  We present an alternative formulation of quantum mechanical angular momentum,
based on spatial wavefunctions that depend on the Euler angles $\phi, \theta,
\chi$. The wavefunctions are Wigner D-functions, $D_{n m}^j (\phi, \theta,
\chi)$, for which the body-fixed projection quantum number $n$ has the unusual
value $n=|j|=\sqrt{j(j+1)}$, or $n=0$. The $D_{\sqrt{j(j+1)},m}^j (\phi,
\theta, \chi)$ wavefunctions are unnormalizable, however we demonstrate a
renormalization procedure that allows the calculation of expectation values:
for example, the states $D_{\sqrt{S(S+1)} M}^S (\phi, \theta, \chi)$ of
elementary particles with spin $S$ give a gyromagnetic ratio of $g=2$ for
$S>0$, and we identify these as the spatial angular-momentum wavefunctions of
known fundamental charged particles with spin. Therefore, we make the case that
the $D_{n m}^j (\phi, \theta, \chi)$ are useful as spatial wavefunctions for
angular momentum. Some implications of the quantum number $n$ for fundamental
particles are discussed, such as the proposed Dirac-fermion nature of the
neutrino, and some proposed dark-matter candidates.
