Understanding the output of a calculation

Question

I have a simple question about the output of a calculation. I have written a function which is a sum of 3j-symbols and Clebsch-Gordan coefficients in which I sum over various indices.

Jsymbol[l1_, l2_, S_, L1_, L2_, J1_, J2_] := 
 3*Sum[(-1)^(ml1 + b + mL)*(2 L + 1)*Sqrt[(2 l1 + 1)*(2 l2 + 1)]*
    ThreeJSymbol[{l1, 0}, {l2, 0}, {L, 0}]*
    ThreeJSymbol[{l1, ml1}, {l2, ml2}, {L, mL}]*
    ThreeJSymbol[{1, 0}, {1, 0}, {L, 0}]*
    ThreeJSymbol[{1, b}, {1, c}, {L, mL}]*
    ClebschGordan[{1, mg1}, {1, i}, {1, b}]*
    ClebschGordan[{1, mg2}, {1, i}, {1, c}]*
    ClebschGordan[{l1, ml1}, {1, mg1}, {L1, ML1}]*
    ClebschGordan[{l2, ml2}, {1, mg2}, {L2, ML2}]*
    ClebschGordan[{S, MS}, {L1, ML1}, {J1, MJ1}]*
    ClebschGordan[{S, MS}, {L2, ML2}, {J2, MJ2}], {ml1, -l1, l1, 
    1}, {ml2, -l2, l2, 1}, {mg1, -1, 1, 1}, {mg2, -1, 1, 1}, {mL, -L, 
    L, 1}, {i, -1, 1, 1}, {b, -1, 1, 1}, {c, -1, 1, 1}, {ML1, -L1, L1,
     1}, {ML2, -L2, L2, 1}, {MS, -S, S, 1}, {MJ1, -J1, J1, 
    1}, {MJ2, -J2, J2, 1}, {L, 0, 2, 1}]

When I plug in certain numbers for the indices I get

Jsymbol[1, 1, 0, 0, 0, 0, 0]
3 (-(2/15) + 2[{-1/30, L>=1, 0 True}])

But how do I interpret this? I tried to perform the sums for the three values of L that I am interested in, $L=0,1,2$, separately, and I get $-1/3,0,-7/15$, respectively, which I can't make agree with the previous answer. What am I misunderstanding?

Thank you in advance.

Answer 1

Clear["Global`*"] (* clear all prior definitions *)

Jsymbol[l1_, l2_, S_, L1_, L2_, J1_, J2_] := 
 3*Sum[(-1)^(ml1 + b + mL)*(2 L + 1)*Sqrt[(2 l1 + 1)*(2 l2 + 1)]*
    ThreeJSymbol[{l1, 0}, {l2, 0}, {L, 0}]*
    ThreeJSymbol[{l1, ml1}, {l2, ml2}, {L, mL}]*
    ThreeJSymbol[{1, 0}, {1, 0}, {L, 0}]*
    ThreeJSymbol[{1, b}, {1, c}, {L, mL}]*
    ClebschGordan[{1, mg1}, {1, i}, {1, b}]*
    ClebschGordan[{1, mg2}, {1, i}, {1, c}]*
    ClebschGordan[{l1, ml1}, {1, mg1}, {L1, ML1}]*
    ClebschGordan[{l2, ml2}, {1, mg2}, {L2, ML2}]*
    ClebschGordan[{S, MS}, {L1, ML1}, {J1, MJ1}]*
    ClebschGordan[{S, MS}, {L2, ML2}, {J2, MJ2}], {ml1, -l1, l1, 
    1}, {ml2, -l2, l2, 1}, {mg1, -1, 1, 1}, {mg2, -1, 1, 1}, {mL, -L, L, 
    1}, {i, -1, 1, 1}, {b, -1, 1, 1}, {c, -1, 1, 1}, {ML1, -L1, L1, 
    1}, {ML2, -L2, L2, 1}, {MS, -S, S, 1}, {MJ1, -J1, J1, 1}, {MJ2, -J2, J2, 
    1}, {L, 0, 2, 1}]

result = Jsymbol[1, 1, 0, 0, 0, 0, 0]

result // InputForm

(* 3*(-2/15 + 2*Piecewise[{{-1/30, L >= 1}}, 0]) *)

The last term is a Piecewise function. The value of the Piecewise is -1/30 if L >= 1 or zero (default) otherwise. To simplify the expression use either

result // Simplify

or

result // PiecewiseExpand

The results for different values of L are

result /. {{L -> 0}, {L -> 1}, {L -> 2}}

(* {-(2/5), -(3/5), -(3/5)} *)

Assigning the values to L prior to calculating Jsymbol produces error messages and presumably erroneous results.

 Table[(L = val;
  Jsymbol[1, 1, 0, 0, 0, 0, 0]), {val, 0, 2}]