mirror of
https://github.com/tgorordo/WignerSymbols.jl.git
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add help and more tests
This commit is contained in:
Jutho
parent
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3 changed files with 169 additions and 52 deletions
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@ -4,7 +4,6 @@ os:
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- linux
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- osx
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julia:
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- 0.6
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- nightly
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notifications:
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email: false
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@ -1,33 +1,76 @@
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module WignerSymbols
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export δ, Δ, clebschgordan, wigner3j, wigner6j
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export δ, Δ, clebschgordan, wigner3j, wigner6j, racahV, racahW
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include("primefactorization.jl")
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const Wigner3j = Dict{Tuple{UInt,UInt,UInt,Int,Int},Tuple{Rational{BigInt},Rational{BigInt}}}()
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const Wigner6j = Dict{NTuple{6,UInt},Tuple{Rational{BigInt},Rational{BigInt}}}()
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clebschgordan(j₁, m₁, j₂, m₂, j₃, m₃ = m₁+m₂) = clebschgordan(Float64, j₁, m₁, j₂, m₂, j₃, m₃)
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function clebschgordan(T::Type{<:AbstractFloat}, j₁, m₁, j₂, m₂, j₃, m₃ = m₁+m₂)
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s = wigner3j(T, j₁, j₂, j₃, m₁, m₂, -m₃)
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iszero(s) && return s
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if isodd(convert(Int,j₁ - j₂ + m₃))
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s *= -one(s)
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end
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s *= sqrt(convert(T, j₃+j₃+one(j₃)))
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return s
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# check integerness and correctness of (j,m) angular momentum
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ϵ(j, m) = (abs(m) <= j && isinteger(j-m) && isinteger(j+m))
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# check triangle condition
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"""
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δ(j₁, j₂, j₃) -> ::Bool
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Checks the triangle conditions `j₃ <= j₁ + j₂`, `j₁ <= j₂ + j₃` and `j₂ <= j₃ + j₁`.
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"""
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function δ(j₁, j₂, j₃)
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j₃ <= j₁ + j₂ || return false
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j₁ <= j₂ + j₃ || return false
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j₂ <= j₃ + j₁ || return false
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return true
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end
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wigner3j(j₁, j₂, j₃, m₁, m₂, m₃ = -m₂-m₃) = wigner3j(Float64, j₁, j₂, j₃, m₁, m₂, m₃)
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function wigner3j(T::Type{<:AbstractFloat}, j₁, j₂, j₃, m₁, m₂, m₃ = -m₂-m₃)
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# check angular momenta and triangle condition
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if !(δ(j₁, j₂, j₃) && ϵ(j₁, m₁) && ϵ(j₂, m₂) && ϵ(j₃, m₃))
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throw(DomainError((j₁, j₂, j₃, m₁, m₂, m₃)))
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# triangle coefficient
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"""
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Δ(T::Type{<:AbstractFloat} = Float64, j₁, j₂, j₃) -> ::T
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Computes the triangle coefficient
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`Δ(j₁, j₂, j₃) = √((j₁+j₂-j₃)!*(j₁-j₂+j₃)!*(j₂+j₃-j₁)! / (j₁+j₂+j₃+1)!)`
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as a type `T` floating point number.
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Returns `zero(T)` if the triangle condition `δ(j₁, j₂, j₃)` is not satisfied, but
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throws a `DomainError` if the `jᵢ`s are not (half)integer
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"""
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Δ(j₁, j₂, j₃) = Δ(Float64, j₁, j₂, j₃)
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function Δ(T::Type{<:AbstractFloat}, j₁, j₂, j₃)
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for jᵢ in (j₁, j₂, j₃)
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(isinteger(2*jᵢ) && jᵢ >= 0) || throw(DomainError("invalid jᵢ", jᵢ))
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end
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if !δ(j₁, j₂, j₃)
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return zero(T)
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end
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n, d = Δ²(j₁, j₂, j₃)
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return sqrt(convert(T, convert(BigInt, n) // convert(BigInt, d)))
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end
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"""
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wigner3j(T::Type{<:AbstractFloat} = Float64, j₁, j₂, j₃, m₁, m₂, m₃ = -m₂-m₁) -> ::T
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Compute the value of the Wigner-3j symbol
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⎛ j₁ j₂ j₃ ⎞
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⎝ m₁ m₂ m₃ ⎠
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as a type `T` floating point number. By default, `T = Float64` and `m₃ = -m₁-m₂`.
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Returns `zero(T)` if the triangle condition `δ(j₁, j₂, j₃)` is not satisfied, but
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throws a `DomainError` if the `jᵢ`s and `mᵢ`s are not (half)integer or `abs(mᵢ) > jᵢ`.
