Spatial Wavefunctions of Spin

Abstract

We present an alternative formulation of quantum mechanical angular momentum, based on spatial wavefunctions that depend on the Euler angles $\phi, \theta, \chi$, and have an additional internal projection $n$. The wavefunctions are Wigner D-functions, $D_{n m}^s (\phi, \theta, \chi)$, for which the body-fixed projection quantum number $n$ has the unusual value $n=|s|=\sqrt{s(s+1)}$, or $n=0$. We show that the states $D_{\sqrt{s(s+1)},m}^s (\phi, \theta, \chi)$ of elementary particles with spin $\sqrt{s(s+1)}$ 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. All known Standard-Model particles can be categorized with either value $n=\sqrt{s(s+1)}$ or $n=0$, and all known particle reactions are consistent with the conservation of its projection in the internal frame, and with internal-frame Clebsch-Gordan coefficients of unity. Therefore, we make the case that the $D_{n m}^s (\phi, \theta, \chi)$ are useful as spatial wavefunctions for angular momentum. Some implications and new predictions related to the quantum number $n$ for fundamental particles are discussed, such as the proposed Dirac-fermion nature of the neutrino, the explanation of some Standard-Model structure, and some proposed dark-matter candidates.