Two-body wave functions and compositeness from scattering amplitudes. I. General properties with schematic models

TL;DR

This paper presents a method to extract two-body wave functions from scattering amplitudes in quantum mechanics, analyzing their properties and the concept of compositeness, especially in relation to energy dependence of interactions.

Contribution

It establishes a general scheme to derive two-body wave functions from scattering amplitudes and explores their properties and normalization in schematic models.

Findings

Wave functions correspond to residues at the bound state pole.

Wave functions from Lippmann-Schwinger and Schrödinger equations coincide for energy-independent interactions.

Compositeness equals unity for energy-independent interactions, deviates otherwise.

Abstract

For a general two-body bound state in quantum mechanics, both in the stable and decaying cases, we establish a way to extract its two-body wave function in momentum space from the scattering amplitude of the constituent two particles. For this purpose, we first show that the two-body wave function of the bound state corresponds to the residue of the off-shell scattering amplitude at the bound state pole. Then, we examine our scheme to extract the two-body wave function from the scattering amplitude in several schematic models. As a result, the two-body wave functions from the Lippmann--Schwinger equation coincides with that from the Schr\"{o}dinger equation for an energy-independent interaction. Of special interest is that the two-body wave function from the scattering amplitude is automatically scaled; the norm of the two-body wave function, to which we refer as the compositeness, is unity for an energy-independent interaction, while the compositeness deviates from unity for an energy-dependent interaction, which can be interpreted to implement missing channel contributions. We also discuss general properties of the two-body wave function and compositeness for bound states in the schematic models.