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

Regge58(1,2)

‘Symmetry Properties of Clebsch-Gordan Coefficients’, T. Regge, Nuovo Cimento, Volume 10, pp. 544 (1958)

Regge59

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

Edmonds74(1,2,3,4,5,6,7,8,9,10)

A. R. Edmonds. Angular momentum in quantum mechanics. Investigations in physics, 4.; Investigations in physics, no. 4. Princeton, N.J., Princeton University Press, 1957.

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)

Liberatodebrito82

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

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

• Oscar Gerardo Lazo Arjona (2017-06-18): added Wigner D matrices

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].

Parameters

j_1, j_2, j_3, m_1, m_2, m_3 :

Integer or half integer.

Returns

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.

Explanation

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.

Parameters

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.

Returns

Rational number times the square root of a rational number

(if prec=None), or real number if a precision is given.

Explanation

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

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

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 \(\mathbb{N}\)

Algorithms

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].

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 \(W(a,b,c,d;e,f)\).

Parameters

a, …, f :

Integer or half integer.

prec :

Precision, default: None. Providing a precision can drastically speed up the calculation.

Returns

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 \(\operatorname{Wigner3j}(j_1,j_2,j_3,m_1,m_2,m_3)\).

Parameters

j_1, j_2, j_3, m_1, m_2, m_3 :

Integer or half integer.

Returns

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].

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 \(\operatorname{Wigner6j}(j_1,j_2,j_3,j_4,j_5,j_6)\).

Parameters

j_1, …, j_6 :

Integer or half integer.

prec :

Precision, default: None. Providing a precision can drastically speed up the calculation.

Returns

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].

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 \(\operatorname{Wigner9j}(j_1,j_2,j_3,j_4,j_5,j_6,j_7,j_8,j_9)\).

Parameters

j_1, …, j_9 :

Integer or half integer.

prec : precision, default

None. Providing a precision can drastically speed up the calculation.

Returns

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].

sympy.physics.wigner.wigner_d(J, alpha, beta, gamma)

Return the Wigner D matrix for angular momentum J.

Returns

A matrix representing the corresponding Euler angle rotation( in the basis

of eigenvectors of \(J_z\)).

\[\mathcal{D}_{\alpha \beta \gamma} = \exp\big( \frac{i\alpha}{\hbar} J_z\big) \exp\big( \frac{i\beta}{\hbar} J_y\big) \exp\big( \frac{i\gamma}{\hbar} J_z\big)\]

The components are calculated using the general form [Edmonds74],

equation 4.1.12.

Explanation

J :

An integer, half-integer, or SymPy symbol for the total angular momentum of the angular momentum space being rotated.

alpha, beta, gamma - Real numbers representing the Euler.

Angles of rotation about the so-called vertical, line of nodes, and figure axes. See [Edmonds74].

Examples

The simplest possible example:

>>> from sympy.physics.wigner import wigner_d
>>> from sympy import Integer, symbols, pprint
>>> half = 1/Integer(2)
>>> alpha, beta, gamma = symbols("alpha, beta, gamma", real=True)
>>> pprint(wigner_d(half, alpha, beta, gamma), use_unicode=True)
⎡  ⅈ⋅α  ⅈ⋅γ             ⅈ⋅α  -ⅈ⋅γ         ⎤
⎢  ───  ───             ───  ─────        ⎥
⎢   2    2     ⎛β⎞       2     2      ⎛β⎞ ⎥
⎢ ℯ   ⋅ℯ   ⋅cos⎜─⎟     ℯ   ⋅ℯ     ⋅sin⎜─⎟ ⎥
⎢              ⎝2⎠                    ⎝2⎠ ⎥
⎢                                         ⎥
⎢  -ⅈ⋅α   ⅈ⋅γ          -ⅈ⋅α   -ⅈ⋅γ        ⎥
⎢  ─────  ───          ─────  ─────       ⎥
⎢    2     2     ⎛β⎞     2      2      ⎛β⎞⎥
⎢-ℯ     ⋅ℯ   ⋅sin⎜─⎟  ℯ     ⋅ℯ     ⋅cos⎜─⎟⎥
⎣                ⎝2⎠                   ⎝2⎠⎦

sympy.physics.wigner.wigner_d_small(J, beta)

Return the small Wigner d matrix for angular momentum J.

Returns

A matrix representing the corresponding Euler angle rotation( in the basis

of eigenvectors of \(J_z\)).

\[\mathcal{d}_{\beta} = \exp\big( \frac{i\beta}{\hbar} J_y\big)\]

The components are calculated using the general form [Edmonds74],

equation 4.1.15.

