# Factorization, Supersymmetry, Coherent States and Classical Trajectories

## TL;DR

This paper explores the relationship between generalized coherent states in supersymmetric quantum mechanics and their corresponding classical trajectories, providing formulas and methods to compare quantum and classical dynamics.

## Contribution

It introduces a method to relate mean position values of supersymmetric coherent states to classical trajectories for simple superpotentials, extending known results for harmonic oscillators.

## Key findings

- Mean position can be expressed as a power series in time with coefficients linked to the superpotential.
- The method reproduces known harmonic oscillator results and extends to other systems.
- The time scale for divergence between quantum mean values and classical trajectories is estimated.

## Abstract

A generalization of coherent states has been developed in the context of supersymmetric quantum mechanics. For many cases, no link has been made with the corresponding classical system. In this work, we consider simple superpotentials and compare the classical trajectories and the mean values of the position operator in these states. The mean value of the position operator can be written as a power series in time with coefficients whose relations to the superpotential are given in the general case. These coefficients imply integrals and recursion formulas. The method used reproduces exactly what is known for the harmonic oscillator. It is extended to study a family of systems which encompasses the harmonic oscillator. We also consider a third degree superpotential. The time scale after which the mean value of the position operator and the classical trajectory begin differing significantly is evaluated. Keywords: Coherent States, SUSYQM.