On the Normalization and Density of 1D Scattering States

TL;DR

This paper reveals that the normalization of 1D scattering states encodes the density of the scattering spectrum, providing formulas to compute it for specific potentials and demonstrating its use in calculating the partition function for interacting particles.

Contribution

It establishes the link between scattering state normalization and density of states, offering practical formulas and applications for specific potentials.

Findings

Formulas for density of states for delta function and square well potentials

Normalization contains spectrum density information, not just wavefunctions

Application to two-particle partition function with point-like interaction

Abstract

The normalization of scattering states is more than a rote step necessary to calculate expectation values. This normalization actually contains important information regarding the density of the scattering spectrum (along with useful details on the bound states). For many applications, this information is more useful than the wavefunctions themselves. In this paper we show that this correspondence between scattering state normalization and the density of states is a consequence of the completeness relation, and we present formulas for calculating the density of states which are applicable to certain potentials. We then apply these formulas to the delta function potential and the square well. We then illustrate how the density of states can be used to calculate the partition function for a system of two particles with a point-like (delta potential) interaction.