle \sum _{m_{1}=-j_{1}}^{j_{1}}\sum _{m_{2}=-j_{2}}^{j_{2}}\sum _{m_{3}=-j_{3}}^{j_{3}}|j_{1}m_{1}\rangle |j_{2}m_{2}\rangle |j_{3}m_{3}\rangle {\begin{pmatrix}j_{1}&j_{2}&j_{3}\\m_{1}&m_{2}&m_{3}\end{pmatrix}}=|0\,0\rangle .}$

Here ${\displaystyle |0\,0\rangle }$ is the zero-angular-momentum state (${\displaystyle j=m=0}$). 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 ${\displaystyle |0\,0\rangle }$ 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:

${\displaystyle {\begin{aligned}&m_{i}\in \{-j_{i},-j_{i}+1,-j_{i}+2,\ldots ,j_{i}\}\quad (i=1,2,3),\\&m_{1}+m_{2}+m_{3}=0,\\&|j_{1}-j_{2}|\leq j_{3}\leq j_{1}+j_{2},\\&(j_{1}+j_{2}+j_{3}){\text{ is an integer (and, moreover, an even integer if }}m_{1}=m_{2}=m_{3}=0{\text{)}}.\\\end{aligned}}}$

Symmetry properties

A 3-j symbol is invariant under an even permutation of its columns:

${\displaystyle {\begin{pmatrix}j_{1}&j_{2}&j_{3}\\m_{1}&m_{2}&m_{3}\end{pmatrix}}={\begin{pmatrix}j_{2}&j_{3}&j_{1}\\m_{2}&m_{3}&m_{1}\end{pmatrix}}={\begin{pmatrix}j_{3}&j_{1}&j_{2}\\m_{3}&m_{1}&m_{2}\end{pmatrix}}.}$

An odd permutation of the columns gives a phase factor:

${\displaystyle {\begin{pmatrix}j_{1}&j_{2}&j_{3}\\m_{1}&m_{2}&m_{3}\end{pmatrix}}=(-1)^{j_{1}+j_{2}+j_{3}}{\begin{pmatrix}j_{2}&j_{1}&j_{3}\\m_{2}&m_{1}&m_{3}\end{pmatrix}}}$

${\displaystyle =(-1)^{j_{1}+j_{2}+j_{3}}{\begin{pmatrix}j_{1}&j_{3}&j_{2}\\m_{1}&m_{3}&m_{2}\end{pmatrix}}=(-1)^{j_{1}+j_{2}+j_{3}}{\begin{pmatrix}j_{3}&j_{2}&j_{1}\\m_{3}&m_{2}&m_{1}\end{pmatrix}}.}$

Changing the sign of the ${\displaystyle m}$ quantum numbers (time reversal) also gives a phase:

${\displaystyle {\begin{pmatrix}j_{1}&j_{2}&j_{3}\\-m_{1}&-m_{2}&-m_{3}\end{pmatrix}}=(-1)^{j_{1}+j_{2}+j_{3}}{\begin{pmatrix}j_{1}&j_{2}&j_{3}\\m_{1}&m_{2}&m_{3}\end{pmatrix}}.}$

The 3-j symbols also have so-called Regge symmetries, which are not due to permutations or time reversal.^[2] These symmetries are:

${\displaystyle {\begin{pmatrix}j_{1}&j_{2}&j_{3}\\m_{1}&m_{2}&m_{3}\end{pmatrix}}={\begin{pmatrix}j_{1}&{\frac {j_{2}+j_{3}-m_{1}}{2}}&{\frac {j_{2}+j_{3}+m_{1}}{2}}\\j_{3}-j_{2}&{\frac {j_{2}-j_{3}-m_{1}}{2}}-m_{3}&{\frac {j_{2}-j_{3}+m_{1}}{2}}+m_{3}\end{pmatrix}},}$

