WignerSymbols.Internal

Functions in this module are subject to change without notice.

Documentation

newtype SignedSqrtRational

Represents a mathematical expression of the form:

s √(n / d)

where

• s is a sign (+, -, or 0),
• n is a nonnegative numerator, and
• d is a positive denominator.

Constructors

SignedSqrtRational Rational

Instances

Eq SignedSqrtRational

Ord SignedSqrtRational

Read SignedSqrtRational

Show SignedSqrtRational

ssr_new

Arguments

:: (Integer, Rational)

(c, r)

-> SignedSqrtRational

Construct a SignedSqrtRational equal to c √r.

ssr_split :: SignedSqrtRational -> (Integer, Rational)

Deconstruct a SignedSqrtRational.

ssr_signum :: SignedSqrtRational -> Integer

Extract the sign of a SignedSqrtRational.

ssr_numerator :: SignedSqrtRational -> Integer

Extract the numerator of a SignedSqrtRational.

ssr_denominator :: SignedSqrtRational -> Integer

Extract the denominator of a SignedSqrtRational.

ssr_approx :: Floating b => SignedSqrtRational -> b

Approximate a SignedSqrtRational as a floating-point number.

clebschGordan

Arguments

:: (Int, Int, Int, Int, Int, Int)

(tj1, tm1, tj2, tm2, tj12, tm12)

-> Double

Calculate a Clebsch-Gordan coefficient:

⟨j1 j2 m1 m2|j1 j2 j12 m12⟩

clebschGordanSq

Arguments

:: (Int, Int, Int, Int, Int, Int)

(tj1, tm1, tj2, tm2, tj12, tm12)

-> SignedSqrtRational

Similar to clebschGordan but exact.

clebschGordanSqSlow :: (Int, Int, Int, Int, Int, Int) -> SignedSqrtRational

Used only as a reference, it implements the formula from Wikipedia, which comes from equation (2.41) on page 172 of Quantum Mechanics: Foundations and Applications (1993) by A. Bohm and M. Loewe (ISBN 0-387-95330-2).

(Note: clebschGordan is not implemented using this function.)

wigner3j

Arguments

:: (Int, Int, Int, Int, Int, Int)

(tj1, tm1, tj2, tm2, tj3, tm3)

-> Double

Calculate a Wigner 3-j symbol:

⎛j1 j2 j3⎞
⎝m1 m2 m3⎠

wigner3jSq

Arguments

:: (Int, Int, Int, Int, Int, Int)

(tj1, tm1, tj2, tm2, tj3, tm3)

-> SignedSqrtRational

Similar to wigner3j but exact.

wigner3jSqRawC

Arguments

:: (Int, Int, Int, Int, Int, Int)

(tj1, tm1, tj2, tm2, tj3, tm3)

-> SignedSqrtRational

Calculate the Wigner 3-j symbol times (−1) ^ (j1 − j2 − m3).

wigner3jSqRaw

Arguments

:: (Int, Int)

(j1 + m1, j2 + m2)

-> (Int, Int, Int, Int, Int, Int)

(tj1, tm1, tj2, tm2, tj3, tm3)

-> SignedSqrtRational

Calculate the Wigner 3-j symbol times (−1) ^ (j1 − j2 − m3). The selection rules are not checked.

wigner6j

Arguments

:: (Int, Int, Int, Int, Int, Int)

(tj11, tj12, tj13, tj21, tj22, tj23)

-> Double

Calculate a Wigner 6-j symbol:

⎧j11 j12 j13⎫
⎩j21 j22 j23⎭

wigner6jSq

Arguments

:: (Int, Int, Int, Int, Int, Int)

(tj11, tj12, tj13, tj21, tj22, tj23)

-> SignedSqrtRational

Similar to wigner6j but exact.

wigner6jSqRaw

Arguments

:: (Int, Int, Int, Int, Int, Int)

(tj11, tj12, tj13, tj21, tj22, tj23)

-> SignedSqrtRational

Calculate the Wigner 6-j symbol. The selection rules are not checked.

