wigner 2025.1

Wigner d functions, 3j symbols, 6j symbols

wigner package for Python

The wigner package for Python contains bindings to the wigner C library for computing Wigner d functions and 3j symbols.

Functions are provided with minimal wrapping:

>>> import wigner
>>> wigner.wigner_3jj(2, 3, -1, 2)
(1.0, 5.0, [-0.308, -4.280e-17, 0.188, -0.197, 0.101])

Installation

The package can be installed with pip:

$ pip install wigner

The only dependency is numpy.

Functions

• legendre_pl – Legendre polynomial as function of l
• wigner_3jj – Wigner 3j symbol as function of l1
• wigner_3jm – Wigner 3j symbol as function of m2
• wigner_6j – Wigner 6j symbol as function of l1
• wigner_dl – Wigner d function as function of l

legendre_pl

legendre_pl(lmin, lmax, x)

Compute the Legendre polynomials P_l(x) for all degrees l = lmin to l = lmax, with x being held fixed. The arguments lmin and lmax must be integers, while the argument x must be float. Returns a numpy array of size lmax-lmin+1.

wigner_3jj

wigner_3jj(l2, l3, m2, m3)

Evaluate the Wigner 3j symbol

/   l1    l2  l3 \
\ -m2-m3  m2  m3 /

for all allowable values of l1, with the other parameters held fixed. For physically meaningful outputs, the arguments must be integer or half-integer, although other inputs are allowed. Returns a tuple l1min, l1max, thrcof where l1min and l1max are the smallest and largest allowable values of l1, and thrcof is a numpy array of size l1max-l1min+1 containing the values of the 3j symbol.

wigner_3jm

wigner_3jm(l1, l2, l3, m1)

Evaluate the Wigner 3j symbol

/ l1  l2    l3  \
\ m1  m2  -m1-m2/

for all allowable values of m2, with the other parameters held fixed. For physically meaningful outputs, the arguments must be integer or half-integer, although other inputs are allowed. Returns a tuple m2min, m2max, thrcof where m2min and m2max are the smallest and largest allowable values of m2, and thrcof is a numpy array of size m2max-m2min+1 containing the values of the 3j symbol.

wigner_6j

wigner_6j(l2, l3, l4, l5, l6)

Evaluate the Wigner 6j symbol

{l1  l2  l3}
{l4  l5  l6}

for all allowable values of l1, with the other parameters held fixed. For physically meaningful outputs, the arguments must be integer or half-integer, although other inputs are allowed. Returns a tuple l1min, l1max, sixcof where l1min and l1max are the smallest and largest allowable values of l1, and sixcof is a numpy array of size l1max-l1min+1 containing the values of the 6j symbol.

wigner_dl

wigner_dl(lmin, lmax, m1, m2, theta)

Compute the Wigner d function d^l_{m1,m2}(theta) for all degrees l = lmin to l = lmax, with m1, m2, and theta being held fixed. The arguments lmin, lmax, m1, m2 must be integers, and the angle theta must be given in radian as float. Returns a numpy array of size lmax-lmin+1.