${\displaystyle {\begin{pmatrix}j_{1}&j_{2}&j_{3}\\m_{1}&m_{2}&m_{3}\end{pmatrix}}=(-1)^{j_{1}+j_{2}+j_{3}}{\begin{pmatrix}{\frac {j_{2}+j_{3}+m_{1}}{2}}&{\frac {j_{1}+j_{3}+m_{2}}{2}}&{\frac {j_{1}+j_{2}+m_{3}}{2}}\\j_{1}-{\frac {j_{2}+j_{3}-m_{1}}{2}}&j_{2}-{\frac {j_{1}+j_{3}-m_{2}}{2}}&j_{3}-{\frac {j_{1}+j_{2}-m_{3}}{2}}\end{pmatrix}}.}$

With the Regge symmetries, the 3-j symbol has a total of 72 symmetries. These are best displayed by the definition of a Regge symbol, which is a one-to-one correspondence between it and a 3-j symbol and assumes the properties of a semi-magic square:^[3]

${\displaystyle R={\begin{array}{|ccc|}\hline -j_{1}+j_{2}+j_{3}&j_{1}-j_{2}+j_{3}&j_{1}+j_{2}-j_{3}\\j_{1}-m_{1}&j_{2}-m_{2}&j_{3}-m_{3}\\j_{1}+m_{1}&j_{2}+m_{2}&j_{3}+m_{3}\\\hline \end{array}},}$

whereby the 72 symmetries now correspond to 3! row and 3! column interchanges plus a transposition of the matrix. These facts can be used to devise an effective storage scheme.^[3]

Orthogonality relations

A system of two angular momenta with magnitudes j₁ and j₂ can be described either in terms of the uncoupled basis states (labeled by the quantum numbers m₁ and m₂), or the coupled basis states (labeled by j₃ and m₃). The 3-j symbols constitute a unitary transformation between these two bases, and this unitarity implies the orthogonality relations

${\displaystyle (2j_{3}+1)\sum _{m_{1}m_{2}}{\begin{pmatrix}j_{1}&j_{2}&j_{3}\\m_{1}&m_{2}&m_{3}\end{pmatrix}}{\begin{pmatrix}j_{1}&j_{2}&j'_{3}\\m_{1}&m_{2}&m'_{3}\end{pmatrix}}=\delta _{j_{3},j'_{3}}\delta _{m_{3},m'_{3}}{\begin{Bmatrix}j_{1}&j_{2}&j_{3}\end{Bmatrix}},}$

${\displaystyle \sum _{j_{3}m_{3}}(2j_{3}+1){\begin{pmatrix}j_{1}&j_{2}&j_{3}\\m_{1}&m_{2}&m_{3}\end{pmatrix}}{\begin{pmatrix}j_{1}&j_{2}&j_{3}\\m_{1}'&m_{2}'&m_{3}\end{pmatrix}}=\delta _{m_{1},m_{1}'}\delta _{m_{2},m_{2}'}.}$

The triangular delta {j₁ j₂ j₃} is equal to 1 when the triad (j₁, j₂, j₃) satisfies the triangle conditions, and is zero otherwise. The triangular delta itself is sometimes confusingly called^[4] a "3-j symbol" (without the m) in analogy to 6-j and 9-j symbols, all of which are irreducible summations of 3-jm symbols where no m variables remain.

Relation to spherical harmonics; Gaunt coefficients

The 3-jm symbols give the integral of the products of three spherical harmonics^[5]

${\displaystyle {\begin{aligned}&\int Y_{l_{1}m_{1}}(\theta ,\varphi )Y_{l_{2}m_{2}}(\theta ,\varphi )Y_{l_{3}m_{3}}(\theta ,\varphi )\,\sin \theta \,\mathrm {d} \theta \,\mathrm {d} \varphi \\&\quad ={\sqrt {\frac {(2l_{1}+1)(2l_{2}+1)(2l_{3}+1)}{4\pi }}}{\begin{pmatrix}l_{1}&l_{2}&l_{3}\\0&0&0\end{pmatrix}}{\begin{pmatrix}l_{1}&l_{2}&l_{3}\\m_{1}&m_{2}&m_{3}\end{pmatrix}}\end{aligned}}}$

with ${\displaystyle l_{1}}$, ${\displaystyle l_{2}}$ and ${\displaystyle l_{3}}$ integers. These integrals are called Gaunt coefficients.

