3-j symbol explained

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

\begin{pmatrix} j1&j2&j3\\ m1&m2&m3 \end{pmatrix} \equiv

j1-j2-m3
(-1)
\sqrt{2j3+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 (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 :

\langlej1m1j2m2|j3m3\rangle =

-j1+j2-m3
(-1)

\sqrt{2j3+1} \begin{pmatrix} j1&j2&j3\\ m1&m2&-m3 \end{pmatrix}.

Explicit expression

\begin{align}\begin{pmatrix} j1&j2&j3\\ m1&m2&m3 \end{pmatrix} &\equiv\delta(m1+m2+m3,0)

j1-j2-m3
(-1)

{}\sqrt{

(j1+j2-j3)!(j1-j2+j3)!(-j1+j2+j3)!
(j1+j2+j3+1)!
}\ \times \\[6pt]&\times\sqrt\ \times \\[6pt]&\times\sum_^N \frac,\endwhere

\delta(i,j)

is the Kronecker delta.

The summation is performed over those integer values for which the argument of each factorial in the denominator is non-negative, i.e. summation limits and are taken equal: the lower one

K=max(0,j2-j3-m1,j1-j3+m2),

the upper one

N=min(j1+j2-j3,j1-m1,j2+m2).

Factorials of negative numbers are conventionally taken equal to zero, so that the values of the 3j symbol at, for example,

j3>j1+j2

or

j1<m1

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:

|j3m3\rangle =

j1
\sum
m1=-j1
j2
\sum
m2=-j2

\langlej1m1j2m2|j3m3\rangle |j1m1j2m2\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:
j1
\sum
m1=-j1
j2
\sum
m2=-j2
j3
\sum
m3=-j3

|j1m1\rangle|j2m2\rangle|j3m3\rangle \begin{pmatrix} j1&j2&j3\\ m1&m2&m3 \end{pmatrix} =|00\rangle.

Here

|00\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

|00\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{align} &mi\in\{-ji,-ji+1,-ji+2,\ldots,ji\}(i=1,2,3),\\ &m1+m2+m3=0,\\ &|j1-j2|\lej3\lej1+j2,\\ &(j1+j2+j3)isaninteger(and,moreover,anevenintegerifm1=m2=m3=0).\\ \end{align}

Symmetry properties

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

\begin{pmatrix} j1&j2&j3\\ m1&m2&m3 \end{pmatrix} = \begin{pmatrix} j2&j3&j1\\ m2&m3&m1 \end{pmatrix} = \begin{pmatrix} j3&j1&j2\\ m3&m1&m2 \end{pmatrix}.

An odd permutation of the columns gives a phase factor:

\begin{pmatrix} j1&j2&j3\\ m1&m2&

j1+j2+j3
m
3 \end{pmatrix} = (-1)

\begin{pmatrix} j2&j1&j3\\ m2&m1&m3 \end{pmatrix}

j1+j2+j3
= (-1)

\begin{pmatrix} j1&j3&j2\\ m1&m3&

j1+j2+j3
m
2 \end{pmatrix} = (-1)

\begin{pmatrix} j3&j2&j1\\ m3&m2&m1 \end{pmatrix}.

Changing the sign of the

m

quantum numbers (time reversal) also gives a phase:

\begin{pmatrix} j1&j2&j3\\ -m1&-m2&

j1+j2+j3
-m
3 \end{pmatrix} = (-1)

\begin{pmatrix} j1&j2&j3\\ m1&m2&m3 \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:

\begin{pmatrix} j1&j2&j3\\ m1&m2&m3 \end{pmatrix} = \begin{pmatrix} j1&

j2+j3-m1
2

&

j2+j3+m1
2

\\ j3-j2&

j2-j3-m1
2

-m3&

j2-j3+m1
2

+m3 \end{pmatrix},

\begin{pmatrix} j1&j2&j3\\ m1&m2&

j1+j2+j3
m
3 \end{pmatrix} = (-1)

\begin{pmatrix}

j2+j3+m1
2

&

j1+j3+m2
2

&

j1+j2+m3
2

\\ j1-

j2+j3-m1
2

&j2-

j1+j3-m2
2

&

j
3-j1+j2-m3
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]

R= \begin{array}{|ccc|} \hline -j1+j2+j3&j1-j2+j3&j1+j2-j3\\ j1-m1&j2-m2&j3-m3\\ j1+m1&j2+m2&j3+m3\\ \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.

