The Vector-Model Wavefunction: spatial description and wavepacket formation of quantum-mechanical angular momenta

Abstract

In quantum mechanics, spatial wavefunctions describe distributions of a particle's position or momentum, but not of angular momentum $j$. In contrast, here we show that a spatial wavefunction, $j_m (\phi,\theta,\chi)=~e^{i m \phi} \delta (\theta - \theta_m) ~e^{i(j+1/2)\chi}$, which treats $j$ in the $|jm>$ state as a three-dimensional entity, is an asymptotic eigenfunction of angular-momentum operators; $\phi$, $\theta$, $\chi$ are the Euler angles, and $cos \theta_m=(m/|j|)$ is the Vector-Model polar angle. The $j_m (\phi,\theta,\chi)$ gives a computationally simple description of particle and orbital-angular-momentum wavepackets (constructed from Gaussian distributions in $j$ and $m$) which predicts the effective wavepacket angular uncertainty relations for $\Delta m \Delta \phi $, $\Delta j \Delta \chi$, and $\Delta\phi\Delta\theta$, and the position of the particle-wavepacket angular motion on the orbital plane. The particle-wavepacket rotation can be experimentally probed through continuous and non-destructive $j$-rotation measurements. We also use the $j_m (\phi,\theta,\chi)$ to determine well-known asymptotic expressions for Clebsch-Gordan coefficients, Wigner d-functions, the gyromagnetic ratio of elementary particles, $g=2$, and the m-state-correlation matrix elements, $<j_3 m_3|j_{1X} j_{2X}|j_3 m_3>$. Interestingly, for low j, even down to $j=1/2$, these expressions are either exact (the last two) or excellent approximations (the first two), showing that $j_m (\phi,\theta,\chi)$ gives a useful spatial description of quantum-mechanical angular momentum, and provides a smooth connection with classical angular momentum.