Approach of the Generating Functions to the Coherent States for Some Quantum Solvable Models

TL;DR

This paper introduces new coherent states for various quantum solvable models derived from generating functions of classical orthogonal polynomials, simplifying their construction without complex algebraic methods.

Contribution

It presents a novel approach to constructing coherent states directly from generating functions, applicable to models lacking algebraic structures or shape invariance.

Findings

States align with Klauder-Perelomove and Barut-Girardello types

Method applicable to non-classical and q-orthogonal polynomials

Extends to systems without specific algebraic symmetries

Abstract

We introduce to this paper new kinds of coherent states for some quantum solvable models: a free particle on a sphere, one-dimensional Calogero-Sutherland model, the motion of spinless electrons subjected to a perpendicular magnetic field B, respectively, in two dimensional flat surface and an infinite flat band. We explain how these states come directly from the generating functions of the certain families of classical orthogonal polynomials without the complexity of the algebraic approaches. We have shown that some examples become consistent with the Klauder- Perelomove and the Barut-Girardello coherent states. It can be extended to the non-classical, q-orthogonal and the exceptional orthogonal polynomials, too. Especially for physical systems that they don't have a specific algebraic structure or involved with the shape invariance symmetries, too.