Coherent states for general potentials. IV. Three-dimensional systems

The minimum-uncertainty coherent-states formalism is extended to higher-dimensional systems. Specifically, for spherically symmetric three-dimensional potentials the formalism looks for coherent states which are products of an angular wave function times a radial wave function. After reviewing the many studies on angular coherent states, I concentrate on the physically distinguishing radial coherent states. The radial formalism is explained in detail and contrasted with the effective one-dimensional formalism. The natural classical variables in the radial formalism are those which vary sinusoidally as $g(E, L)\theta \mathrm{}\left(t\right)\mathrm{}$, where $\theta \mathrm{}\left(t\right)\mathrm{}$ is the real azimuthal angular variable and $g(E, L)$ is the number of oscillations between apsidal distances per classical orbit. When changed to natural quantum operators, these operators can be given as the Hermitian sums and differences of the "$l$" raising and lowering operators. The formalism is applied to the three-dimensional harmonic-oscillator and Coulomb problems.