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 [Rasch03].

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

REFERENCES:

[Rasch03](1, 2, 3, 4, 5, 6) J. Rasch and A. C. H. Yu, ‘Efficient Storage Scheme for Pre-calculated Wigner 3j, 6j and Gaunt Coefficients’, SIAM J. Sci. Comput. Volume 25, Issue 4, pp. 1416-1428 (2003)

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)

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

The reference for this function is [Edmonds74].

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))
1
sage: clebsch_gordan(1.5,0.5,1, 1.5,-0.5,1)
1/2*sqrt(3)
sage: clebsch_gordan(3/2,1/2,1, -1/2,1/2,0)
-sqrt(3)*sqrt(1/6)

NOTES:

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

\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}

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)

Calculate the Gaunt coefficient.

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

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}

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: 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 trac ticket #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

NOTES:

The Gaunt coefficient obeys the following symmetry rules:

  • invariant under any permutation of the columns

    Y(l_1,l_2,l_3,m_1,m_2,m_3)
=Y(l_3,l_1,l_2,m_3,m_1,m_2) \hspace{3em} \\ \hspace{3em}
=Y(l_2,l_3,l_1,m_2,m_3,m_1)
=Y(l_3,l_2,l_1,m_3,m_2,m_1) \\ \hspace{3em}
=Y(l_1,l_3,l_2,m_1,m_3,m_2)
=Y(l_2,l_1,l_3,m_2,m_1,m_3)

  • invariant under space inflection, i.e.

    Y(l_1,l_2,l_3,m_1,m_2,m_3)
=Y(l_1,l_2,l_3,-m_1,-m_2,-m_3)

  • symmetric with respect to the 72 Regge symmetries as inherited for the 3-j symbols [Regge58]

  • zero for l_1, l_2, l_3 not fulfilling triangle relation

  • zero for violating any one of the conditions: l_1 \ge |m_1|, l_2 \ge |m_2|, l_3 \ge |m_3|

  • non-zero only for an even sum of the l_i, i.e. J=l_1+l_2+l_3=2n for n in \Bold{N}

ALGORITHM:

This function uses the algorithm of [Liberatodebrito82] 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 [Rasch03].

REFERENCES:

[Liberatodebrito82]‘FORTRAN program for the integral of three spherical harmonics’, A. Liberato de Brito, Comput. Phys. Commun., Volume 25, pp. 81-85 (1982)

AUTHORS:

  • Jens Rasch (2009-03-24): initial version for Sage
sage.functions.wigner.racah(aa, bb, cc, dd, ee, ff, prec=None)

Calculate 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)
-1/14

NOTES:

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

\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)

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

ALGORITHM:

This function uses the algorithm of [Edmonds74] 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 [Rasch03].

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)

Calculate 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)
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)
sqrt(1/6)
sage: wigner_3j(40, 100, 60, -10, 60, -50)
95608/18702538494885*sqrt(21082735836735314343364163310/220491455010479533763)
sage: wigner_3j(2500, 2500, 5000, 2488, 2400, -4888, prec=64)
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

NOTES:

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{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}

  • invariant under space inflection, i.e.

    \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}

  • symmetric with respect to the 72 additional symmetries based on the work by [Regge58]

  • 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 [Edmonds74] 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 [Rasch03].

REFERENCES:

[Regge58](1, 2) ‘Symmetry Properties of Clebsch-Gordan Coefficients’, T. Regge, Nuovo Cimento, Volume 10, pp. 544 (1958)
[Edmonds74](1, 2, 3, 4, 5) ‘Angular Momentum in Quantum Mechanics’, A. R. Edmonds, Princeton University Press (1974)

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)

Calculate 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: 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

NOTES:

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

\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)

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

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

    \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}

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

    \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}

  • additional 6 symmetries [Regge59] 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 [Edmonds74] 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 [Rasch03].

REFERENCES:

[Regge59]‘Symmetry Properties of Racah Coefficients’, T. Regge, Nuovo Cimento, Volume 11, pp. 116 (1959)
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)

Calculate 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: 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)
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 [Edmonds74] 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 [Rasch03].