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"""
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wigner3j(j₁, j₂, j₃, m₁, m₂, m₃ = -m₁-m₂) = wigner3j(Float64, j₁, j₂, j₃, m₁, m₂, m₃)
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function wigner3j(T::Type{<:AbstractFloat}, j₁, j₂, j₃, m₁, m₂, m₃ = -m₁-m₂)
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# check angular momenta
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for (jᵢ,mᵢ) in ((j₁, m₁), (j₂, m₂), (j₃, m₃))
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ϵ(jᵢ, mᵢ) || throw(DomainError("invalid (jᵢ, mᵢ)", (jᵢ, mᵢ) ))
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end
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# check triangle condition and m₁+m₂+m₃ == 0
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if !δ(j₁, j₂, j₃) || !iszero(m₁+m₂+m₃)
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return zero(T)
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end
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iszero(m₁+m₂+m₃) || return zero(T)
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# we reorder such that j₁ >= j₂ >= j₃ and m₁ >= 0 or m₁ == 0 && m₂ >= 0
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j₁, j₂, j₃, m₁, m₂, m₃, sgn = reorder3j(j₁, j₂, j₃, m₁, m₂, m₃)
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# do we also want to use Regge symmetries?
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# TODO: do we also want to use Regge symmetries?
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α₁ = convert(Int, j₂ - m₁ - j₃ ) # can be negative
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α₂ = convert(Int, j₁ + m₂ - j₃ ) # can be negative
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β₁ = convert(Int, j₁ + j₂ - j₃ )
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@ -41,12 +84,11 @@ function wigner3j(T::Type{<:AbstractFloat}, j₁, j₂, j₃, m₁, m₂, m₃ =
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if haskey(Wigner3j, (β₁, β₂, β₃, α₁, α₂))
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r, s = Wigner3j[(β₁, β₂, β₃, α₁, α₂)]
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else
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s1 = Δ²(j₁, j₂, j₃)
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s2 = prod(map(primefactorial, (β₂, β₁ - α₁, β₁ - α₂, β₃, β₃ - α₁, β₂ - α₂)))
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s1n, s1d = Δ²(j₁, j₂, j₃)
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s2n = prod(map(primefactorial, (β₂, β₁ - α₁, β₁ - α₂, β₃, β₃ - α₁, β₂ - α₂)))
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snum, rnum = splitsquare(s1.num*s2)
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sden, rden = splitsquare(s1.den)
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snum, rnum = splitsquare(s1n*s2n)
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sden, rden = splitsquare(s1d)
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s = convert(BigInt, snum) // convert(BigInt, sden)
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r = convert(BigInt, rnum) // convert(BigInt, rden)
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s *= compute3jseries(β₁, β₂, β₃, α₁, α₂)
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@ -56,17 +98,67 @@ function wigner3j(T::Type{<:AbstractFloat}, j₁, j₂, j₃, m₁, m₂, m₃ =
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return sgn*sqrt(convert(T, r))*convert(T, s)
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end
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"""
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clebschgordan(T::Type{<:AbstractFloat} = Float64, j₁, j₂, j₃, m₁, m₂, m₃ = m₁+m₂) -> ::T
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Compute the value of the Clebsch-Gordan coefficient <j₁, m₁; j₂, m₂ | j₃, m₃ >
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as a type `T` floating point number. By default, `T = Float64` and `m₃ = m₁+m₂`.
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Returns `zero(T)` if the triangle condition `δ(j₁, j₂, j₃)` is not satisfied, but
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throws a `DomainError` if the `jᵢ`s and `mᵢ`s are not (half)integer or `abs(mᵢ) > jᵢ`.
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"""
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clebschgordan(j₁, m₁, j₂, m₂, j₃, m₃ = m₁+m₂) = clebschgordan(Float64, j₁, m₁, j₂, m₂, j₃, m₃)
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function clebschgordan(T::Type{<:AbstractFloat}, j₁, m₁, j₂, m₂, j₃, m₃ = m₁+m₂)
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s = wigner3j(T, j₁, j₂, j₃, m₁, m₂, -m₃)
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iszero(s) && return s
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s *= sqrt(convert(T, j₃+j₃+one(j₃)))
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return isodd(convert(Int,j₁ - j₂ + m₃)) ? -s : s
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end
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"""
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racahV(T::Type{<:AbstractFloat} = Float64, j₁, j₂, j₃, m₁, m₂, m₃ = -m₁-m₂) -> ::T
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Compute the value of Racah's V-symbol
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V(j₁, j₂, j₃; m₁, m₂, m₃) = (-1)^(-j₁+j₂+j₃) * ⎛ j₁ j₂ j₃ ⎞
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⎝ m₁ m₂ m₃ ⎠
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as a type `T` floating point number. By default, `T = Float64` and `m₃ = -m₁-m₂`.
