Angular momentum eigenstates of the isotropic 3-D harmonic oscillator: Phase-space distributions and coalescence probabilities

The isotropic 3-dimensional harmonic oscillator potential can serve as an approximate description of many systems in atomic, solid state, nuclear, and particle physics. In particular, the question of 2 particles binding (or coalescing) into angular momentum eigenstates in such a potential has interesting applications. We compute the probabilities for coalescence of two distinguishable, non-relativistic particles into such a bound state, where the initial particles are represented by generic wave packets of given average positions and momenta. We use a phase-space formulation and hence need the Wigner distribution functions of angular momentum eigenstates in isotropic 3-dimensional harmonic oscillators. These distribution functions have been discussed in the literature before but we utilize an alternative approach to obtain these functions. Along the way, we derive a general formula that expands angular momentum eigenstates in terms of products of 1-dimensional harmonic oscillator eigenstates. (c) 2022 Elsevier Inc. All rights reserved.

Automated summary

The isotropic 3-dimensional harmonic oscillator potential is an approximate description of various systems in different branches of physics, and it has interesting applications in studying the binding of two particles into angular momentum eigenstates. In this study, we compute the probabilities for the coalescence of two distinguishable particles into bound states in this potential, using wave packets and the phase-space formulation. We also derive a general formula for expanding angular momentum eigenstates based on 1-dimensional harmonic oscillator eigenstates.