WignerSymbols
Compute Wigner's 3j and 6j symbols, and related quantities such as ClebschGordan 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 selfexplanatory, 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 = m₂m₁) > ::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 = m₁m₂) > ::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) 376384 (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 BigInt
s) 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 9j symbols, as explained in [1] and based on

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