Re: Wigner matrices Dpq(a) implementation
- To: mathgroup at smc.vnet.net
- Subject: [mg15943] Re: Wigner matrices Dpq(a) implementation
- From: Paul Abbott <paul at physics.uwa.edu.au>
- Date: Wed, 17 Feb 1999 23:34:07 -0500
- Organization: University of Western Australia
- References: <7a2ae3$1r1@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
Roberto Pratolongo wrote: > I'm studying a branch of molecular dynamics, so I need to be familiar > with rotating reference systems, and so son. Spherical harmonics are > standard in Math, I know,and I've found the additional package about 3D > rotation, but I also need to work with so-called Wigner matrices > Dpq(a), that are a sort of spherical tensors of rank a=1,2,... > I'm novice with the argument, but they have almost nothing to do with > Wigner's 3J Symbol ,page 730 of Math3 manual, I suspect. There are very important inter-relationships. > Has anybody a reference for a suitable implementation of them. I personally like A R Edmonds, "Angular Momentum in Quantum Mechanics", Princeton University Press, 1974. Implementing them is straightforward using the definition (4.1.23) of Edmonds: In[1]:= d[m_, n_][j_][x_] = Sqrt[((j + m)! (j - m)!)/ ((j + n)! (j - n)!)] Cos[x/2]^(m + n) Sin[x/2]^(m - n)* JacobiP[j - m, m - n, m + n, Cos[x]]; For example, In[2]:= FullSimplify[d[1, 1][4][x]] Out[2]= 1 -- (9 Cos[x] + 2 Cos[2 x] + 7 (Cos[3 x] + 2 Cos[4 x])) 32 Cheers, Paul ____________________________________________________________________ Paul Abbott Phone: +61-8-9380-2734 Department of Physics Fax: +61-8-9380-1014 The University of Western Australia Nedlands WA 6907 mailto:paul at physics.uwa.edu.au AUSTRALIA http://www.physics.uwa.edu.au/~paul God IS a weakly left-handed dice player ____________________________________________________________________