Two-particle coalescence conditions revisited

TL;DR

This paper revisits the conditions for two-particle coalescence in quantum systems, deriving new constraints from the local energy concept using a simplified radial model to understand wave function behavior near coalescence points.

Contribution

It introduces a novel analysis of n-th order two-particle coalescence conditions using the local energy framework with a radial model, providing insights into wave function expansions.

Findings

Derived energy-independent coalescence constraints.

Identified state-specific conditions for local energy regularity.

Established relations between wave function expansion coefficients.

Abstract

The notion of the n-th order local energy, generated by the n-th power of the Hamiltonian, has been introduced. The n-th order two-particle coalescence conditions have been derived from the requirements that the n-th order local energy at the coalescence point is non-singular and equal to the n-th power of the Hamiltonian eigenvalue. The first condition leads to energy-independent constraints. The second one is state-specific. The analysis has been done using a radial, one-dimensional, model Hamiltonian. The model is valid in the asymptotic region of r ~ 0. The coalescence conditions set the relations between the expansion coefficients of the radial wave function into a power series with respect to r.