Wigner, Clebsch-Gordan, Racah, and Gaunt coefficients#

Collection of functions for calculating Wigner 3-\(j\), 6-\(j\), 9-\(j\), Clebsch-Gordan, Racah as well as Gaunt coefficients exactly, all evaluating to a rational number times the square root of a rational number [RH2003].

Please see the description of the individual functions for further details and examples.

AUTHORS:

  • Jens Rasch (2009-03-24): initial version for Sage

  • Jens Rasch (2009-05-31): updated to sage-4.0

sage.functions.wigner.clebsch_gordan(j_1, j_2, j_3, m_1, m_2, m_3, prec=None)#

Return the Clebsch-Gordan coefficient \(\langle j_1 m_1 \; j_2 m_2 | j_3 m_3 \rangle\).

The reference for this function is [Ed1974].

INPUT:

  • j_1, j_2, j_3, m_1, m_2, m_3 – integer or half integer

  • prec – precision, default: None. Providing a precision can drastically speed up the calculation.

OUTPUT:

Rational number times the square root of a rational number (if prec=None), or real number if a precision is given.

EXAMPLES:

sage: simplify(clebsch_gordan(3/2,1/2,2, 3/2,1/2,2))                            # needs sage.symbolic
1
sage: clebsch_gordan(1.5,0.5,1, 1.5,-0.5,1)                                     # needs sage.symbolic
1/2*sqrt(3)
sage: clebsch_gordan(3/2,1/2,1, -1/2,1/2,0)                                     # needs sage.symbolic
-sqrt(3)*sqrt(1/6)

Note

The Clebsch-Gordan coefficient will be evaluated via its relation to Wigner 3-\(j\) symbols:

\[\begin{split}\langle j_1 m_1 \; j_2 m_2 | j_3 m_3 \rangle =(-1)^{j_1-j_2+m_3} \sqrt{2j_3+1} \begin{pmatrix} j_1 & j_2 & j_3 \\ m_1 & m_2 & -m_3 \end{pmatrix}\end{split}\]

See also the documentation on Wigner 3-\(j\) symbols which exhibit much higher symmetry relations than the Clebsch-Gordan coefficient.

AUTHORS:

  • Jens Rasch (2009-03-24): initial version

sage.functions.wigner.gaunt(l_1, l_2, l_3, m_1, m_2, m_3, prec=None)#

Return the Gaunt coefficient.

The Gaunt coefficient is defined as the integral over three spherical harmonics:

\[\begin{split}Y(l_1,l_2,l_3,m_1,m_2,m_3) \hspace{12em} \\ =\int Y_{l_1,m_1}(\Omega) \ Y_{l_2,m_2}(\Omega) \ Y_{l_3,m_3}(\Omega) \ d\Omega \hspace{5em} \\ =\sqrt{\frac{(2l_1+1)(2l_2+1)(2l_3+1)}{4\pi}} \hspace{6.5em} \\ \times \begin{pmatrix} l_1 & l_2 & l_3 \\ 0 & 0 & 0 \end{pmatrix} \begin{pmatrix} l_1 & l_2 & l_3 \\ m_1 & m_2 & m_3 \end{pmatrix}\end{split}\]

INPUT:

  • l_1, l_2, l_3, m_1, m_2, m_3 – integer

  • prec – precision, default: None. Providing a precision can drastically speed up the calculation.

OUTPUT:

Rational number times the square root of a rational number (if prec=None), or real number if a precision is given.

EXAMPLES:

sage: # needs sage.symbolic
sage: gaunt(1,0,1,1,0,-1)
-1/2/sqrt(pi)
sage: gaunt(1,0,1,1,0,0)
0
sage: gaunt(29,29,34,10,-5,-5)
1821867940156/215552371055153321*sqrt(22134)/sqrt(pi)
sage: gaunt(20,20,40,1,-1,0)
28384503878959800/74029560764440771/sqrt(pi)
sage: gaunt(12,15,5,2,3,-5)
91/124062*sqrt(36890)/sqrt(pi)
sage: gaunt(10,10,12,9,3,-12)
-98/62031*sqrt(6279)/sqrt(pi)
sage: gaunt(1000,1000,1200,9,3,-12).n(64)
0.00689500421922113448

If the sum of the \(l_i\) is odd, the answer is zero, even for Python ints (see github issue #14766):

sage: gaunt(1,2,2,1,0,-1)
0
sage: gaunt(int(1),int(2),int(2),1,0,-1)
0

It is an error to use non-integer values for \(l\) or \(m\):

sage: gaunt(1.2,0,1.2,0,0,0)
Traceback (most recent call last):
...
TypeError: Attempt to coerce non-integral RealNumber to Integer
sage: gaunt(1,0,1,1.1,0,-1.1)
Traceback (most recent call last):
...
TypeError: Attempt to coerce non-integral RealNumber to Integer

ALGORITHM:

This function uses the algorithm of [LdB1982] to calculate the value of the Gaunt coefficient exactly. Note that the formula contains alternating sums over large factorials and is therefore unsuitable for finite precision arithmetic and only useful for a computer algebra system [RH2003].

