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
____________________________________________________________________

```

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