Harmonic oscillator coherent states from the orbit theory standpoint

TL;DR

This paper explores harmonic oscillator coherent states using orbit theory and noncommutative integration, linking symmetry properties and Lie group representations to construct and analyze these states.

Contribution

It introduces a novel approach to coherent states via orbit theory and noncommutative integration, connecting Lie algebra representations to quantum states.

Findings

Coherent states can be expressed through solutions of differential equations on Lie groups.

The method relates coherent states to irreducible Lie algebra representations.

Orbit geometry provides a new perspective on quantum harmonic oscillator states.

Abstract

We study the known coherent states of a quantum harmonic oscillator from the standpoint of the original developed noncommutative integration method for linear partial differential equations. The application of the method is based on the symmetry properties of the Schr\"odinger equation and on the orbit geometry of the coadjoint representation of Lie groups. We have shown that analogs of coherent states constructed by the noncommutative integration can be expressed in terms of the solution of a system of differential equations on the Lie group of the oscillatory Lie algebra. The solutions constructed are directly related to irreducible representation of the Lie algebra on the Hilbert space functions on the Lagrangian submanifold to the orbit of the coadjoint representation.