AUTHORS:

  • Jens Rasch (2009-03-24): initial version for Sage

sage.functions.wigner.racah(aa, bb, cc, dd, ee, ff, prec=None)#

Return the Racah symbol \(W(aa,bb,cc,dd;ee,ff)\).

INPUT:

  • aa, …, ff – integer or half integer

  • prec – precision, default: None. Providing a precision can drastically speed up the calculation.

OUTPUT:

Rational number times the square root of a rational number (if prec=None), or real number if a precision is given.

EXAMPLES:

sage: racah(3,3,3,3,3,3)                                                        # needs sage.symbolic
-1/14

Note

The Racah symbol is related to the Wigner 6-\(j\) symbol:

\[\begin{split}\begin{Bmatrix} j_1 & j_2 & j_3 \\ j_4 & j_5 & j_6 \end{Bmatrix} =(-1)^{j_1+j_2+j_4+j_5} W(j_1,j_2,j_5,j_4;j_3,j_6)\end{split}\]

Please see the 6-\(j\) symbol for its much richer symmetries and for additional properties.

ALGORITHM:

This function uses the algorithm of [Ed1974] to calculate the value of the 6-\(j\) symbol exactly. Note that the formula contains alternating sums over large factorials and is therefore unsuitable for finite precision arithmetic and only useful for a computer algebra system [RH2003].

AUTHORS:

  • Jens Rasch (2009-03-24): initial version

sage.functions.wigner.wigner_3j(j_1, j_2, j_3, m_1, m_2, m_3, prec=None)#

Return the Wigner 3-\(j\) symbol \(\begin{pmatrix} j_1 & j_2 & j_3 \\ m_1 & m_2 & m_3 \end{pmatrix}\).

INPUT:

  • j_1, j_2, j_3, m_1, m_2, m_3 – integer or half integer

  • prec – precision, default: None. Providing a precision can drastically speed up the calculation.

OUTPUT:

Rational number times the square root of a rational number (if prec=None), or real number if a precision is given.

EXAMPLES:

sage: wigner_3j(2, 6, 4, 0, 0, 0)                                               # needs sage.symbolic
sqrt(5/143)
sage: wigner_3j(2, 6, 4, 0, 0, 1)
0
sage: wigner_3j(0.5, 0.5, 1, 0.5, -0.5, 0)                                      # needs sage.symbolic
sqrt(1/6)
sage: wigner_3j(40, 100, 60, -10, 60, -50)                                      # needs sage.symbolic
95608/18702538494885*sqrt(21082735836735314343364163310/220491455010479533763)
sage: wigner_3j(2500, 2500, 5000, 2488, 2400, -4888, prec=64)                   # needs sage.rings.real_mpfr
7.60424456883448589e-12

It is an error to have arguments that are not integer or half integer values:

sage: wigner_3j(2.1, 6, 4, 0, 0, 0)
Traceback (most recent call last):
...
ValueError: j values must be integer or half integer
sage: wigner_3j(2, 6, 4, 1, 0, -1.1)
Traceback (most recent call last):
...
ValueError: m values must be integer or half integer

The Wigner 3-\(j\) symbol obeys the following symmetry rules:

  • invariant under any permutation of the columns (with the exception of a sign change where \(J=j_1+j_2+j_3\)):

    \[\begin{split}\begin{pmatrix} j_1 & j_2 & j_3 \\ m_1 & m_2 & m_3 \end{pmatrix} =\begin{pmatrix} j_3 & j_1 & j_2 \\ m_3 & m_1 & m_2 \end{pmatrix} =\begin{pmatrix} j_2 & j_3 & j_1 \\ m_2 & m_3 & m_1 \end{pmatrix} \hspace{10em} \\ =(-1)^J \begin{pmatrix} j_3 & j_2 & j_1 \\ m_3 & m_2 & m_1 \end{pmatrix} =(-1)^J \begin{pmatrix} j_1 & j_3 & j_2 \\ m_1 & m_3 & m_2 \end{pmatrix} =(-1)^J \begin{pmatrix} j_2 & j_1 & j_3 \\ m_2 & m_1 & m_3 \end{pmatrix}\end{split}\]

