Operator Methods for the Time Evolution of Wave Functions

TL;DR

This paper introduces an operator-based method for solving the time evolution of wave functions across various potentials, linking classical and quantum chaos, and deriving general solutions for different systems.

Contribution

It develops a novel operator approach for wave function evolution, including exact and approximate solutions, and explores the connection between classical and quantum chaos.

Findings

Derived general solutions for wave function evolution in multiple potentials

Established a link between classical chaos and quantum chaos

Developed methods for exact and approximate operator solutions

Abstract

A method based off of operator consideration for solving the time evolution of a wave function is developed. The method is applied to free space, constant force and harmonic oscillator potentials where general solutions are derived for the wave function time evolutions for these potentials. Solving the Heisenberg equations of motion is discussed. A method to obtain exact solutions to Heisenberg's equations of motion is developed based off of solutions of classical equations of motion. Approximation methods for the time evolution of operators and wave functions are developed. A connection between classical and quantum chaos is made where it is realized that whenever a classical system behaves chaotically, the quantum system will as well.