A New Expansion of the Heisenberg Equation of Motion with Projection Operator

TL;DR

This paper introduces a generalized expansion of the Heisenberg equation of motion using an improved projection operator method that allows for arbitrary initial states, demonstrated on a damped harmonic oscillator model.

Contribution

The authors extend the projection operator method to accommodate arbitrary initial states, overcoming previous limitations, and validate it with an exactly solvable model.

Findings

Successfully generalized the projection operator method.

Confirmed the validity using the damped harmonic oscillator model.

Enabled calculation of quantum system dynamics with arbitrary initial states.

Abstract

We derive a new expansion of the Heisenberg equation of motion based on the projection operator method proposed by Shibata, Hashitsume and Shing\=u. In their projection operator method, a certain restriction is imposed on the initial state. As a result, one cannot prepare arbitrary initial states, for example a coherent state, to calculate the time development of quantum systems. In this paper, we generalize the projection operator method by relaxing this restriction. We explain our method in the case of a Hamiltonian both with and without explicit time dependence. Furthermore, we apply it to an exactly solvable model called the damped harmonic oscillator model and confirm the validity of our method.