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Returns `zero(T)` if the triangle condition `δ(j₁, j₂, j₃)` is not satisfied, but
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throws a `DomainError` if the `jᵢ`s and `mᵢ`s are not (half)integer or `abs(mᵢ) > jᵢ`.
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"""
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racahV(j₁, j₂, j₃, m₁, m₂, m₃ = -m₁-m₂) = racahV(Float64, j₁, j₂, j₃, m₁, m₂, m₃)
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function racahV(T::Type{<:AbstractFloat}, j₁, j₂, j₃, m₁, m₂, m₃ = -m₁-m₂)
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s = wigner3j(T, j₁, j₂, j₃, m₁, m₂, m₃)
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return isodd(convert(Int, -j₁ + j₂ + j₃)) ? -s : s
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end
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"""
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wigner6j(T::Type{<:AbstractFloat} = Float64, j₁, j₂, j₃, j₄, j₅, j₆) -> ::T
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Compute the value of the Wigner-6j symbol
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_⎧ j₁ j₂ j₃ ⎫_
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⎩ j₄ j₅ j₆ ⎭
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as a type `T` floating point number. By default, `T = Float64`.
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Returns `zero(T)` if any of triangle conditions `δ(j₁, j₂, j₃)`, `δ(j₁, j₆, j₅)`,
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`δ(j₂, j₄, j₆)`, `δ(j₃, j₄, j₅)` are not satisfied, but throws a `DomainError` if
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the `jᵢ`s are not integer or halfinteger.
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"""
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wigner6j(j₁, j₂, j₃, j₄, j₅, j₆) = wigner6j(Float64, j₁, j₂, j₃, j₄, j₅, j₆)
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function wigner6j(T::Type{<:AbstractFloat}, j₁, j₂, j₃, j₄, j₅, j₆)
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# check validity of `jᵢ`s
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for jᵢ in (j₁, j₂, j₃, j₄, j₅, j₆)
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(isinteger(2*jᵢ) && jᵢ >= 0) || throw(DomainError("invalid jᵢ", jᵢ))
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end
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α̂₁ = (j₁, j₂, j₃)
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α̂₂ = (j₁, j₅, j₆)
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α̂₂ = (j₁, j₆, j₅)
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α̂₃ = (j₂, j₄, j₆)
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α̂₄ = (j₃, j₄, j₅)
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# check triangle conditions
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if !(δ(α̂₁...) && δ(α̂₂...) && δ(α̂₃...) && δ(α̂₄...))
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return zero(T)
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# throw(DomainError())
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end
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# reduce
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α₁ = convert(UInt, +(α̂₁...))
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@ -86,13 +178,13 @@ function wigner6j(T::Type{<:AbstractFloat}, j₁, j₂, j₃, j₄, j₅, j₆)
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r, s = Wigner6j[(β₁, β₂, β₃, α₁, α₂, α₃)]
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else
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# order irrelevant: product remains the same under action of reorder6j
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Δ₁ = Δ²(α̂₁...)
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Δ₂ = Δ²(α̂₂...)
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Δ₃ = Δ²(α̂₃...)
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Δ₄ = Δ²(α̂₄...)
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n₁, d₁ = Δ²(α̂₁...)
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n₂, d₂ = Δ²(α̂₂...)
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n₃, d₃ = Δ²(α̂₃...)
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n₄, d₄ = Δ²(α̂₄...)
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snum, rnum = splitsquare(Δ₁.num * Δ₂.num * Δ₃.num * Δ₄.num)
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sden, rden = splitsquare(Δ₁.den * Δ₂.den * Δ₃.den * Δ₄.den)
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snum, rnum = splitsquare(n₁ * n₂ * n₃ * n₄)
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sden, rden = splitsquare(d₁ * d₂ * d₃ * d₄)
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s = convert(BigInt, snum) // convert(BigInt, sden)
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r = convert(BigInt, rnum) // convert(BigInt, rden)
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s *= compute6jseries(β₁, β₂, β₃, α₁, α₂, α₃, α₄)
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@ -103,30 +195,23 @@ function wigner6j(T::Type{<:AbstractFloat}, j₁, j₂, j₃, j₄, j₅, j₆)
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return sqrt(convert(T, r))*convert(T, s)
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end
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"""
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racahW(T::Type{<:AbstractFloat} = Float64, j₁, j₂, J, j₃, J₁₂, J₂₃) -> ::T
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# check angular momentum
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ϵ(j, m) = (abs(m) <= j && isinteger(j-m) && isinteger(j+m))
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Compute the value of Racah's W coefficient
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`W(j₁, j₂, J, j₃; J₁₂, J₂₃) = <(j₁,(j₂j₃)J₂₃)J | ((j₁j₂)J₁₂,j₃)J> / sqrt((2J₁₂+1)*(2J₁₃+1))`
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as a type `T` floating point number. By default, `T = Float64`.
