wigxj
Fast Wigner 3j, 6j, 9j symbols and Clebsch-Gordan coefficients for Python.
- Fast: a CPython extension around wigxjpf (exact prime-factorisation arithmetic), about 0.3 µs per symbol for small j.
- Simple: pass spins as plain numbers (
0.5,1,1.5); no doubled integers, no manual setup. - Always works: if the C extension cannot be built, a pure-Python backend with exact integer arithmetic takes over, with the same API.
- Typed: ships type stubs (
py.typed).
Installation
pip install wigxj
Building from source needs a C compiler (GCC, Clang or MSVC). Without one, installation still succeeds and wigxj uses the pure-Python backend. To check which backend is active:
import wigxj
print(wigxj._backend) # "C" or "pure"
Quick start
from wigxj import clebsch_gordan, wigner3j, wigner6j, wigner9j
wigner3j(1, 1, 2, 0, 0, 0) # 0.3651... = sqrt(2/15)
wigner6j(1, 1, 0, 1, 1, 0) # 0.3333... = 1/3
clebsch_gordan(0.5, 0.5, 0.5, -0.5, 0, 0) # 0.7071... = 1/sqrt(2) (spin singlet)
wigner9j(1, 1, 1, 1, 1, 1, 1, 1, 0) # 0.0555... = 1/18
API
All functions take integer or half-integer arguments and return a float.
Arguments that violate the selection rules (triangle conditions, m1 + m2 + m3 != 0, ...)
return 0.0.
| Function | Signature | Symbol |
|---|---|---|
wigner3j |
(j1, j2, j3, m1, m2, m3) |
(j1 j2 j3 / m1 m2 m3) |
wigner6j |
(j1, j2, j3, j4, j5, j6) |
{j1 j2 j3 / j4 j5 j6} |
wigner9j |
(j1, j2, j3, j4, j5, j6, j7, j8, j9) |
{j1 j2 j3 / j4 j5 j6 / j7 j8 j9} |
clebsch_gordan |
(j1, m1, j2, m2, j3, m3) |
⟨j1 m1 j2 m2 | j3 m3⟩ |
init |
(max_j) |
resize the C backend's tables (see below) |
clebsch_gordan uses the Condon-Shortley convention:
$$\langle j_1 m_1 j_2 m_2 | j_3 m_3 \rangle = (-1)^{-j_1+j_2-m_3} \sqrt{2j_3+1} \begin{pmatrix} j_1 & j_2 & j_3 \ m_1 & m_2 & -m_3 \end{pmatrix}$$
Large j
The C backend precomputes tables for j up to 200. Larger values raise a ValueError
until you extend the tables:
from wigxj import init, wigner3j
init(500) # tables for j up to 500
wigner3j(400, 400, 0, 0, 0, 0)
The pure-Python backend has no tables and no limit on j.
Performance
Time per call (best of 5, single machine, Python 3.15):
| Symbol | wigxj (C) | wigxj (pure) | sympy.physics.wigner |
|---|---|---|---|
| 3j, small (2 2 2 / 1 -1 0) | 0.27 µs | 17 µs | 22 µs |
| 3j, j = 50 (50 50 50 / 0 0 0) | 6.2 µs | 218 µs | 180 µs |
| 6j, small {2 2 2 / 2 2 2} | 0.34 µs | 8.6 µs | 122 µs |
| 6j, j = 30 {30 30 30 / 30 30 30} | 5.2 µs | 103 µs | 427 µs |
Why wigxj?
| Package | When to use it |
|---|---|
| wigxj | Fast floating-point 3j/6j/9j/Clebsch-Gordan with a plain-number API. |
| pywigxjpf | wigxjpf's own binding: same core, but arguments are doubled integers and tables are set up manually. |
| sympy.physics.wigner | You need exact symbolic results (rationals and square roots), and speed does not matter. |
| wigner | You need Wigner d-functions. |
Examples
See examples/, e.g. a dipole matrix element calculation.
Development
The development environment is managed with pixi:
pixi run test # build the C extension, run the tests on both backends
pixi run lint # ruff
pixi run format-check # ruff format
pixi run typecheck # mypy
The tests run every check against both the C and the pure-Python backend, and use Hypothesis to verify orthogonality relations, symmetries and agreement with sympy's exact results on random inputs.
License
wigxj is MIT-licensed (see LICENSE). It bundles
wigxjpf 1.15 by H. T. Johansson and
C. Forssén, licensed under LGPL-3.0-or-later (see
src/wigxj/_ext/wigxjpf/COPYING.LESSER).
If you use wigxj in your research, please cite wigxjpf: H. T. Johansson and C. Forssén, Fast and Accurate Evaluation of Wigner 3j, 6j, and 9j Symbols Using Prime Factorization and Multiword Integer Arithmetic, SIAM J. Sci. Comput. 38, A376 (2016), doi:10.1137/15M1021908.
1 maintainer
Metadata
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| Size | 83.5 kB |
| Tags | CPython 3.10 macOS 11.0+ ARM64 |
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