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

(2j3+

1)\sum
m1m2

\begin{pmatrix} j1&j2&j3\\ m1&m2&m3 \end{pmatrix} \begin{pmatrix} j1&j2&j'3\\ m1&m2&m'3 \end{pmatrix} =

\delta
j3,j'3
\delta
m3,m'3

\begin{Bmatrix}j1&j2&j3\end{Bmatrix},

\sum
j3m3

(2j3+1) \begin{pmatrix} j1&j2&j3\\ m1&m2&m3 \end{pmatrix} \begin{pmatrix} j1&j2&j3\\ m1'&m2'&m3 \end{pmatrix} =

\delta
m1,m1'
\delta
m2,m2'

.

The triangular delta 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 variables remain.

Relation to spherical harmonics; Gaunt coefficients

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

\begin{align} &\int

Y
l1m1

(\theta,\varphi)

Y
l2m2

(\theta,\varphi)

Y
l3m3

(\theta,\varphi)\sin\thetad\thetad\varphi\\ &=\sqrt{

(2l1+1)(2l2+1)(2l3+1)
4\pi
}\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\endwith

l1

,

l2

and

l3

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

s1+s2+s3=0

:

\begin{align} &\intd\hatn

s1
Y
j1m1

(\hatn)

s2
Y
j2m2

(\hatn)

s3
Y
j3m3

(\hatn)\\ &=\sqrt{

(2j1+1)(2j2+1)(2j3+1)
4\pi
}\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{align} &{-}\sqrt{(l3\mps3)(l3\pms3+1)}\begin{pmatrix} l1&l2&l3\\ s1&s2&s3\pm1 \end{pmatrix}= \\ &=\sqrt{(l1\mps1)(l1\pms1+1)}\begin{pmatrix} l1&l2&l3\\ s1\pm1&s2&s3 \end{pmatrix} +\sqrt{(l2\mps2)(l2\pms2+1)}\begin{pmatrix} l1&l2&l3\\ s1&s2\pm1&s3 \end{pmatrix}. \end{align}

Asymptotic expressions

For

l1\lll2,l3

a non-zero 3-j symbol is

\begin{pmatrix} l1&l2&l3\\ m1&m2&m3 \end{pmatrix}

l3+m3
(-1)
l1
d(\theta)
m1,l3-l2
\sqrt{2l3+1
},where

\cos(\theta)=-2m3/(2l3+1)

, and
l
d
mn
is a Wigner function. Generally a better approximation obeying the Regge symmetry is given by

\begin{pmatrix} l1&l2&l3\\ m1&m2&m3 \end{pmatrix}

l3+m3
(-1)
l1
d
m1,l3-l2
(\theta)
\sqrt{l2+l3+1
},where

\cos(\theta)=(m2-m3)/(l2+l3+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]

\begin{pmatrix} j\\ mm' \end{pmatrix} :=\sqrt{2j+1} \begin{pmatrix} j&0&j\\ m&0&m' \end{pmatrix} =(-1)j\deltam,.

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

Special cases and other properties

\summ(-1)j\begin{pmatrix} j&j&J\\ m&-m&0 \end{pmatrix}=\sqrt{2j+1}\deltaJ,.

From equation (3.7.9) in [6]

\begin{pmatrix} j&j&0\\ m&-m&0 \end{pmatrix}=

1
\sqrt{2j+1
} (-1)^.
1
2
1
\int
-1
P
l1

(x)

P
l2

(x)Pl(x)dx=\begin{pmatrix} l&l1&l2\\ 0&0&0 \end{pmatrix}2,

where P are Legendre polynomials.

Relation to Racah -coefficients

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

V(j1j2j3;m1m2m3)=

j1-j2-j3
(-1)

\begin{pmatrix} j1&j2&j3\\ m1&m2&m3 \end{pmatrix}.

Relation to group theory

This section essentially recasts the definitional relationin the language of group theory.