Explanation

JAn integer, half-integer, or SymPy symbol for the total angular

momentum of the angular momentum space being rotated.

betaA real number representing the Euler angle of rotation about

the so-called line of nodes. See [Edmonds74].

Examples

>>> from sympy import Integer, symbols, pi, pprint
>>> from sympy.physics.wigner import wigner_d_small
>>> half = 1/Integer(2)
>>> beta = symbols("beta", real=True)
>>> pprint(wigner_d_small(half, beta), use_unicode=True)
⎡   ⎛β⎞      ⎛β⎞⎤
⎢cos⎜─⎟   sin⎜─⎟⎥
⎢   ⎝2⎠      ⎝2⎠⎥
⎢               ⎥
⎢    ⎛β⎞     ⎛β⎞⎥
⎢-sin⎜─⎟  cos⎜─⎟⎥
⎣    ⎝2⎠     ⎝2⎠⎦

>>> pprint(wigner_d_small(2*half, beta), use_unicode=True)
⎡        2⎛β⎞              ⎛β⎞    ⎛β⎞           2⎛β⎞     ⎤
⎢     cos ⎜─⎟        √2⋅sin⎜─⎟⋅cos⎜─⎟        sin ⎜─⎟     ⎥
⎢         ⎝2⎠              ⎝2⎠    ⎝2⎠            ⎝2⎠     ⎥
⎢                                                        ⎥
⎢       ⎛β⎞    ⎛β⎞       2⎛β⎞      2⎛β⎞        ⎛β⎞    ⎛β⎞⎥
⎢-√2⋅sin⎜─⎟⋅cos⎜─⎟  - sin ⎜─⎟ + cos ⎜─⎟  √2⋅sin⎜─⎟⋅cos⎜─⎟⎥
⎢       ⎝2⎠    ⎝2⎠        ⎝2⎠       ⎝2⎠        ⎝2⎠    ⎝2⎠⎥
⎢                                                        ⎥
⎢        2⎛β⎞               ⎛β⎞    ⎛β⎞          2⎛β⎞     ⎥
⎢     sin ⎜─⎟        -√2⋅sin⎜─⎟⋅cos⎜─⎟       cos ⎜─⎟     ⎥
⎣         ⎝2⎠               ⎝2⎠    ⎝2⎠           ⎝2⎠     ⎦

From table 4 in [Edmonds74]

>>> pprint(wigner_d_small(half, beta).subs({beta:pi/2}), use_unicode=True)
⎡ √2   √2⎤
⎢ ──   ──⎥
⎢ 2    2 ⎥
⎢        ⎥
⎢-√2   √2⎥
⎢────  ──⎥
⎣ 2    2 ⎦

>>> pprint(wigner_d_small(2*half, beta).subs({beta:pi/2}),
... use_unicode=True)
⎡       √2      ⎤
⎢1/2    ──   1/2⎥
⎢       2       ⎥
⎢               ⎥
⎢-√2         √2 ⎥
⎢────   0    ── ⎥
⎢ 2          2  ⎥
⎢               ⎥
⎢      -√2      ⎥
⎢1/2   ────  1/2⎥
⎣       2       ⎦

>>> pprint(wigner_d_small(3*half, beta).subs({beta:pi/2}),
... use_unicode=True)
⎡ √2    √6    √6   √2⎤
⎢ ──    ──    ──   ──⎥
⎢ 4     4     4    4 ⎥
⎢                    ⎥
⎢-√6   -√2    √2   √6⎥
⎢────  ────   ──   ──⎥
⎢ 4     4     4    4 ⎥
⎢                    ⎥
⎢ √6   -√2   -√2   √6⎥
⎢ ──   ────  ────  ──⎥
⎢ 4     4     4    4 ⎥
⎢                    ⎥
⎢-√2    √6   -√6   √2⎥
⎢────   ──   ────  ──⎥
⎣ 4     4     4    4 ⎦

>>> pprint(wigner_d_small(4*half, beta).subs({beta:pi/2}),
... use_unicode=True)
⎡             √6            ⎤
⎢1/4   1/2    ──   1/2   1/4⎥
⎢             4             ⎥
⎢                           ⎥
⎢-1/2  -1/2   0    1/2   1/2⎥
⎢                           ⎥
⎢ √6                     √6 ⎥
⎢ ──    0    -1/2   0    ── ⎥
⎢ 4                      4  ⎥
⎢                           ⎥
⎢-1/2  1/2    0    -1/2  1/2⎥
⎢                           ⎥
⎢             √6            ⎥
⎢1/4   -1/2   ──   -1/2  1/4⎥
⎣             4             ⎦

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