# Solving the Schrodinger Equation by Reduction to a First-order Differential Operator through a Coherent States Transform

## TL;DR

This paper introduces a method to simplify certain quantum Hamiltonians into first-order differential operators using coherent state transforms, enabling explicit solutions for quantum dynamics.

## Contribution

It generalizes the geometric dynamics approach to quantum systems, identifying Hamiltonians reducible via Gaussian and Airy beam transforms and providing explicit solutions.

## Key findings

- Reduction of specific quantum Hamiltonians to first-order PDEs
- Explicit solutions for systems involving Gaussian and Airy beams
- Extension of harmonic oscillator dynamics to broader quantum systems

## Abstract

The Legendre transform expresses dynamics of a classical system through first-order Hamiltonian equations. We consider coherent state transforms with a similar effect in quantum mechanics: they reduce certain quantum Hamiltonians to first-order partial differential operators. Therefore, the respective dynamics can be explicitly solved through a flow of points in extensions of the phase space. This generalises the geometric dynamics of a harmonic oscillator in the Fock space. We describe all Hamiltonians which are geometrised (in the above sense) by Gaussian and Airy beams and write down explicit solutions for such systems.