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# check triangle condition
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function δ(j₁, j₂, j₃)
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j₃ <= j₁ + j₂ || return false
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j₁ <= j₂ + j₃ || return false
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j₂ <= j₃ + j₁ || return false
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isinteger(j₁ + j₂ - j₃) || return false
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isinteger(j₁ - j₂ + j₃) || return false
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isinteger(j₁ - j₂ - j₃) || return false
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return true
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Returns `zero(T)` if any of triangle conditions `δ(j₁, j₂, J₁₂)`, `δ(j₂, j₃, J₂₃)`,
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`δ(j₁, J₂₃, J)`, `δ(J₁₂, j₃, J)` are not satisfied, but throws a `DomainError` if
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the `jᵢ`s and `J`s are not integer or halfinteger.
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"""
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racahW(j₁, j₂, J, j₃, J₁₂, J₂₃) = racahW(Float64, j₁, j₂, J, j₃, J₁₂, J₂₃)
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function racahW(T::Type{<:AbstractFloat}, j₁, j₂, J, j₃, J₁₂, J₂₃)
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s = wigner6j(T, j₁, j₂, J₁₂, j₃, J, J₂₃)
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return isodd(convert(Int, j₁ + j₂ + j₃ + J)) ? -s : s
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end
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# triangle coefficient
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Δ(j₁, j₂, j₃) = Δ(Float64, j₁, j₂, j₃)
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function Δ(T::Type{<:AbstractFloat}, j₁, j₂, j₃)
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if !δ(j₁, j₂, j₃)
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throw(DomainError())
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end
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v = Δ²(j₁, j₂, j₃)
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return sqrt(convert(T, convert(BigInt, v.num) // convert(BigInt, v.den)))
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end
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# squared triangle coefficient
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function Δ²(j₁, j₂, j₃)
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@ -136,7 +221,7 @@ function Δ²(j₁, j₂, j₃)
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n3 = primefactorial( convert(UInt, - j₁ + j₂ + j₃) )
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d = primefactorial( convert(UInt, j₁ + j₂ + j₃ + 1) )
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# result
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return (n1*n2*n3)//d
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return (n1*n2*n3), d
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end
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# reorder parameters determining the 3j symbol to canonical order:
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@ -94,3 +94,36 @@ for k = 1:10
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@test (X*wigner6j(BigFloat,j+1,j2,j3,l1,l2,l3) + Z*wigner6j(BigFloat,j-1,j2,j3,l1,l2,l3))≈(-Y*wigner6j(BigFloat,j,j2,j3,l1,l2,l3)) atol=tol
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end
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end
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# test recoupling, i.e. relation between 3j and 6j symbols (but use Clebsch-Gordan and RacahW)
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#---------------------------------------------------------------------------------------------
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smallerjlist = 0:1//2:5
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for j1 in smallerjlist, j2 in smallerjlist, j3 in smallerjlist
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@show (j1,j2,j3)
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m1range = -j1:j1
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m2range = -j2:j2
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m3range = -j3:j3
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V1 = Array{Float64}(length(m1range),length(m2range),length(m3range))
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V2 = Array{Float64}(length(m1range),length(m2range),length(m3range))
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for J in max(abs(j1-j2-j3),abs(j1-j2+j3),abs(j1+j2-j3)):(j1+j2+j3)
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J12range = max(abs(j1-j2),abs(J-j3)):min((j1+j2),(J+j3))
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J23range = max(abs(j2-j3),abs(j1-J)):min((j2+j3),(j1+J))
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for J12 in J12range, J23 in J23range
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M = J # only test for J, should be independent
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fill!(V1,0)
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fill!(V2,0)
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for (k1,m1) in enumerate(m1range)
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for (k2,m2) in enumerate(m2range)
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abs(m1+m2)<=J12 || continue
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for (k3,m3) in enumerate(m3range)
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abs(m2+m3)<=J23 || continue
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||||
m1+m2+m3==M || continue
|
||||
V1[k1,k2,k3] = clebschgordan(j1,m1,j2,m2,J12)*clebschgordan(J12,m1+m2,j3,m3,J)
|
||||
V2[k1,k2,k3] = clebschgordan(j2,m2,j3,m3,J23)*clebschgordan(j1,m1,J23,m2+m3,J)
|
||||
end
|
||||
end
|
||||
end
|
||||
@test racahW(j1,j2,J,j3,J12,J23) ≈ vecdot(V2,V1)/sqrt((2*J12+1)*(2*J23+1)) atol=eps(Float64)
|
||||
end
|
||||
end
|
||||
end
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue