Using the rotation operator to solve for eigenstates upon a general basis

[infamous80518]

Homework Statement

I need to express the rotation operator as follows

R(uj) = cos(u/2) + 2i(\hbar) S_y sin(u/2)

given the fact that

R(uj)= e^(iuS_y/(\hbar))

using |+-z> as a basis,
expanding R in a taylor series
express S_y^2 as a matrix

Homework Equations

I know
e^(ix)=cos(x)+isin(x)

using this alone I can show this equivalence

The Attempt at a Solution

e^(ix)=cos(x)+isin(x)

which implies

R(uj)= e^(iuS_y/(\hbar)) = cos(uS_y/(\hbar)+isin(uS_y/(\hbar)

S_y = (\hbar)/2

Therefore

R(uj)= e^(iuS_y/(\hbar)) = cos(u/2)+iS_y*sin(u/2)



... What's this about finding the matrix representation of S_y^2 ?

 

[member 553083]

Ok, S_y^2 = (\hbar)^2/4The matrix representation of S_y^2 is then S_y^2 = (\hbar)^2/4 * I where I is the identity matrix.