Re: addition of three angular momenta

• To: mathgroup at smc.vnet.net
• Subject: [mg47749] Re: [mg47735] addition of three angular momenta
• From: Hugh Walker <hwalker at gvtc.com>
• Date: Sun, 25 Apr 2004 05:13:21 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

```On Saturday, April 24, 2004, at 03:15 AM, Francesco Siano 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.
> -Francesco

Francesco, here is a notebook containing the basic functions for spin
coupling. I have the same functions in package format which I can email
upon request. Regards...
==========

(*
SpinCoupling[X_,Y_][j1_,j2_][J_,M_] uses the coefficients defined by
ClebschGordan[{j1,m1},{j2,m2},{J,M}] functions to couple particles  X
and Y with spins j1 and j2, respectively, to give the coupled states
j1+j2, j1+j2-1, j1+j2-2, ... Abs[j1-j2] according to quantum laws.

SpinCoupling[X_,Y_,Z_][j1_,j2_][j12_,j3_][J_,M_] couples j1 and j2 to
result in j12, which is then coupled to j3.
*)

(* Turn off ClebschGordan warnings *)
Off[ClebschGordan::phy]

(* Spin Coupling functions *)

SpinCoupling[X_, Y_][j1_, j2_][J_, M_] :=
Sum[X[j1, m1]*Y[j2, m2]*ClebschGordan[{j1, m1}, {j2, m2}, {J, M}],
{m2, j2, -j2, -1}, {m1, j1, -j1, -1}]

SpinCoupling[X_, Y_, Z_][j1_, j2_][j12_, j3_][J_, M_] :=
ExpandAll[SpinCoupling[SpinCoupling[X, Y][j1, j2], Z][j12, j3][J, M]]

==========
Hugh Walker
Gnarly Oaks

```

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