# Infinite square-well, trigonometric P\"oschl-Teller and other potential wells with a moving barrier

## TL;DR

This paper develops new quantum potential wells with moving barriers using point transformations and time-dependent supersymmetry, providing explicit solutions for their Schrödinger equations, expanding the class of exactly solvable time-dependent quantum systems.

## Contribution

It introduces a systematic method to generate and solve new infinite potential wells with moving barriers using advanced transformation techniques.

## Key findings

- Explicit solutions for time-dependent Schrödinger equations are provided.
- New classes of potential wells with moving barriers are constructed.
- The methods unify and extend previous approaches to quantum well problems.

## Abstract

Using mainly two techniques, a point transformation and a time dependent supersymmetry, we construct in sequence several quantum infinite potential wells with a moving barrier. We depart from the well known system of a one-dimensional particle in a box. With a point transformation, an infinite square-well potential with a moving barrier is generated. Using time dependent supersymmetry, the latter leads to a trigonometric P\"oschl-Teller potential with a moving barrier. Finally, a confluent time dependent supersymmetry transformation is implemented to generate new infinite potential wells, all of them with a moving barrier. For all systems, solutions of the corresponding time dependent Schr\"odinger equation fulfilling boundary conditions are presented in a closed form.