Relation to integrals of spin-weighted spherical harmonics

Similar relations exist for the spin-weighted spherical harmonics if ${\displaystyle s_{1}+s_{2}+s_{3}=0}$:

${\displaystyle {\begin{aligned}&\int d\mathbf {\hat {n}} \,_{s_{1}}\!Y_{j_{1}m_{1}}(\mathbf {\hat {n}} )\,_{s_{2}}\!Y_{j_{2}m_{2}}(\mathbf {\hat {n}} )\,_{s_{3}}\!Y_{j_{3}m_{3}}(\mathbf {\hat {n}} )\\&\quad ={\sqrt {\frac {(2j_{1}+1)(2j_{2}+1)(2j_{3}+1)}{4\pi }}}{\begin{pmatrix}j_{1}&j_{2}&j_{3}\\m_{1}&m_{2}&m_{3}\end{pmatrix}}{\begin{pmatrix}j_{1}&j_{2}&j_{3}\\-s_{1}&-s_{2}&-s_{3}\end{pmatrix}}.\end{aligned}}}$

Recursion relations

${\displaystyle {\begin{aligned}&{-}{\sqrt {(l_{3}\mp s_{3})(l_{3}\pm s_{3}+1)}}{\begin{pmatrix}l_{1}&l_{2}&l_{3}\\s_{1}&s_{2}&s_{3}\pm 1\end{pmatrix}}=\\&\quad ={\sqrt {(l_{1}\mp s_{1})(l_{1}\pm s_{1}+1)}}{\begin{pmatrix}l_{1}&l_{2}&l_{3}\\s_{1}\pm 1&s_{2}&s_{3}\end{pmatrix}}+{\sqrt {(l_{2}\mp s_{2})(l_{2}\pm s_{2}+1)}}{\begin{pmatrix}l_{1}&l_{2}&l_{3}\\s_{1}&s_{2}\pm 1&s_{3}\end{pmatrix}}.\end{aligned}}}$

Asymptotic expressions

For ${\displaystyle l_{1}\ll l_{2},l_{3}}$ a non-zero 3-j symbol is

${\displaystyle {\begin{pmatrix}l_{1}&l_{2}&l_{3}\\m_{1}&m_{2}&m_{3}\end{pmatrix}}\approx (-1)^{l_{3}+m_{3}}{\frac {d_{m_{1},l_{3}-l_{2}}^{l_{1}}(\theta )}{\sqrt {2l_{3}+1}}},}$

where ${\displaystyle \cos(\theta )=-2m_{3}/(2l_{3}+1)}$, and ${\displaystyle d_{mn}^{l}}$ is a Wigner function. Generally a better approximation obeying the Regge symmetry is given by

${\displaystyle {\begin{pmatrix}l_{1}&l_{2}&l_{3}\\m_{1}&m_{2}&m_{3}\end{pmatrix}}\approx (-1)^{l_{3}+m_{3}}{\frac {d_{m_{1},l_{3}-l_{2}}^{l_{1}}(\theta )}{\sqrt {l_{2}+l_{3}+1}}},}$

where ${\displaystyle \cos(\theta )=(m_{2}-m_{3})/(l_{2}+l_{3}+1)}$.

Metric tensor

The following quantity acts as a metric tensor in angular-momentum theory and is also known as a Wigner 1-jm symbol:^[1]

${\displaystyle {\begin{pmatrix}j\\m\quad m'\end{pmatrix}}:={\sqrt {2j+1}}{\begin{pmatrix}j&0&j\\m&0&m'\end{pmatrix}}=(-1)^{j-m'}\delta _{m,-m'}.}$

It can be used to perform time reversal on angular momenta.

