Equation of Motion for the Quantum Characteristic Functions

TL;DR

This paper derives simple, unified equations of motion for various quantum characteristic functions, facilitating comparison with classical dynamics and enabling efficient numerical simulations.

Contribution

It introduces a unified, straightforward approach to quantum characteristic function dynamics applicable to general Hamiltonians, bridging quantum and classical descriptions.

Findings

Equations recover classical dynamics as 6b0 6b0 limit.

Unified equations for normal, symmetric, and antinormal-order functions.

Potential for improved numerical simulation methods.

Abstract

In this paper, we derive equations of motion for the normal-order, the symmetric-order and the antinormal-order quantum characteristic functions, applicable for general Hamiltonian systems. We do this by utilizing the `characteristic form' of both quantum states and Hamiltonians. The equations of motion we derive here are rather simple in form and in essence, and as such have a number of attractive features. As we shall see, our approach enables the descriptions of quantum and classical time evolutions in one unified language. It allows for a direct comparison between quantum and classical dynamics, providing insight into the relations between quantum and classical behavior, while also revealing a smooth transition between quantum and classical time evolutions. In particular, the $\hbar\to 0$ limit of the quantum equations of motion instantly recovers their classical counterpart. We also argue that the derived equations may prove to be very useful in numerical simulations.