A group representation of a group is a homomorphism of the group intoa group of linear transformations over some vector space. The lineartransformations 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 representationis 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 twoirreps, the tensor direct product can be reduced to asum 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 thegroup representation theory of the groups SU(2) and SO(3)as discussed above. However, manyother 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 Wignerwas not restricted to SO(3)/SU(2)but instead focussed on simply reducible (SR) groups.These are groups in which

X

is a member of a class then so is

X-1

For SR groups, every irrep is equivalent to its complex conjugate,and under permutations of the columns the absolute value of thesymbol is invariant and the phase of each can be chosen so thatthey at most change sign under odd permutations and remainunchanged 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 topologyand many of the Lie groups.

General compact groups will neither be ambivalent nor multiplicity free.Derome and Sharp[8] and Derome[9] examined the

3j

symbolfor the general case using the relation to the Clebsch-Gordon coefficients of

\begin{pmatrix} j1&j2&j3\\ m1&m2&m3 \end{pmatrix} \equiv

1
[j3]

\langlej1m1j2m2|

*
j
3

m3\rangle.

where

[j]

is the dimension of the representation space of

j

and
*
j
3
is the complex conjugaterepresentation to

j3

.

By examining permutations of columns of the

3j

symbol, they showed three cases:

j1,j2,j3

are inequivalent then the

3j

symbol may be chosen to be invariant under any permutation of its columns

S3

[10] showed that these correspond to the representations

[2]

or

[12]

of the symmetric group

S2

. Cyclic permutations leave the

3j

symbol invariant.

S3

. Wreath group representations corresponding to

[3]

are invariant under transpositions of the columns, corresponding to

[13]

change sign under transpositions, while a pair corresponding to the two dimensional representation

[21]

transform according to that.

Further research into

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

3j

orequivalent symbol[12] [13] [14] [15] [16] [17] [18] [19]

The

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

3j

symbols or Clebsch-Gordon coefficients for the finite crystallographic point groupsand the double point groupsThe 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 usingWigner'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

j1j2

is generally smaller than the multiplicity of the trivial corepresentation in the tripleproduct

j1j2j3

, leading to significant differences between the Clebsch-Gordoncoefficients and the

3j

symbol.

The

3j

symbols have been examined for the grey groups[25] [26] and for the magnetic point groups[27]

