Wigner 3-j Symbols

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. 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
:$\begin j_1 & j_2 & j_3 \\ m_1 & m_2 & m_3 \end \equiv \frac \langle j_1 \, m_1 \, j_2 \, m_2 , j_3 \, (-m_3) \rangle.$
The ''j'' and ''m'' components are angular-momentum quantum numbers, i.e., every (and every corresponding ) 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 :
:$\langle j_1 \, m_1 \, j_2 \, m_2 , j_3 \, m_3 \rangle = (-1)^ \sqrt \begin j_1 & j_2 & j_3 \\ m_1 & m_2 & -m_3 \end.$

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:
:$, j_3\, m_3\rangle = \sum_^ \sum_^ \langle j_1 \, m_1 \, j_2 \, m_2 , j_3 \, m_3 \rangle , j_1 \, m_1 \, j_2 \, m_2 \rangle.$
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:
:$\sum_^ \sum_^ \sum_^ , j_1 m_1\rangle , j_2 m_2\rangle , j_3 m_3\rangle \begin j_1 & j_2 & j_3\\ m_1 & m_2 & m_3 \end = , 0 \, 0\rangle.$
Here $, 0 \, 0\rangle$ is the zero-angular-momentum state ($j = m = 0$). 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 $, 0 \, 0\rangle$ 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:

:$\begin & m_i \in \ \quad (i = 1, 2, 3), \\ & m_1 + m_2 + m_3 = 0, \\ & , j_1 - j_2, \le j_3 \le j_1 + j_2, \\ & (j_1 + j_2 + j_3) \text m_1 = m_2 = m_3 = 0 \text. \\ \end$

Symmetry properties

A 3-''j'' symbol is invariant under an even permutation of its columns:
:$\begin j_1 & j_2 & j_3\\ m_1 & m_2 & m_3 \end = \begin j_2 & j_3 & j_1\\ m_2 & m_3 & m_1 \end = \begin j_3 & j_1 & j_2\\ m_3 & m_1 & m_2 \end.$
An odd permutation of the columns gives a phase factor:
:$\begin j_1 & j_2 & j_3\\ m_1 & m_2 & m_3 \end = (-1)^ \begin j_2 & j_1 & j_3\\ m_2 & m_1 & m_3 \end = (-1)^ \begin j_1 & j_3 & j_2\\ m_1 & m_3 & m_2 \end = (-1)^ \begin j_3 & j_2 & j_1\\ m_3 & m_2 & m_1 \end.$
Changing the sign of the $m$ quantum numbers ( time reversal) also gives a phase:
:$\begin j_1 & j_2 & j_3\\ -m_1 & -m_2 & -m_3 \end = (-1)^ \begin j_1 & j_2 & j_3\\ m_1 & m_2 & m_3 \end.$
The 3-''j'' symbols also have so-called Regge symmetries, which are not due to permutations or time reversal. These symmetries are:
:$\begin j_1 & j_2 & j_3\\ m_1 & m_2 & m_3 \end = \begin j_1 & \frac & \frac\\ j_3-j_2 & \frac-m_3 & \frac+m_3 \end,$
:$\begin j_1 & j_2 & j_3\\ m_1 & m_2 & m_3 \end = (-1)^ \begin \frac & \frac & \frac\\ j_1 - \frac & j_2 - \frac & j_3-\frac \end.$
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:
:$R= \begin \hline -j_1+j_2+j_3 & j_1-j_2+j_3 & j_1+j_2-j_3\\ j_1-m_1 & j_2-m_2 & j_3-m_3\\ j_1+m_1 & j_2+m_2 & j_3+m_3\\ \hline \end,$
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.

Orthogonality relations

A system of two angular momenta with magnitudes and can be described either in terms of the uncoupled basis states (labeled by the quantum numbers and ), or the coupled basis states (labeled by and ). The 3-''j'' symbols constitute a unitary transformation between these two bases, and this unitarity implies the orthogonality relations
:$(2 j_3 + 1)\sum_ \begin j_1 & j_2 & j_3\\ m_1 & m_2 & m_3 \end \begin j_1 & j_2 & j'_3\\ m_1 & m_2 & m'_3 \end = \delta_ \delta_ \begin j_1 & j_2 & j_3 \end,$
:$\sum_ (2 j_3 + 1) \begin j_1 & j_2 & j_3\\ m_1 & m_2 & m_3 \end \begin j_1 & j_2 & j_3\\ m_1' & m_2' & m_3 \end = \delta_ \delta_.$
The ''triangular delta'' is equal to 1 when the triad (''j''₁, ''j''₂, ''j''₃) satisfies the triangle conditions, and is zero otherwise. The triangular delta itself is sometimes confusingly called 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 variables remain.