  • invariant under space inflection, i.e.

    \[\begin{split}\begin{pmatrix} j_1 & j_2 & j_3 \\ m_1 & m_2 & m_3 \end{pmatrix} =(-1)^J \begin{pmatrix} j_1 & j_2 & j_3 \\ -m_1 & -m_2 & -m_3 \end{pmatrix}\end{split}\]

  • symmetric with respect to the 72 additional symmetries based on the work by [Reg1958]

  • zero for \(j_1\), \(j_2\), \(j_3\) not fulfilling triangle relation

  • zero for \(m_1 + m_2 + m_3 \neq 0\)

  • zero for violating any one of the conditions \(j_1 \ge |m_1|\), \(j_2 \ge |m_2|\), \(j_3 \ge |m_3|\)

ALGORITHM:

This function uses the algorithm of [Ed1974] to calculate the value of the 3-\(j\) symbol exactly. Note that the formula contains alternating sums over large factorials and is therefore unsuitable for finite precision arithmetic and only useful for a computer algebra system [RH2003].

AUTHORS:

  • Jens Rasch (2009-03-24): initial version

sage.functions.wigner.wigner_6j(j_1, j_2, j_3, j_4, j_5, j_6, prec=None)#

Return the Wigner 6-\(j\) symbol \(\begin{Bmatrix} j_1 & j_2 & j_3 \\ j_4 & j_5 & j_6 \end{Bmatrix}\).

INPUT:

  • j_1, …, j_6 – integer or half integer

  • prec – precision, default: None. Providing a precision can drastically speed up the calculation.

OUTPUT:

Rational number times the square root of a rational number (if prec=None), or real number if a precision is given.

EXAMPLES:

sage: # needs sage.symbolic
sage: wigner_6j(3,3,3,3,3,3)
-1/14
sage: wigner_6j(5,5,5,5,5,5)
1/52
sage: wigner_6j(6,6,6,6,6,6)
309/10868
sage: wigner_6j(8,8,8,8,8,8)
-12219/965770
sage: wigner_6j(30,30,30,30,30,30)
36082186869033479581/87954851694828981714124
sage: wigner_6j(0.5,0.5,1,0.5,0.5,1)
1/6
sage: wigner_6j(200,200,200,200,200,200, prec=1000)*1.0
0.000155903212413242

It is an error to have arguments that are not integer or half integer values or do not fulfill the triangle relation:

sage: wigner_6j(2.5,2.5,2.5,2.5,2.5,2.5)
Traceback (most recent call last):
...
ValueError: j values must be integer or half integer and fulfill the triangle relation
sage: wigner_6j(0.5,0.5,1.1,0.5,0.5,1.1)
Traceback (most recent call last):
...
ValueError: j values must be integer or half integer and fulfill the triangle relation

The Wigner 6-\(j\) symbol is related to the Racah symbol but exhibits more symmetries as detailed below.

\[\begin{split}\begin{Bmatrix} j_1 & j_2 & j_3 \\ j_4 & j_5 & j_6 \end{Bmatrix} =(-1)^{j_1+j_2+j_4+j_5} W(j_1,j_2,j_5,j_4;j_3,j_6)\end{split}\]

The Wigner 6-\(j\) symbol obeys the following symmetry rules:

  • Wigner 6-\(j\) symbols are left invariant under any permutation of the columns:

    \[\begin{split}\begin{Bmatrix} j_1 & j_2 & j_3 \\ j_4 & j_5 & j_6 \end{Bmatrix} =\begin{Bmatrix} j_3 & j_1 & j_2 \\ j_6 & j_4 & j_5 \end{Bmatrix} =\begin{Bmatrix} j_2 & j_3 & j_1 \\ j_5 & j_6 & j_4 \end{Bmatrix} \hspace{7em} \\ =\begin{Bmatrix} j_3 & j_2 & j_1 \\ j_6 & j_5 & j_4 \end{Bmatrix} =\begin{Bmatrix} j_1 & j_3 & j_2 \\ j_4 & j_6 & j_5 \end{Bmatrix} =\begin{Bmatrix} j_2 & j_1 & j_3 \\ j_5 & j_4 & j_6 \end{Bmatrix} \hspace{3em}\end{split}\]

  • They are invariant under the exchange of the upper and lower arguments in each of any two columns, i.e.

