Analyzing the Time Evolution of Wave Functions by Decomposing the Hamiltonian into State-Preserving and State-Changing Hamiltonians

TL;DR

This paper introduces a novel Hamiltonian decomposition method to analyze the time evolution of wave functions, simplifying the Schrödinger equation and enabling exact solutions for nonspreading wave packets and general systems.

Contribution

The paper proposes a new Hamiltonian decomposition approach that separates state-preserving and state-changing components, simplifying the analysis of quantum dynamics.

Findings

Exact solutions for nonspreading wave packets

Simplified Schrödinger equation form

Applicable to general Hamiltonian systems

Abstract

We show a new method for analyzing the time evolution of the Schrodinger wave function Psi(x,t). We propose the decomposition of the Hamiltonian as: H(t)=Hp(t)+Hc(t), where Hp(t) is the Hamiltonian such that Psi(x,t) is its instantaneous eigenfunction, and Hc(t) the Hamiltonian which changes the state Psi. With this decomposition, the action of H(t) on the wave function is simplified and the Schrodinger equation is in a simpler form which can be solved more easily. We illustrate this method by exactly solving the Schrodinger equation for cases of nonspreading wave packets. This method can be applied as well to analyzing the time evolution of general Hamiltonian systems.