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Re: Mathematica


A local researcher wants to know how to simplify expressions
involving the structures in his message below.  Anyone have
any experience with this?

Steve C. -moderator]

OK, here is an input file.  The rules at the beginning helped somewhat, 
but I think that if you evaluate the expressions that follow the rules, 
you'll see what I mean.  The y functions are spherical harmonics, yet to 
be defined (i.e. arbitrary functions for now).  Eventually I'll need more 
complicated expressions, and the outlook didn't seem terribly hopeful.


srule =  Sqrt[x_] Sqrt[y_] -> Sqrt[x y]
sdrule = 1/(Sqrt[x_] Sqrt[y_]) -> 1/(Sqrt[x y])
scrule = Sqrt[x_ y_]/Sqrt[x_] -> Sqrt[y]
srules = {srule,sdrule,scrule}
trule = Sqrt[x_ y_] -> Sqrt[Expand[x y]]
minusrule1 = (-1)^(2^x_) -> -1

t1[s_, sp_, l_] := ThreeJSymbol[{l,m-sp},{1/2,sp},{j,-m}] (2 j +1) (
-1)^(m-sp) yl2 ( 2 s (m-s) yl1 ThreeJSymbol[{l,m-s},{1/2,s},
{j,-m}] +
Sqrt[ l (l+1) - (m-s-1) (m-s) ] ThreeJSymbol[{l,m-s-1},{1/2,s+1},{j,-m}] yl1 + 
Sqrt[l (l+1) - (m-s+1) (m-s) ] ThreeJSymbol[{l,m-s+1},{1/2,s-1},
{j,-m}] yl1 )

t2[s_, sp_, l_] := ThreeJSymbol[{l,m-s},{1/2,s},{j,-m}] (2 j +1) yl1 ( 
(-1)^(m-sp) 2 sp (sp-m) yl2 ThreeJSymbol[{l,m-sp},{1/2,sp},{j,-m}] + 
(-1)^(m-sp-1) Sqrt[l (l+1) - (m-sp+1) (m-sp) ] ThreeJSymbol[{l,m-sp+1},{1/2,
sp-1},{j,-m}] yl2 + (-1)^(m-sp+1) Sqrt[l (l+1) - (m-sp-1)
(m-sp) ] ThreeJSymbol[{l,m-sp-1},{1/2,sp+1},{j,-m}] yl2 )

L[mu_, l_, m_] := Sqrt[ l (l+1) ] ClebschGordan[{l,m},{1,mu},{l,m+mu}]

sigl[mu_, s_] := (-1)^(1/2-s) Sqrt[6] ThreeJSymbol[{1/2,-s},{1,mu},{1/2,s-mu}]

f15[s_, sp_, l_] := Sum[(-1)^mu Simplify[ClebschGordan[{l,m-sp},{1/2,sp},
{j,m}]] Simplify[ClebschGordan[{l,m-s-mu},{1/2,s+mu},{j,m}]] Simplify[
(-1)^(m-sp)*L[mu,l,m-s-mu] sigl[-mu,s]] yl1 yl2 , {mu,-1,1,1}]

f16[s_, sp_, l_] := -Sum[(-1)^mu Simplify[ClebschGordan[{l,m-s},{1/2,s},
{j,m}]] Simplify[ClebschGordan[{l,m-sp+mu},{1/2,sp-mu},{j,m}]] Simplify[
(-1)^(m-sp+mu)*L[mu,l,m-s] sigl[-mu,sp-mu]] y1[l, m-s+mu] y2[l, sp-m-mu] ,

f25[s_, sp_, l_] := -Sum[(-1)^mu Simplify[ClebschGordan[{l,m-sp},{1/2,sp},
{j,m}]] Simplify[ClebschGordan[{l,m-s-mu},{1/2,s+mu},{j,m}]] Simplify[
(-1)^(m-sp)*L[mu,l,sp-m] sigl[-mu,s]] y1[l,m-s-mu] y2[l,sp-m+mu] , {mu,-1,1,1}]

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