WignerSymbols

Compute Wigner's 3j and 6j symbols, and related quantities such as Clebsch-Gordan coefficients and Racah's symbols.

Installation

Install with the new package manager via ]add WignerSymbols or

using Pkg
Pkg.add("WignerSymbols")

Available functions

While the following function signatures are probably self-explanatory, you can query help for them in the Julia REPL to get further details.

• wigner3j(T::Type{<:Real} = RationalRoot{BigInt}, j₁, j₂, j₃, m₁, m₂, m₃ = -m₂-m₁) -> ::T
• wigner6j(T::Type{<:Real} = RationalRoot{BigInt}, j₁, j₂, j₃, j₄, j₅, j₆) -> ::T
• clebschgordan(T::Type{<:Real} = RationalRoot{BigInt}, j₁, m₁, j₂, m₂, j₃, m₃ = m₁+m₂) -> ::T
• racahV(T::Type{<:Real} = RationalRoot{BigInt}, j₁, j₂, j₃, m₁, m₂, m₃ = -m₁-m₂) -> ::T
• racahW(T::Type{<:Real} = RationalRoot{BigInt}, j₁, j₂, J, j₃, J₁₂, J₂₃) -> ::T
• δ(j₁, j₂, j₃) -> ::Bool
• Δ(T::Type{<:Real} = RationalRoot{BigInt}, j₁, j₂, j₃) -> ::T

The package relies on HalfIntegers.jl to support and use arithmetic with half integer numbers, and, since v1.1, on RationalRoots.jl to return the result exactly as the square root of a Rational{BigInt}, which will then be automatically converted to a suitable floating point value upon further arithmetic, using the AbstractIrrational interface from Julia Base.

Implementation

Largely based on reading the paper (but not the code):

[1] H. T. Johansson and C. Forssén, SIAM Journal on Scientific Compututing 38 (2016) 376-384 (arXiv:1504.08329)

with some additional modifications to further improve efficiency for large j (angular momenta quantum numbers).

In particular, 3j and 6j symbols are computed exactly, in the format √(r) * s where r and s are exactly computed as Rational{BigInt}, using an intermediate representation based on prime number factorization. As a consequence thereof, all of the above functions can be called requesting BigFloat precision for the result. There is currently no convenient syntax for obtaining r and s directly (see TODO).

Most intermediate calculations (prime factorizations of numbers and their factorials, conversion between prime powers and BigInts) are cached to improve the efficiency, but this can result in large use of memory when querying Wigner symbols for large values of j.

Also uses ideas from

[2] J. Rasch and A. C. H. Yu, SIAM Journal on Scientific Compututing 25 (2003), 1416–1428

for caching the computed 3j and 6j symbols.

Todo

• Wigner 9-j symbols, as explained in [1] and based on

  [3] L. Wei, New formula for 9-j symbols and their direct calculation, Computers in Physics, 12 (1998), 632–634.

• Convenient syntax to get the exact results in the √(r) * s format, but in such a way that it can be used by the Julia type system and can be converted afterwards. Solved in v1.1 by the package RationalRoots.jl, the implementation of which was initiated by @w-vdh in PR #9.