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Wigner Symbols¶

Wigner, Clebsch-Gordan, Racah, and Gaunt coefficients

Collection of functions for calculating Wigner 3j, 6j, 9j, 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)

This code was taken from Sage with the permission of all authors:

AUTHORS:

• Jens Rasch (2009-03-24): initial version for Sage
• Jens Rasch (2009-05-31): updated to sage-4.0

Copyright (C) 2008 Jens Rasch <jyr2000@gmail.com>

sympy.physics.wigner.clebsch_gordan(j_1, j_2, j_3, m_1, m_2, m_3, prec=None)[source]

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:

>>> from sympy import S
>>> from sympy.physics.wigner import clebsch_gordan
>>> clebsch_gordan(S(3)/2, S(1)/2, 2, S(3)/2, S(1)/2, 2)
1
>>> clebsch_gordan(S(3)/2, S(1)/2, 1, S(3)/2, -S(1)/2, 1)
sqrt(3)/2
>>> clebsch_gordan(S(3)/2, S(1)/2, 1, -S(1)/2, S(1)/2, 0)
-sqrt(2)/2


NOTES:

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

$\langle j_1 m_1 \; j_2 m_2 | j_3 m_3 \rangle =(-1)^{j_1-j_2+m_3} \sqrt{2j_3+1} \; Wigner3j(j_1,j_2,j_3,m_1,m_2,-m_3)$

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

AUTHORS:

• Jens Rasch (2009-03-24): initial version
sympy.physics.wigner.gaunt(l_1, l_2, l_3, m_1, m_2, m_3, prec=None)[source]

Calculate the Gaunt coefficient.

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

$Y(j_1,j_2,j_3,m_1,m_2,m_3) =\int Y_{l_1,m_1}(\Omega) Y_{l_2,m_2}(\Omega) Y_{l_3,m_3}(\Omega) d\Omega =\sqrt{(2l_1+1)(2l_2+1)(2l_3+1)/(4\pi)} \; Y(j_1,j_2,j_3,0,0,0) \; Y(j_1,j_2,j_3,m_1,m_2,m_3)$

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

>>> from sympy.physics.wigner import gaunt
>>> gaunt(1,0,1,1,0,-1)
-1/(2*sqrt(pi))
>>> gaunt(1000,1000,1200,9,3,-12).n(64)
0.00689500421922113448...


It is an error to use non-integer values for $$l$$ and $$m$$:

sage: gaunt(1.2,0,1.2,0,0,0)
Traceback (most recent call last):
...
ValueError: l values must be integer
sage: gaunt(1,0,1,1.1,0,-1.1)
Traceback (most recent call last):
...
ValueError: m values must be integer

NOTES:

The Gaunt coefficient obeys the following symmetry rules:

• invariant under any permutation of the columns

$Y(j_1,j_2,j_3,m_1,m_2,m_3) =Y(j_3,j_1,j_2,m_3,m_1,m_2) =Y(j_2,j_3,j_1,m_2,m_3,m_1) =Y(j_3,j_2,j_1,m_3,m_2,m_1) =Y(j_1,j_3,j_2,m_1,m_3,m_2) =Y(j_2,j_1,j_3,m_2,m_1,m_3)$
• invariant under space inflection, i.e.

$Y(j_1,j_2,j_3,m_1,m_2,m_3) =Y(j_1,j_2,j_3,-m_1,-m_2,-m_3)$
• symmetric with respect to the 72 Regge symmetries as inherited for the $$3j$$ 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 $$\mathbb{N}$$

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
sympy.physics.wigner.racah(aa, bb, cc, dd, ee, ff, prec=None)[source]

Calculate the Racah symbol $$W(a,b,c,d;e,f)$$.

INPUT:

• a, ..., f - 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

>>> from sympy.physics.wigner import racah
>>> racah(3,3,3,3,3,3)
-1/14


NOTES:

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

$Wigner6j(j_1,j_2,j_3,j_4,j_5,j_6) =(-1)^{j_1+j_2+j_4+j_5} W(j_1,j_2,j_5,j_4,j_3,j_6)$

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

ALGORITHM:

This function uses the algorithm of [Edmonds74] to calculate the value of the 6j 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
sympy.physics.wigner.wigner_3j(j_1, j_2, j_3, m_1, m_2, m_3, prec=None)[source]

Calculate the Wigner 3j symbol $$Wigner3j(j_1,j_2,j_3,m_1,m_2,m_3)$$.

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

>>> from sympy.physics.wigner import wigner_3j
>>> wigner_3j(2, 6, 4, 0, 0, 0)
sqrt(715)/143
>>> wigner_3j(2, 6, 4, 0, 0, 1)
0


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 3j symbol obeys the following symmetry rules:

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

$Wigner3j(j_1,j_2,j_3,m_1,m_2,m_3) =Wigner3j(j_3,j_1,j_2,m_3,m_1,m_2) =Wigner3j(j_2,j_3,j_1,m_2,m_3,m_1) =(-1)^J Wigner3j(j_3,j_2,j_1,m_3,m_2,m_1) =(-1)^J Wigner3j(j_1,j_3,j_2,m_1,m_3,m_2) =(-1)^J Wigner3j(j_2,j_1,j_3,m_2,m_1,m_3)$
• invariant under space inflection, i.e.

$Wigner3j(j_1,j_2,j_3,m_1,m_2,m_3) =(-1)^J Wigner3j(j_1,j_2,j_3,-m_1,-m_2,-m_3)$
• 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 3j 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
sympy.physics.wigner.wigner_6j(j_1, j_2, j_3, j_4, j_5, j_6, prec=None)[source]

Calculate the Wigner 6j symbol $$Wigner6j(j_1,j_2,j_3,j_4,j_5,j_6)$$.

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

>>> from sympy.physics.wigner import wigner_6j
>>> wigner_6j(3,3,3,3,3,3)
-1/14
>>> wigner_6j(5,5,5,5,5,5)
1/52


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

$Wigner6j(j_1,j_2,j_3,j_4,j_5,j_6) =(-1)^{j_1+j_2+j_4+j_5} W(j_1,j_2,j_5,j_4,j_3,j_6)$

The Wigner 6j symbol obeys the following symmetry rules:

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

$Wigner6j(j_1,j_2,j_3,j_4,j_5,j_6) =Wigner6j(j_3,j_1,j_2,j_6,j_4,j_5) =Wigner6j(j_2,j_3,j_1,j_5,j_6,j_4) =Wigner6j(j_3,j_2,j_1,j_6,j_5,j_4) =Wigner6j(j_1,j_3,j_2,j_4,j_6,j_5) =Wigner6j(j_2,j_1,j_3,j_5,j_4,j_6)$
• They are invariant under the exchange of the upper and lower arguments in each of any two columns, i.e.

$Wigner6j(j_1,j_2,j_3,j_4,j_5,j_6) =Wigner6j(j_1,j_5,j_6,j_4,j_2,j_3) =Wigner6j(j_4,j_2,j_6,j_1,j_5,j_3) =Wigner6j(j_4,j_5,j_3,j_1,j_2,j_6)$
• 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 6j 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)
sympy.physics.wigner.wigner_9j(j_1, j_2, j_3, j_4, j_5, j_6, j_7, j_8, j_9, prec=None)[source]

Calculate the Wigner 9j symbol $$Wigner9j(j_1,j_2,j_3,j_4,j_5,j_6,j_7,j_8,j_9)$$.

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

>>> from sympy.physics.wigner import wigner_9j
>>> wigner_9j(1,1,1, 1,1,1, 1,1,0 ,prec=64) # ==1/18
0.05555555...


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 3j 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].

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