Special cases and other properties

${\displaystyle \sum _{m}(-1)^{j-m}{\begin{pmatrix}j&j&J\\m&-m&0\end{pmatrix}}={\sqrt {2j+1}}\,\delta _{J,0}.}$

From equation (3.7.9) in ^[6]

${\displaystyle {\begin{pmatrix}j&j&0\\m&-m&0\end{pmatrix}}={\frac {1}{\sqrt {2j+1}}}(-1)^{j-m}.}$

${\displaystyle {\frac {1}{2}}\int _{-1}^{1}P_{l_{1}}(x)P_{l_{2}}(x)P_{l}(x)\,dx={\begin{pmatrix}l&l_{1}&l_{2}\\0&0&0\end{pmatrix}}^{2},}$

where P are Legendre polynomials.

Relation to Racah V-coefficients

Wigner 3-j symbols are related to Racah V-coefficients^[7] by a simple phase:

${\displaystyle V(j_{1}\,j_{2}\,j_{3};m_{1}\,m_{2}\,m_{3})=(-1)^{j_{1}-j_{2}-j_{3}}{\begin{pmatrix}j_{1}&j_{2}&j_{3}\\m_{1}&m_{2}&m_{3}\end{pmatrix}}.}$

Relation to group theory

This section essentially recasts the definitional relation in the language of group theory.

A group representation of a group is a homomorphism of the group into a group of linear transformations over some vector space. The linear transformations can be given by a group of matrices with respect to some basis of the vector space.

The group of transformations leaving angular momenta invariant is the three dimensional rotation group SO(3). When "spin" angular momenta are included, the group is its double covering group, SU(2).

A reducible representation is one where a change of basis can be applied to bring all the matrices into block diagonal form. A representation is irreducible (irrep) if no such transformation exists.

For each value of j, the 2j+1 kets form a basis for an irreducible representation (irrep) of SO(3)/SU(2) over the complex numbers. Given two irreps, the tensor direct product can be reduced to a sum of irreps, giving rise to the Clebcsh-Gordon coefficients, or by reduction of the triple product of three irreps to the trivial irrep 1 giving rise to the 3j symbols.

3j symbols for other groups

The ${\displaystyle 3j}$ symbol has been most intensely studied in the context of the coupling of angular momentum. For this, it is strongly related to the group representation theory of the groups SU(2) and SO(3) as discussed above. However, many other groups are of importance in physics and chemistry, and there has been much work on the ${\displaystyle 3j}$ symbol for these other groups. In this section, some of that work is considered.

Simply reducible groups

The original paper by Wigner^[1] was not restricted to SO(3)/SU(2) but instead focussed on simply reducible (SR) groups. These are groups in which

• all classes are ambivalent i.e. if ${\displaystyle X}$ is a member of a class then so is ${\displaystyle X^{-1}}$
• the Kronecker product of two irreps is multiplicity free i.e. does not contain any irrep more than once.

For SR groups, every irrep is equivalent to its complex conjugate, and under permutations of the columns the absolute value of the symbol is invariant and the phase of each can be chosen so that they at most change sign under odd permutations and remain unchanged under even permutations.

General compact groups

Compact groups form a wide class of groups with topological structure. They include the finite groups with added discrete topology and many of the Lie groups.

General compact groups will neither be ambivalent nor multiplicity free. Derome and Sharp^[8] and Derome^[9] examined the ${\displaystyle 3j}$ symbol for the general case using the relation to the Clebsch-Gordon coefficients of

${\displaystyle {\begin{pmatrix}j_{1}&j_{2}&j_{3}\\m_{1}&m_{2}&m_{3}\end{pmatrix}}\equiv {\frac {1}{[j_{3}]}}\langle j_{1}\,m_{1}\,j_{2}\,m_{2}|j_{3}^{*}\,m_{3}\rangle .}$

where ${\displaystyle [j]}$ is the dimension of the representation space of ${\displaystyle j}$ and ${\displaystyle j_{3}^{*}}$ is the complex conjugate representation to ${\displaystyle j_{3}}$.