    \[\begin{split}\begin{Bmatrix} j_1 & j_2 & j_3 \\ j_4 & j_5 & j_6 \end{Bmatrix} =\begin{Bmatrix} j_1 & j_5 & j_6 \\ j_4 & j_2 & j_3 \end{Bmatrix} =\begin{Bmatrix} j_4 & j_2 & j_6 \\ j_1 & j_5 & j_3 \end{Bmatrix} =\begin{Bmatrix} j_4 & j_5 & j_3 \\ j_1 & j_2 & j_6 \end{Bmatrix}\end{split}\]

  • additional 6 symmetries [Reg1959] giving rise to 144 symmetries in total

  • only non-zero if any triple of \(j\)’s fulfill a triangle relation

ALGORITHM:

This function uses the algorithm of [Ed1974] to calculate the value of the 6-\(j\) symbol exactly. Note that the formula contains alternating sums over large factorials and is therefore unsuitable for finite precision arithmetic and only useful for a computer algebra system [RH2003].

sage.functions.wigner.wigner_9j(j_1, j_2, j_3, j_4, j_5, j_6, j_7, j_8, j_9, prec=None)#

Return the Wigner 9-\(j\) symbol \(\begin{Bmatrix} j_1 & j_2 & j_3 \\ j_4 & j_5 & j_6 \\ j_7 & j_8 & j_9 \end{Bmatrix}\).

INPUT:

  • j_1, …, j_9 – integer or half integer

  • prec – precision, default: None. Providing a precision can drastically speed up the calculation.

OUTPUT:

Rational number times the square root of a rational number (if prec=None), or real number if a precision is given.

EXAMPLES:

A couple of examples and test cases, note that for speed reasons a precision is given:

sage: # needs sage.symbolic
sage: wigner_9j(1,1,1, 1,1,1, 1,1,0, prec=64) # ==1/18
0.0555555555555555555
sage: wigner_9j(1,1,1, 1,1,1, 1,1,1)
0
sage: wigner_9j(1,1,1, 1,1,1, 1,1,2, prec=64) # ==1/18
0.0555555555555555556
sage: wigner_9j(1,2,1, 2,2,2, 1,2,1, prec=64) # ==-1/150
-0.00666666666666666667
sage: wigner_9j(3,3,2, 2,2,2, 3,3,2, prec=64) # ==157/14700
0.0106802721088435374
sage: wigner_9j(3,3,2, 3,3,2, 3,3,2, prec=64) # ==3221*sqrt(70)/(246960*sqrt(105)) - 365/(3528*sqrt(70)*sqrt(105))
0.00944247746651111739
sage: wigner_9j(3,3,1, 3.5,3.5,2, 3.5,3.5,1, prec=64) # ==3221*sqrt(70)/(246960*sqrt(105)) - 365/(3528*sqrt(70)*sqrt(105))
0.0110216678544351364
sage: wigner_9j(100,80,50, 50,100,70, 60,50,100, prec=1000)*1.0
1.05597798065761e-7
sage: wigner_9j(30,30,10, 30.5,30.5,20, 30.5,30.5,10, prec=1000)*1.0 # ==(80944680186359968990/95103769817469)*sqrt(1/682288158959699477295)
0.0000325841699408828
sage: wigner_9j(64,62.5,114.5, 61.5,61,112.5, 113.5,110.5,60, prec=1000)*1.0
-3.41407910055520e-39
sage: wigner_9j(15,15,15, 15,3,15, 15,18,10, prec=1000)*1.0
-0.0000778324615309539
sage: wigner_9j(1.5,1,1.5, 1,1,1, 1.5,1,1.5)
0

It is an error to have arguments that are not integer or half integer values or do not fulfill the triangle relation:

sage: wigner_9j(0.5,0.5,0.5, 0.5,0.5,0.5, 0.5,0.5,0.5,prec=64)
Traceback (most recent call last):
...
ValueError: j values must be integer or half integer and fulfill the triangle relation
sage: wigner_9j(1,1,1, 0.5,1,1.5, 0.5,1,2.5,prec=64)                            # needs sage.rings.real_mpfr
Traceback (most recent call last):
...
ValueError: j values must be integer or half integer and fulfill the triangle relation

ALGORITHM:

This function uses the algorithm of [Ed1974] to calculate the value of the 3-\(j\) symbol exactly. Note that the formula contains alternating sums over large factorials and is therefore unsuitable for finite precision arithmetic and only useful for a computer algebra system [RH2003].