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Re: addition of three angular momenta

  • To: mathgroup at
  • Subject: [mg47788] Re: addition of three angular momenta
  • From: Paul Abbott <paul at>
  • Date: Tue, 27 Apr 2004 04:46:29 -0400 (EDT)
  • Organization: The University of Western Australia
  • References: <c6d8fi$jhh$>
  • Sender: owner-wri-mathgroup at

In article <c6d8fi$jhh$1 at>,
 "Francesco Siano" <fsiano at> wrote:

> I am trying to use ClebschGordan[...] to generate the table of  
> coefficients for the addition of three or more angular momenta.
> For the addition of two angular momenta J=J1+J2 I can do
> Table[ClebschGordan[{J1, m1}, {J2, m2}, {J, m}], {J, J1 + J2, Abs[J1 -  
> J2], -1}, {m, J, -J, -1}, {m1, J1, -J1, -1}, {m2, J2, -J2, -1}]
> For more than two angular momenta, J=J1+J2+J3 I should first add J1 and  
> J2, and then, for any Jtemp=J1+J2,J1+J2-1,...,Abs[J1-J2] add Jtemp and J3  
> and so on.
> Any help would be greatly appreciated.

The 6-j symbols (built-in as SixJSymbol) give the couplings of three 
quantum mechanical angular momentum states. 

Since both ThreeJSymbol (effectively equivalent to ClebschGordan) and 
SixJSymbol can be evaluated for symbolic arguments, why do you want to 
generate a table of values? For example,

  Flatten[Table[{{{l, l + k, n}, {m, -p - m, p}},
   FullSimplify[ThreeJSymbol[{l, m}, {l + k, -p - m}, {n, p}]]}, 
    {n, 0, 2, 1/2}, {k, Mod[n, 1], n}, {p, n, Mod[n, 1], -1}], 2]

generates table 3 from Brink and Satchler. So why generate a table of 
values if a closed-form exists for a particular class of coefficient?

Mathematica code for 9-j symbols -- coupling 4 angular momenta -- also 
exists but there is, AFAIK, no closed form (except for a triple sum) for 
the 9-j symbols.


Paul Abbott                                   Phone: +61 8 9380 2734
School of Physics, M013                         Fax: +61 8 9380 1014
The University of Western Australia      (CRICOS Provider No 00126G)         
35 Stirling Highway
Crawley WA 6009                      mailto:paul at 

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