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
} \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
=
\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)
{}\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
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 3
j symbol at, for example,
or
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
=
\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:
|j1m1\rangle|j2m2\rangle|j3m3\rangle
\begin{pmatrix}
j1&j2&j3\\
m1&m2&m3
\end{pmatrix}
=|00\rangle.
Here
is the zero-angular-momentum state (
). 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
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}
\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
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&
&
\\
j3-j2&
-m3&
+m3
\end{pmatrix},
\begin{pmatrix}
j1&j2&j3\\
m1&m2&
| j1+j2+j3 |
m | |
| 3
\end{pmatrix}
=
(-1) |
\begin{pmatrix}
&
&
\\
j1-
&j2-
&
\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+
\begin{pmatrix}
j1&j2&j3\\
m1&m2&m3
\end{pmatrix}
\begin{pmatrix}
j1&j2&j'3\\
m1&m2&m'3
\end{pmatrix}
=
\begin{Bmatrix}j1&j2&j3\end{Bmatrix},
(2j3+1)
\begin{pmatrix}
j1&j2&j3\\
m1&m2&m3
\end{pmatrix}
\begin{pmatrix}
j1&j2&j3\\
m1'&m2'&m3
\end{pmatrix}
=
.
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
(\theta,\varphi)
(\theta,\varphi)
(\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
,
and
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
:
\begin{align}
&\intd\hatn
(\hatn)
(\hatn)
(\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
a non-zero 3-
j symbol is
\begin{pmatrix}
l1&l2&l3\\
m1&m2&m3
\end{pmatrix}
≈
},where
\cos(\theta)=-2m3/(2l3+1)
, and
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}
≈
},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\\
m m'
\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)^.
(x)
(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)=
\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
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
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
- all classes are ambivalent i.e. if
is a member of a class then so is
- 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 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
symbolfor the general case using the relation to the Clebsch-Gordon coefficients of
\begin{pmatrix}
j1&j2&j3\\
m1&m2&m3
\end{pmatrix}
\equiv
\langlej1m1j2m2|
m3\rangle.
where
is the dimension of the representation space of
and
is the complex conjugaterepresentation to
.
By examining permutations of columns of the
symbol, they showed three cases:
are inequivalent then the
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
[10] showed that these correspond to the
representations
or
of the symmetric group
. Cyclic permutations leave the
symbol invariant.
- if all three are equivalent, the behaviour is dependent on the representations of the symmetric group
. Wreath group representations corresponding to
are invariant under transpositions of the columns, corresponding to
change sign under transpositions, while a pair corresponding to the two dimensional representation
transform according to that.
Further research into
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
orequivalent symbol
[12] [13] [14] [15] [16] [17] [18] [19] The
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
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
in the direct product of the irreducible corepresentations
is generally smaller than the multiplicity of the trivial corepresentation in the tripleproduct
, leading to significant differences between the Clebsch-Gordoncoefficients and the
symbol.
The
symbols have been examined for the grey groups
[25] [26] and for the magnetic point groups
[27] See also
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