See also

References

  1. Book: Wigner, E. P. . The Collected Works of Eugene Paul Wigner . 1993 . 978-3-642-08154-5 . Wightman . Arthur S. . A/1 . 608–654 . On the Matrices Which Reduce the Kronecker Products of Representations of S. R. Groups . 10.1007/978-3-662-02781-3_42.
  2. T. . Regge . Symmetry Properties of Clebsch-Gordan Coefficients . Nuovo Cimento . 1958 . 10 . 3 . 544 . 10.1007/BF02859841 . 1958NCim...10..544R. 122299161 .
  3. J. . Rasch . A. C. H. . Yu . Efficient Storage Scheme for Pre-calculated Wigner 3j, 6j and Gaunt Coefficients . SIAM J. Sci. Comput. . 25 . 4 . 2003 . 1416–1428 . 10.1137/s1064827503422932.
  4. Angular Momentum Diagrams . P. E. S. Wormer . J. Paldus . Advances in Quantum Chemistry . Elsevier . 51 . 59–124 . 2006 . 0065-3276 . 10.1016/S0065-3276(06)51002-0 . 2006AdQC...51...59W . 9780120348510.
  5. Cruzan . Orval R. . 1962 . Translational addition theorems for spherical vector wave functions . Quarterly of Applied Mathematics . en . 20 . 1 . 33–40 . 10.1090/qam/132851 . 0033-569X. free .
  6. Book: Edmonds . Alan . Angular Momentum in Quantum Mechanics . 1957 . Princeton University Press.
  7. G. . Racah . Theory of Complex Spectra II . . 62 . 9–10 . 438–462 . 1942 . 10.1103/PhysRev.62.438 . 1942PhRv...62..438R .
  8. J-R. Derome. W. T.. Sharp. Racah Algebra for an Arbitrary Group. J. Math. Phys.. 6. 10. 1965. 1584–1590. 10.1063/1.1704698. 1965JMP.....6.1584D.
  9. J-R. Derome. Symmetry Properties of the 3j Symbols for an Arbitrary Group. J. Math. Phys.. 7. 4. 1966. 612–615. 10.1063/1.1704973. 1966JMP.....7..612D.
  10. J. D.. Newmarch. On the 3j symmetries. J. Math. Phys.. 24. 4. 1983. 757–764. 10.1063/1.525771. 1983JMP....24..757N.
  11. P. H.. Butler. B. G.. Wybourne. Calculation of j and jm Symbols forArbitrary Compact Groups. I. Methodology. Int. J. Quantum Chem.. X. 1976. 4. 581–598. 10.1002/qua.560100404.
  12. Marcos. Moshinsky. Wigner coefficients for the SU3 group and some applications. Rev. Mod. Phys.. 34. 1962. 813. 10.1103/RevModPhys.34.813. 4. 1962RvMP...34..813M .
  13. S. J.. P. McNamee. Frank. Chilton. Tables of Clebsch-Gordan coefficients of SU3. Rev. Mod. Phys.. 36. 1964. 1005. 10.1103/RevModPhys.36.1005. 4. 1964RvMP...36.1005M.
  14. J. P.. Draayer. Yoshimi. Akiyama. Wigner and Racah coefficients for SU3. J. Math. Phys.. 14. 1973. 1904. 10.1063/1.1666267. 12. 1973JMP....14.1904D . 2027.42/70151. free.
  15. Yoshimi. Akiyama. J. P.. Draayer. A users' guide to fortran programs for Wigner and Racah coefficients of SU3. Comput. Phys. Commun.. 5. 6. 1973. 405. 10.1016/0010-4655(73)90077-5. 1973CoPhC...5..405A . 2027.42/24983. free.
  16. R. P.. Bickerstaff. P. H.. Butler. M. B.. Butts. R. w.. Haase. M. F.. Reid. 3jm and 6j tables for some bases of SU6 and SU3. J. Phys. A. 15. 4. 1982. 1087. 10.1088/0305-4470/15/4/014. 1982JPhA...15.1087B.
  17. J. J.. Swart de. The octet model and its Glebsch-Gordan coefficients. Rev. Mod. Phys.. 35. 1963. 916. 10.1103/RevModPhys.35.916. 4. 1963RvMP...35..916D .
  18. J-R. Derome. Symmetry Properties of the 3j Symbols for SU(3). J. Math. Phys.. 8. 4. 1967. 714–716. 10.1063/1.1705269. 1967JMP.....8..714D.
  19. K. T.. Hecht. SU3 recoupling and fractional parentage in the 2s-1d shell. Nucl. Phys.. 62. 1965. 1. 10.1016/0029-5582(65)90068-4. 1. 1965NucPh..62....1H. 2027.42/32049. free.
  20. K. T.. Hecht. Sing Ching. Pang. On the Wigner Supermultiplet Scheme. J. Math. Phys.. 10. 1969. 1571. 10.1063/1.1665007. 9. 1969JMP....10.1571H . 2027.42/70485. free.
  21. E. M.. Haacke. J. W.. Moffat. P.. Savaria. A calculation of SU(4) Glebsch-Gordan coefficients. J. Math. Phys.. 17. 1976. 2041. 10.1063/1.522843. 11. 1976JMP....17.2041H .
  22. G. E.. Baird. L. C.. Biedenharn. On the representation of the semisimple Lie Groups. II. J. Math. Phys.. 4. 12. 1963. 1449. 10.1063/1.1703926. 1963JMP.....4.1449B.
  23. G. E.. Baird. L. C.. Biedenharn. On the representations of the semisimple Lie Groups. III. The explicit conjugation Operation for SUn. J. Math. Phys.. 5. 12. 1964. 1723. 10.1063/1.1704095. 1964JMP.....5.1723B .
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  25. Newmarch. J. D.. 1981. The Racah Algebra for Groups with Time Reversal Symmetry. University of New South Wales.
  26. J. D.. Newmarch. R. M.. Golding. The Racah Algebra for Groups with Time Reversal Symmetry. J. Math. Phys.. 22. 2. 1981. 233–244. 10.1063/1.524894. 1981JMP....22..233N. 1959.4/69692. free.
  27. J. N.. Kotsev. M. I.. Aroyo. M. N.. Angelova. Tables of Spectroscopic Coefficients for Magnetic Point Group Symmetry. J. Mol. Structure. 115. 1984. 123–128. 10.1016/0022-2860(84)80030-7.

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