Spherical harmonics

See examples in Ylm.mw

References: Handbook on special functions

Wikipedia: Spherical harmonics and their tables, Wigner D-matrix, 3-j symbol

Acknowledgements: Dennis Isbister for Wigner3j/6j code

Notations

• phi azimutal angle
• m::integer azimutal number; in matrices, complex and trigonometric basises are ordered as m=0,1,-1,2,-2,... and 1,cos(p),sin(p),cos(2p),... respectively
• complex::boolean:=complexTrigBasis if true then the exp(I*m*phi) basis is used, otherwise the real trigonometric basis is used: cos(m*phi) for nonnegative m and sin(m*phi) for negative m. The complexTrigBasis can be changed by Setup command.
• theta polar angle
• Theta either theta or list of hyperspherical non-azimutal angles beginning with theta
• l::nonnegint order of a spherical harmonic
• L::{nonnegint,list(nonnegint)} either l or list of indexes of Ylm beginning with l
• convention::string:=conventionY one of [standard,complex,real]; the conventionY can be changed by Setup command. The "standard" convention is used in quantum mechanics, "complex" convention is used in mathematics.
• normalized::boolean if true then the function is normalized

Procedures

• Phim(m,phi,{complex},$) trigonometric basis
• NPhim2(m,{complex},$) squared norm of Phim
• Pnma(n::nonnegint,m::nonnegint, opt a:=0,theta) hyperspherical quasipolynomial as defined in the referenced handbook; it differs from associated Legendre polynomial by (-1)^m. Note that a>-1/2
• NPnma2(n,m,a) squared norm of Pnma (with proper weight)
• Ylm(L,m,Theta,phi,/convention,{normalized:=normalizedY},$) hyperspherical harmonic; normalizedY can be changed by Setup command
• Yxyz(l,m::{integer,string},x,y,z,$) normalized harmonic polynomial. If m is integer then trigonometric basis is used, otherwise it can be one of [trig,cubic,A1,A2,E,F1,F2]. In the case of [trig,cubic] the list of all basis functions is returned.
• YxyzM(l,m::{integer,string},/convention,$)::{Vector,Matrix} expansion of harmonic polynomials in spherical harmonics, i.e. <Ylm|Yxyz>+
• YxyzR(l)::list(string) list of irreducible representations of the cubic group
• YxyzL(l)::list(string) list of labels of functions of the cubic group
• WignerD(l,m1,m2,alpha,beta,gamma) Wigner's rotation function (gives representation of the rotation group and rotates spherical harmonics)
• Wignerd(l,m1,m2,cos(beta/2),sin(beta/2)) auxiliary function
• Wigner3j(j1,j2,j3,m1,m2,m3) Wigner's 3j symbols
• Wigner6j(j1,j2,j3,l1,l2,l3) Wigner's 6j symbols
• MatElemYlm(l1,m1,l,m,l2,m2) matrix elements of spherical harmonics on complex spherical harmonics
• MatElemY(l1,m1,f,l2,m2,/convention,$) matrix elements of any function on complex or real spherical harmonics