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)

Credits and Copyright

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

https://groups.google.com/forum/#!topic/sage-devel/M4NZdu-7O38

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)

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

The reference for this function is [Edmonds74].

INPUT:

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

OUTPUT:

Rational number times the square root of a rational number.

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:

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

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.dot_rot_grad_Ynm(j, p, l, m, theta, phi)

Returns dot product of rotational gradients of spherical harmonics.

This function returns the right hand side of the following expression:

\[\vec{R}Y{_j^{p}} \cdot \vec{R}Y{_l^{m}} = (-1)^{m+p} \sum\limits_{k=|l-j|}^{l+j}Y{_k^{m+p}} * \alpha_{l,m,j,p,k} * \frac{1}{2} (k^2-j^2-l^2+k-j-l)\]

Arguments

j, p, l, m …. indices in spherical harmonics (expressions or integers) theta, phi …. angle arguments in spherical harmonics

Example

>>> from sympy import symbols
>>> from sympy.physics.wigner import dot_rot_grad_Ynm
>>> theta, phi = symbols("theta phi")
>>> dot_rot_grad_Ynm(3, 2, 2, 0, theta, phi).doit()
3*sqrt(55)*Ynm(5, 2, theta, phi)/(11*sqrt(pi))

sympy.physics.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:

\[\begin{split}\begin{aligned} \operatorname{Gaunt}(l_1,l_2,l_3,m_1,m_2,m_3) &=\int Y_{l_1,m_1}(\Omega) Y_{l_2,m_2}(\Omega) Y_{l_3,m_3}(\Omega) \,d\Omega \\ &=\sqrt{\frac{(2l_1+1)(2l_2+1)(2l_3+1)}{4\pi}} \operatorname{Wigner3j}(l_1,l_2,l_3,0,0,0) \operatorname{Wigner3j}(l_1,l_2,l_3,m_1,m_2,m_3) \end{aligned}\end{split}\]

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

>>> 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

  \[\begin{split}\begin{aligned} Y(l_1,l_2,l_3,m_1,m_2,m_3) &=Y(l_3,l_1,l_2,m_3,m_1,m_2) \\ &=Y(l_2,l_3,l_1,m_2,m_3,m_1) \\ &=Y(l_3,l_2,l_1,m_3,m_2,m_1) \\ &=Y(l_1,l_3,l_2,m_1,m_3,m_2) \\ &=Y(l_2,l_1,l_3,m_2,m_1,m_3) \end{aligned}\end{split}\]

• 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 \(3j\) 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. \(L = l_1 + l_2 + l_3 = 2n\) 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)

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:

\[\operatorname{Wigner6j}(j_1,j_2,j_3,j_4,j_5,j_6) =(-1)^{j_1+j_2+j_4+j_5} W(j_1,j_2,j_5,j_4,j_3,j_6)\]

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)

Calculate the Wigner 3j symbol \(\operatorname{Wigner3j}(j_1,j_2,j_3,m_1,m_2,m_3)\).

INPUT:

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

OUTPUT:

Rational number times the square root of a rational number.

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 \(J:=j_1+j_2+j_3\)):

  \[\begin{split}\begin{aligned} \operatorname{Wigner3j}(j_1,j_2,j_3,m_1,m_2,m_3) &=\operatorname{Wigner3j}(j_3,j_1,j_2,m_3,m_1,m_2) \\ &=\operatorname{Wigner3j}(j_2,j_3,j_1,m_2,m_3,m_1) \\ &=(-1)^J \operatorname{Wigner3j}(j_3,j_2,j_1,m_3,m_2,m_1) \\ &=(-1)^J \operatorname{Wigner3j}(j_1,j_3,j_2,m_1,m_3,m_2) \\ &=(-1)^J \operatorname{Wigner3j}(j_2,j_1,j_3,m_2,m_1,m_3) \end{aligned}\end{split}\]

• invariant under space inflection, i.e.

  \[\operatorname{Wigner3j}(j_1,j_2,j_3,m_1,m_2,m_3) =(-1)^J \operatorname{Wigner3j}(j_1,j_2,j_3,-m_1,-m_2,-m_3)\]

• 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)

Calculate the Wigner 6j symbol \(\operatorname{Wigner6j}(j_1,j_2,j_3,j_4,j_5,j_6)\).

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.

\[\operatorname{Wigner6j}(j_1,j_2,j_3,j_4,j_5,j_6) =(-1)^{j_1+j_2+j_4+j_5} W(j_1,j_2,j_5,j_4,j_3,j_6)\]

The Wigner 6j symbol obeys the following symmetry rules:

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

  \[\begin{split}\begin{aligned} \operatorname{Wigner6j}(j_1,j_2,j_3,j_4,j_5,j_6) &=\operatorname{Wigner6j}(j_3,j_1,j_2,j_6,j_4,j_5) \\ &=\operatorname{Wigner6j}(j_2,j_3,j_1,j_5,j_6,j_4) \\ &=\operatorname{Wigner6j}(j_3,j_2,j_1,j_6,j_5,j_4) \\ &=\operatorname{Wigner6j}(j_1,j_3,j_2,j_4,j_6,j_5) \\ &=\operatorname{Wigner6j}(j_2,j_1,j_3,j_5,j_4,j_6) \end{aligned}\end{split}\]

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

  \[\operatorname{Wigner6j}(j_1,j_2,j_3,j_4,j_5,j_6) =\operatorname{Wigner6j}(j_1,j_5,j_6,j_4,j_2,j_3) =\operatorname{Wigner6j}(j_4,j_2,j_6,j_1,j_5,j_3) =\operatorname{Wigner6j}(j_4,j_5,j_3,j_1,j_2,j_6)\]

• 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)

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

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...

>>> wigner_9j(1/2,1/2,0, 1/2,3/2,1, 0,1,1 ,prec=64) # ==1/6
0.1666666...

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].