Wigner-d matrices for higher (than 1/2) spins I’ve been reading 
¨Halzen, F., and A. D. Martin. Quarks and Leptons. New York: Wiley Text Books, January 1984. ISBN: 9780471887416¨,
and I’d like some clarification of a concept, please: I’m looking at Problem 2.6, and the question asks us to show that the rotation matrices are represented by certain values, depending on the j-value used, for the equation: $d^{j}_{m’m}(\theta)=\langle{jm’}\lvert{e^{-i{\theta}{J_2}}}\lvert{jm}\rangle$. $J_2$ is represented as a rotation generator, $\theta$ is obviously the angle of rotation, $j$ represents the Eigenstate, and $m$-values are the different available states for each $j$-value. The question said that if $j={1 \over 2}$, we’d have:
$$j = {1 \over 2} 
\begin{cases}
d_{++} = d_{--} = \cos({1 \over 2}\theta) \\
d_{+-} = -d_{-+} = \sin({1 \over 2}\theta)
\end{cases}$$
, where $\pm$ designates $m = \pm {1 \over 2}$- values. 
For $j=1$, I’m supposed to be able to find that: 
$$j = 1 
\begin{cases}
d_{01} = -d_{10} = -d_{0-1} = d_{-10} = \sqrt{1 \over 2} sin(\theta) \\
d_{11} = d_{-1-1} = {1 \over 2} (1 + \cos\theta) \\
d_{-11} = d_{1-1} = {1 \over 2} (1 - \cos\theta) \\ 
d_{00} = \cos\theta
\end{cases}$$
Now, I’ve finished the first part, where we need to find the values for $j= {1 \over 2}$; I basically just used Euler’s Rule ($e^{ix} = \cos (x) + i\sin (x)$) to break down the exponential, and then considered odd and even-integer solutions. 
I have several questions about the second part of the question:


*

*How do I, in general, manipulate a higher-spin system, using a given operator? I feel like this has something to do with SU(3) or expanding SU(2) to SU(3), or perhaps I’m mistaken? 

*I looked at the solution to the first part in the book, and the authors used the Pauli Spin Matrices in order to solve it. Aren’t I doing the same thing with my own solution (as I described above), just without the explicit use of the Pauli Spin Matrices?
 A: *

*Your feeling looks very misguided. Whatever you do, stay away from SU(3) for rotations. The rotation group and its Lie algebra are always linked to SO(3) ~ SU(2), to avoid formal forays into double covers and half angles. Read up on the spin matrices for any representation of the very same group (any spin).

*There are, in fact, simple systematic generalizations for the simple Euler-like exponential of the Pauli matrices, for spin 1 and for all representations of SU(2), but this is distinctly egregious overkill, for your purpose. Wigner's little d rotation matrices in the spherical representation solves the problem simply and in full generality. The text you are referring to assumes the reader has taken a good QM course where all this is covered quite nicely. 

*In any case, since it is safely late to do homework for one, recall the expression for $J_2$ for spin one,
$$
 J_2 = \frac{1}{\sqrt{2}}
    \begin{bmatrix}
      0 &-i &0\\
      i &0  &-i\\
      0 &i  &0
    \end{bmatrix} \equiv iK,
$$ 
which you may easily exponentiate
$$
e^{-i\theta J_2}=e^{\theta K}=  I + (\sin\theta) K  + (1-\cos\theta) K^2 ~,
$$ 
since you can confirm that $K^3=-K$, so $K^2$ behaves like i in combinatoric terms in the series for the exponential, when multiplied by K. 
Now, since 
$$
K^2=  \begin{bmatrix}
      -1/2 &0 &0\\
      0 &-1  &0\\
      0 &0  &-1/2
    \end{bmatrix} ,
$$
the net rotation in the exponential is just 
$$
e^{-i\theta J_2}= \begin{bmatrix}
      \frac{1+\cos\theta}{2} &-\sin\theta /\sqrt{2} & \frac{1-\cos\theta}{2}\\
      \sin\theta /\sqrt{2} &\cos \theta  &-\sin\theta /\sqrt{2}\\
       \frac{1-\cos\theta}{2} &\sin\theta /\sqrt{2}  &\frac{1+\cos\theta}{2}
    \end{bmatrix} .
$$
You are done. From this, you simply read off the spherical basis matrix elements you have for the destination Wigner d-matrix stated.
