Coherent States of Systems with Pure Continuous Energy Spectra

TL;DR

This paper develops a method to construct coherent states for systems with continuous energy spectra using orthogonal polynomials and Bayesian decomposition, linking to the Gazeau-Klauder framework, exemplified by the free particle Hamiltonian.

Contribution

It introduces a novel tridiagonal orthogonal polynomial approach to define coherent states for continuous spectra, connecting them to established formalisms.

Findings

Coherent states obeying Glauber-type conditions are constructed for continuous spectra.

The method is exemplified with the $ ext{l}$-wave free particle Hamiltonian.

The states can be expressed within the Gazeau-Klauder formalism.

Abstract

While dealing with a Hamiltonian with continuous spectrum we use a tridiagonal method involving orthogonal polynomials to construct a set of coherent states obeying a Glauber-type condition. We perform a Bayesian decomposition of the weight function of the orthogonality measure to show that the obtained coherent states can be recast in the Gazeau-Klauder approach. The Hamiltonian of the $\ell$-wave free particle is treated as an example to illustrate the method.