Angular Momentum Eigenstates of the Isotropic 3-D Harmonic Oscillator: Phase-Space Distributions and Coalescence Probabilities

TL;DR

This paper calculates phase-space distributions and coalescence probabilities for two particles forming angular momentum eigenstates in an isotropic 3D harmonic oscillator, using an alternative approach to derive Wigner functions.

Contribution

It introduces a new method to derive Wigner distribution functions for angular momentum eigenstates and applies it to compute coalescence probabilities.

Findings

Derived general formula for angular momentum eigenstates in terms of 1D harmonic oscillator states

Computed phase-space distributions for angular momentum eigenstates

Calculated coalescence probabilities for two particles forming bound states

Abstract

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.