= \pmatrix{\cos{\phi\over2}& \ell\sin{\phi\over2}\cr
        \noalign{\smallskip}
        \ell\sin{\phi\over2}& \cos{\phi\over2}\cr}$$ corresponding to rotations by the angle $\phi$ about the $z$, $y$, and $x$ axes, respectively.

The infinitesimal generators of the Lie algebra $su(2)$ are obtained by differentiating these group elements, via $$L_a = {d R_a\over d\phi} \Bigg|_{\phi=0}$$ where as before $a=x,y,z$. For reasons which will become apparent, we have [*not*]{} multiplied these generators by $-\ell$ to obtain Hermitian matrices. We have instead $$2 L_z = \pmatrix{\ell& 0\cr \noalign{\smallskip} 0& -\ell\cr}
\qquad
2 L_y = \pmatrix{0& 1\cr \noalign{\smallskip} -1& 0\cr}
\qquad
2 L_x = \pmatrix{0& \ell\cr \noalign{\smallskip} \ell& 0\cr}$$ which satisfy the commutation relations $$\left[ L_a , L_b \right] = \epsilon_{abc} L_c$$ where $\epsilon$ is completely antisymmetric and $$\epsilon_{xyz} = 1$$

Spin eigenstates are usually obtained as eigenvectors of the Hermitian matrix $-\ell L_z$, with real eigenvalues. Here we must be careful to multiply by $\ell$ in the correct place. We define $${\widehat}{L}_z \psi := - L_z \psi \ell$$ which is well-defined by alternativity, so that $${\widehat}{L}_z = - \ell_R \circ L_z$$ where the operator $\ell_R$ denotes right multiplication by $\ell$ and where $\circ$ denotes composition. The operators ${\widehat}{L}_a$ are self-adjoint with respect to the inner product $$\label{SelfAdj}
\langle \psi,\chi \rangle = \pi \!\left( \psi^\dagger\chi 