# Ty

The y component of $\vec{T}$ is defined as: $$T_y = S_y - 3 y\left(x S_x + y S_y + z S_z\right)/r^2.$$ The equivalent operator in Quanty is created by:

Example.Quanty
OppTy = NewOperator("Ty", NF, IndexUp, IndexDn)

The operator can alternatively be created with the following function:

Example.Quanty
function Ty(indexup, indexdn, NF)
if #indexup ~= #indexdn then
error("Length of index up must be equal to length of index dn and equal to 2l+1 in Ty")
end
local l=(#indexup-1)/2
if not IntegerQ(l) then
error("Length of index must be equal to 2l+1 in Ty")
end
local function Sx(m1,m2)
return NewOperator("Number",NF,{indexup[m1+l+1],indexdn[m1+l+1]},
{indexdn[m2+l+1],indexup[m2+l+1]},{1/2,1/2})
end
local function Sy(m1,m2)
return NewOperator("Number",NF,{indexup[m1+l+1],indexdn[m1+l+1]},
{indexdn[m2+l+1],indexup[m2+l+1]},{-I/2,I/2})
end
local function Sz(m1,m2)
return NewOperator("Number",NF,{indexup[m1+l+1],indexdn[m1+l+1]},
{indexup[m2+l+1],indexdn[m2+l+1]},{1/2,-1/2})
end
local opp = 0 * NewOperator("Number",NF,0,0)
for m1 = -l,l do
for m2 = -l,l do
opp = opp -2 * (-sqrt(3/8)*SlaterCoefficientC({l,m1},{2,-2},{l,m2})
-sqrt(3/8)*SlaterCoefficientC({l,m1},{2, 2},{l,m2})
-sqrt(2/8)*SlaterCoefficientC({l,m1},{2, 0},{l,m2})
)*Sy(m1,m2)
-I * ( sqrt(3/2)*SlaterCoefficientC({l,m1},{2,-2},{l,m2})
-sqrt(3/2)*SlaterCoefficientC({l,m1},{2, 2},{l,m2})
)*Sx(m1,m2)
-I * ( sqrt(3/2)*SlaterCoefficientC({l,m1},{2,-1},{l,m2})
+sqrt(3/2)*SlaterCoefficientC({l,m1},{2, 1},{l,m2})
)*Sz(m1,m2)
end
end
opp.Chop()
opp.Name="Ty"
return opp
end