Comparison between time-independent and time-dependent quantum systems in the context of energy, Heisenberg uncertainty, average energy, force, average force and thermodynamic quantities

TL;DR

This paper compares time-independent and time-dependent quantum systems by analyzing energy, uncertainty, force, and thermodynamic properties, providing exact solutions and new insights into their differences and similarities.

Contribution

It introduces a comprehensive comparison framework for quantum systems using point transformation, separation of variables, and supersymmetric quantum mechanics, highlighting novel analytical results.

Findings

Exact solutions for time-dependent Schrödinger equation are obtained.

Analytical thermodynamic quantities are derived at high temperature.

Comparison of different quantum systems reveals their similarities and differences.

Abstract

Exact solutions of time-dependent Schr\"odinger equation in presence of time-dependent potential is defined by point transformation and separation of variables. Energy and Heisenberg uncertainty relation are pursued for time-independent potential whereas average energy and Heisenberg uncertainty relation are defined for time-dependent potential. Forces acting on a fixed boundary wall as well as average force acting on moving boundary wall are presented along various trajectories. For high temperature, analytical forms of partition function and the corresponding thermodynamic quantities are derived following the Euler-Maclaurin summation formula over a finite as well as an infinite domain for accurate presentation. Three quantum systems are generated with the help of point transformation, separation of variables and super-symmetric quantum mechanics from one quantum system and the corresponding results are compared among all systems, where two of them are time-independent and another two are time-dependent.