wigner9j

Arguments

:: (Int, Int, Int, Int, Int, Int, Int, Int, Int)

(tj11, tj12, tj13, tj21, tj22, tj23, tj31, tj32, tj33)

-> Double

Calculate a Wigner 9-j symbol:

⎧j11 j12 j13⎫
⎨j21 j22 j23⎬
⎩j31 j32 j33⎭

wigner9jSq

Arguments

:: (Int, Int, Int, Int, Int, Int, Int, Int, Int)

(tj11, tj12, tj13, tj21, tj22, tj23, tj31, tj32, tj33)

-> SignedSqrtRational

Similar to wigner9j but exact.

wigner9jSqRaw

Arguments

:: (Int, Int, Int, Int, Int, Int, Int, Int, Int)

(tj11, tj12, tj13, tj21, tj22, tj23, tj31, tj32, tj33)

-> SignedSqrtRational

Calculate the Wigner 9-j symbol. The selection rules are not checked.

factorial

Arguments

:: Int

n

-> Integer

Calculate the factorial n!.

fallingFactorial

Arguments

:: Int

n

-> Int

k

-> Integer

Calculate the falling factorial, i.e. the product of the integers [n, k).

binomial

Arguments

:: Int

n

-> Int

k

-> Integer

Calculate the binomial coefficient C(n, k).

minusOnePow

Arguments

:: Int

n

-> Int

Calculate (−1) ^ n.

triangleCondition

Arguments

:: (Int, Int, Int)

(tj1, tj2, tj3).

-> Bool

Check |j1 − j2| ≤ j3 ≤ j1 + j2 and j1 + j2 + j3 ∈ ℤ.

triangularFactor

Arguments

:: (Int, Int, Int)

(tj1, tj2, tj3)

-> Rational

Calculate the triangular factor:

Δ(j1, j2, j3) = (−j1 + j2 _ j3)! (j1 − j2 + j3)! (j1 + j2 − j3)!
              / (j1 + j2 + j3 + 1)!

triangularFactorRaw

Arguments

:: Int

jd

-> (Int, Int, Int)

(ja, jb, jc)

-> Rational

Calculate ja! jb! jc! / jd!.

tetrahedralSum

Arguments

:: (Int, Int, Int, Int, Int, Int)

(tj11, tj12, tj13, tj21, tj22, tj23)

-> Integer

Calculate the symbol in the paper by L. Wei that is enclosed in square brackets:

⎡j11 j12 j13⎤
⎣j21 j22 j23⎦

This is essentially a Wigner 6-j symbol without the triangular factors, although the ordering of the arguments is a bit funky here.

getTriangularTjs

Arguments

:: Int

tjmax

-> (Int, Int)

(tj1, tj2)

-> [Int]

Get all angular momenta that satisfy the triangle condition with the given pair of angular momenta, up to a maximum of jmax.

getBitriangularTjs

Arguments

:: Int

tjmax

-> ((Int, Int), (Int, Int))

((tj11, tj12), (tj21, tj22))

-> [Int]

Get all angular momenta that satisfy the triangle condition with each of the given two pairs of angular momenta, up to a maximum of jmax.

getTms

Arguments

:: Int

tj

-> [Int]

Get all projection quantum numbers that lie within the multiplet of j.

get3tjms

Arguments

:: Int

tjmax

-> [(Int, Int, Int, Int, Int, Int)]

Get all possible arguments of the Wigner 3-j symbol that satisfy the selection rules up to a maximum of jmax.

get6tjs :: Int -> [(Int, Int, Int, Int, Int, Int)]

Get all possible arguments of the Wigner 6-j symbol that satisfy the selection rules up to a maximum of jmax.

get9tjs :: Int -> [(Int, Int, Int, Int, Int, Int, Int, Int, Int)]

Get all possible arguments of the Wigner 9-j symbol that satisfy the selection rules up to a maximum of jmax.

tuple6ToList :: (a, a, a, a, a, a) -> [a]

Convert a 6-tuple into a list.

tuple9ToList :: (a, a, a, a, a, a, a, a, a) -> [a]

Convert a 9-tuple into a list.