By examining permutations of columns of the ${\displaystyle 3j}$ symbol, they showed three cases:

• if all of ${\displaystyle j_{1},j_{2},j_{3}}$ are inequivalent then the ${\displaystyle 3j}$ symbol may be chosen to be invariant under any permutation of its columns
• if exactly two are equivalent, then transpositions of its columns may be chosen so that some symbols will be invariant while others will change sign. An approach using a wreath product of the group with ${\displaystyle S_{3}}$^[10] showed that these correspond to the representations ${\displaystyle [2]}$ or ${\displaystyle [1^{2}]}$ of the symmetric group ${\displaystyle S_{2}}$. Cyclic permutations leave the ${\displaystyle 3j}$ symbol invariant.
• if all three are equivalent, the behaviour is dependent on the representations of the symmetric group${\displaystyle S_{3}}$. Wreath group representations corresponding to ${\displaystyle [3]}$ are invariant under transpositions of the columns, corresponding to ${\displaystyle [1^{3}]}$ change sign under transpositions, while a pair corresponding to the two dimensional representation ${\displaystyle [21]}$ transform according to that.

Further research into ${\displaystyle 3j}$ symbols for compact groups has been performed based on these principles. ^[11]

SU(n)

The Special unitary group SU(n) is the Lie group of n × n unitary matrices with determinant 1.

The group SU(3) is important in particle theory. There are many papers dealing with the ${\displaystyle 3j}$ or equivalent symbol ^{[12] [13] [14] [15] [16] [17] [18] [19]}

The ${\displaystyle 3j}$ symbol for the group SU(4) has been studied ^{[20] [21]} while there is also work on the general SU(n) groups ^{[22] [23]}

Crystallographic point groups

There are many papers dealing with the ${\displaystyle 3j}$ symbols or Clebsch-Gordon coefficients for the finite crystallographic point groups and the double point groups The book by Butler ^[24] references these and details the theory along with tables.

Magnetic groups

Magnetic groups include antilinear operators as well as linear operators. They need to be dealt with using Wigner's theory of corepresentations of unitary and antiunitary groups. A significant departure from standard representation theory is that the multiplicity of the irreducible corepresentation ${\displaystyle j_{3}^{*}}$ in the direct product of the irreducible corepresentations ${\displaystyle j_{1}\otimes j_{2}}$ is generally smaller than the multiplicity of the trivial corepresentation in the triple product ${\displaystyle j_{1}\otimes j_{2}\otimes j_{3}}$, leading to significant differences between the Clebsch-Gordon coefficients and the ${\displaystyle 3j}$ symbol.

The ${\displaystyle 3j}$ symbols have been examined for the grey groups ^{[25] [26]} and for the magnetic point groups ^[27]

See also

• Clebsch–Gordan coefficients
• Spherical harmonics
• 6-j symbol
• 9-j symbol

References

External links

• Stone, Anthony. "Wigner coefficient calculator".
• Volya, A. "Clebsch-Gordan, 3-j and 6-j Coefficient Web Calculator". Archived from the original on 2007-09-29. (Numerical)
• Stevenson, Paul (2002). "Clebsch-O-Matic". Computer Physics Communications. 147 (3): 853–858. Bibcode:2002CoPhC.147..853S. doi:10.1016/S0010-4655(02)00462-9.
• 369j-symbol calculator at the Plasma Laboratory of Weizmann Institute of Science (Numerical)
• Frederik J Simons: Matlab software archive, the code THREEJ.M
• Sage (mathematics software) Gives exact answer for any value of j, m
• Johansson, H. T.; Forssén, C. "(WIGXJPF)". (accurate; C, fortran, python)
• Johansson, H. T. "(FASTWIGXJ)". (fast lookup, accurate; C, fortran)