A direct approach to coherent states of billiards using a quantum algebra framework

TL;DR

This paper extends the Generalized Heisenberg Algebra framework to quantum billiards, enabling the construction of coherent states for both separable and non-separable billiard geometries, advancing quantum state analysis in complex systems.

Contribution

It introduces a novel algebraic approach to quantum billiards, developing coherent states for new geometries using the GHA framework, which was not previously applied to such systems.

Findings

Constructed one-dimensional coherent states for square billiards.

Extended the formalism to two-dimensional coherent states.

Demonstrated applicability to non-separable equilateral triangle billiards.

Abstract

Quantum billiards are a key focus in quantum mechanics, offering a simple yet powerful model to study complex quantum features. While the development of algebras for quantum systems is traced from one-dimensional integrable models to quantum groups and the Generalized Heisenberg Algebra (GHA). The primary focus of this work is to extend the GHA to quantum billiards, showcasing its application to separable and non-separable billiards. We apply the formalism to a square billiard, first generating one-dimensional coherent states with specific quantum numbers and exploring their time evolution.Then, we extend this approach to develop two-dimensional coherent states for the square billiards. We also demonstrate its applicability in a non-separable equilateral triangle billiard, describing their algebra generators and associated one-dimensional coherent states.