Relation to spherical harmonics; Gaunt coefficients

The 3-''jm'' symbols give the integral of the products of three spherical harmonics
:$\begin & \int Y_(\theta, \varphi) Y_(\theta, \varphi) Y_(\theta, \varphi)\,\sin\theta\,\mathrm\theta\,\mathrm\varphi \\ &\quad = \sqrt \begin l_1 & l_2 & l_3 \\ 0 & 0 & 0 \end \begin l_1 & l_2 & l_3\\ m_1 & m_2 & m_3 \end \end$
with $l_1$, $l_2$ and $l_3$ 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 $s_1 + s_2 + s_3 = 0$:
:$\begin & \int d\mathbf \,_\!Y_(\mathbf) \,_\!Y_(\mathbf) \,_\!Y_(\mathbf) \\ &\quad = \sqrt \begin j_1 & j_2 & j_3\\ m_1 & m_2 & m_3 \end \begin j_1 & j_2 & j_3\\ -s_1 & -s_2 & -s_3 \end. \end$

Recursion relations

:$\begin & \sqrt \begin l_1 & l_2 & l_3 \\ s_1 & s_2 & s_3 \pm 1 \end= \\ &\quad = \sqrt \begin l_1 & l_2 & l_3 \\ s_1 \pm 1 & s_2 & s_3 \end + \sqrt \begin l_1 & l_2 & l_3 \\ s_1 & s_2 \pm 1 & s_3 \end. \end$

Asymptotic expressions

For $l_1 \ll l_2, l_3$ a non-zero 3-''j'' symbol is
:$\begin l_1 & l_2 & l_3\\ m_1 & m_2 & m_3 \end \approx (-1)^ \frac,$
where $\cos(\theta) = -2m_3 / (2l_3 + 1)$, and $d^l_$ is a Wigner function. Generally a better approximation obeying the Regge symmetry is given by
:$\begin l_1 & l_2 & l_3\\ m_1 & m_2 & m_3 \end \approx (-1)^ \frac,$
where $\cos(\theta) = (m_2 - m_3)/(l_2 + l_3 + 1)$.

Metric tensor

The following quantity acts as a metric tensor in angular-momentum theory and is also known as a ''Wigner 1-jm symbol'':
:$\begin j \\ m \quad m' \end := \sqrt \begin j & 0 & j \\ m & 0 & m' \end = (-1)^ \delta_.$
It can be used to perform time reversal on angular momenta.

Special cases and other properties

:$\sum_m (-1)^ \begin j & j & J \\ m & -m & 0 \end = \sqrt \, \delta_.$

From equation (3.7.9) in
:$\begin j & j & 0 \\ m & -m & 0 \end = \frac (-1)^.$

:$\frac \int_^1 P_(x) P_(x) P_(x) \, dx = \begin l & l_1 & l_2 \\ 0 & 0 & 0 \end^2,$
where ''P'' are Legendre polynomials.

Relation to Racah -coefficients

Wigner 3-''j'' symbols are related to Racah -coefficients by a simple phase:

:$V(j_1 \, j_2 \, j_3; m_1 \, m_2 \, m_3) = (-1)^ \begin j_1 & j_2 & j_3 \\ m_1 & m_2 & m_3 \end.$

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 2''j''+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 $3j$ 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 $3j$ symbol for these other groups.
In this section, some of that work is considered.

Simply reducible groups

The original paper by Wigner
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 $X$ is a member of a class then so is $X^$
* 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
and Derome examined the $3j$ symbol
for the general case using the relation to the Clebsch-Gordon coefficients of
:$\begin j_1 & j_2 & j_3 \\ m_1 & m_2 & m_3 \end \equiv \frac \langle j_1 \, m_1 \, j_2 \, m_2 , j_3^* \, m_3 \rangle.$
where /math> is the dimension of the representation space of $j$ and $j_3^*$ is the complex conjugate
representation to $j_3$.

By examining permutations of columns of the $3j$ symbol, they showed three cases:
* if all of $j_1, j_2, j_3$ are inequivalent then the $3j$ 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 $S_3$ showed that these correspond to the representations /math> or ^2/math> of the symmetric group $S_2$. Cyclic permutations leave the $3j$ symbol invariant.
* if all three are equivalent, the behaviour is dependant on the representations of the symmetric group$S_3$. Wreath group representations corresponding to /math> are invariant under transpositions of the columns, corresponding to ^3/math> change sign under transpositions, while a pair corresponding to the two dimensional representation 1/math> transform according to that.

Further research into $3j$ symbols for compact groups has been performed based on these principles.

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 $3j$ or
equivalent symbol

The $3j$ symbol for the group SU(4) has been studied

while there is also work on the general SU(n) groups

Crystallographic point groups

There are many papers dealing with the $3j$ symbols or Clebsch-Gordon coefficients for the finite crystallographic point groups
and the double point groups
The book by Butler

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 $j_3^*$
in the direct product of the irreducible corepresentations $j_1 \otimes j_2$
is generally smaller than the multiplicity of the trivial corepresentation in the triple
product $j_1 \otimes j_2 \otimes j_3$, leading to significant differences between the Clebsch-Gordon
coefficients and the $3j$ symbol.

The $3j$ symbols have been examined for the grey groups

and for the magnetic point groups

See also

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

References

* L. C. Biedenharn and J. D. Louck, ''Angular Momentum in Quantum Physics'', volume 8 of Encyclopedia of Mathematics, Addison-Wesley, Reading, 1981.
* D. M. Brink and G. R. Satchler, ''Angular Momentum'', 3rd edition, Clarendon, Oxford, 1993.
* A. R. Edmonds, ''Angular Momentum in Quantum Mechanics'', 2nd edition, Princeton University Press, Princeton, 1960.
*
*
*

*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*

External links

*
* (Numerical)
*

369j-symbol calculator at the Plasma Laboratory of Weizmann Institute of Science
(Numerical)


* ttp://www.sagemath.org/ Sage (mathematics software)Gives exact answer for any value of j, m
* (accurate; C, fortran, python)
* {{cite web
, first1=H. T.
, last1=Johansson
, title=(FASTWIGXJ)
, url=http://fy.chalmers.se/subatom/fastwigxj/
(fast lookup, accurate; C, fortran)

Rotational symmetry
Representation theory of Lie groups
Quantum mechanics