Overlap integral of stationary scattering states

TL;DR

This paper analyzes the overlap integrals of stationary scattering states in finite potentials, revealing conditions under which superpositions behave as isolated particles or exhibit time-dependent properties.

Contribution

It provides a detailed expression of overlap integrals, highlighting the role of diagonal and nondiagonal terms in stationary scattering states.

Findings

Diagonal terms are proportional to δ(E₁ - E₂).

Nondiagonal terms cause time-dependent norms in superpositions.

In certain potentials, superpositions with different energies represent isolated particles.

Abstract

The overlap integrals of scattering states in potentials of finite widths are expressed with their asymptotic behaviors and those of energies $E_1$ and $E_2$ consist of diagonal terms that are proportional to $\delta(E_1-E_2)$ and nondiagonal terms. Owing to the composition of nondiagonal terms, superpositions of stationary states have time-dependent norms and finite probability currents. These do not represent isolate states. In various exceptional potentials and in free theory, nondiagonal terms do not exist, and the superpositions of states with different energies represent isolate particles that exactly describe scattering processes.