# How To Define Indexed Vectors

I have to define two 2-D vectors to be multiplied by a matrix and multiplied by several Clebsch-Gordan coefficients. Below are my definitions and my attempt at computing the sum.

sigma = {{1, 1 - I}, {1 + I, -1}}

Out[1]= {{1, 1 - I}, {1 + I, -1}}

Subscript[chi, -1/2] = {0, 1}

Out[28]= {0, 1}

Subscript[chi, 1/2] = {1, 0}

Out[21]= {1, 0}

Sum[ClebschGordan[{S2, MS2}, {L2, ML2}, {J2, MJ2}] ClebschGordan[{S2,
MS2}, {1/2, m2}, {1/2,
m2bar}] *((Subscript[chi, m2]).sigma.Subscript[chi, m1])*
ClebschGordan[{S1, MS1}, {L1, ML1}, {J1, MJ1}] ClebschGordan[{S1,
MS1}, {1/2, m1}, {1/2, m1bar}], {m1, -1/2, 1/2, 1}, {m2, -1/2,
1/2, 1}, {m1bar, -1/2, 1/2, 1}, {m2bar, -1/2, 1/2, 1}, {MS1, -S1,
S1, 1}, {MS2, -S2, S2, 1}, {ML1, -L1, L1, 1}, {ML2, -L2, L2, 1}]


However, this way of defining the indices of the chi vectors does not seem to work. It seems that Mathematica does not know to assign the values of -1/2 or 1/2 to the indices m1 and m2. It gives me

Sum[2 (-1)^(-1 - L1 - L2 + m1bar + m2bar + MJ1 + MJ2 + 2 S1 + 2 S2)
Sqrt[1 + 2 J1] Sqrt[1 + 2 J2]
Subscript[chi, m2].{{1, 1 - I}, {1 + I, -1}}.Subscript[chi, m1]
ThreeJSymbol[{S1, MS1}, {1/2, m1}, {1/
2, -m1bar}] ThreeJSymbol[{S1, MS1}, {L1,
ML1}, {J1, -MJ1}] ThreeJSymbol[{S2, MS2}, {1/2, m2}, {1/
2, -m2bar}] ThreeJSymbol[{S2, MS2}, {L2,
ML2}, {J2, -MJ2}], {m1, -(1/2), 1/2, 1}, {m2, -(1/2), 1/2,
1}, {m1bar, -(1/2), 1/2, 1}, {m2bar, -(1/2), 1/2, 1}, {MS1, -S1, S1,
1}, {MS2, -S2, S2, 1}, {ML1, -L1, L1, 1}, {ML2, -L2, L2, 1}]


That is, id merely re-writes the vectors times the matrix. But if I do it manually for a specific case it works.

(Subscript[chi, 1/2]).sigma.Subscript[chi, -1/2]


produces

1 - I


as expected. And even computing a table works.

Table[Subscript[chi, a].sigma.Subscript[chi, b], {a, -1/2, 1/2,
1}, {b, -1/2, 1/2, 1}]


produces

{{-1, 1 + I}, {1 - I, 1}}


Could anybody tell me what I am doing wrong and how to properly define the indices of the subscripts for the chi vectors?

• YOu haven't specified what L1, L2, S1, S2, J1, J2, and MJ1 and MJ2 are, so the sum cannot be done by Mathematica. Also, can you be more specific about the physical problem you are solving? There are L's and J's and S's, indicating that you're doing some angular momentum coupling, but you have only a two-element state, so that's strange. Can you clarify? Commented Aug 27, 2021 at 23:22
• When you specify the values above, the sum can be done. Of course, most of the time the sum is zero, but I'm sure you have some notion for what values of the L1, L2, etc. the sum should be non-zero Commented Aug 27, 2021 at 23:24
• (Point being, I don't think that the indexing of the vectors is the problem here.) Commented Aug 27, 2021 at 23:30
• Thank you for your reply. I have three-body problem and the CG coefficients that appear are the ones that I get when I couple the three angular momenta. S3, L3 etc do not appear because one of my angular integrals sets those indices equal to each other. Commented Aug 27, 2021 at 23:35
• And you are right, I do know that the S1=S2=1. I should have put that is there explicitely. Commented Aug 27, 2021 at 23:36