Quantization of Damping Particle Based On New Variational Principles

TL;DR

This paper introduces a novel variational approach to quantize damped mechanical systems, extending Feynman's path integral method by using a Lagrangian density, enabling quantization of non-Hamiltonian systems.

Contribution

It proposes a new variational principle and modifies Feynman's path integral to quantize systems with damping, which are not describable by traditional Hamiltonian mechanics.

Findings

Successfully quantized a damped particle using the new method

The approach reduces to standard Feynman propagator for conservative systems

Detailed analysis of a frictional particle demonstrates the method's effectiveness

Abstract

In this paper a new approach is proposed to quantize mechanical systems whose equations of motion can not be put into Hamiltonian form. This approach is based on a new type of variational principle, which is adopted to a describe a relation: a damping particle may shares a common phase curve with a free particle, whose Lagrangian in the new variational principle can be considered as a Lagrangian density in phase space. According to Feynman's theory, the least action principle is adopted to modify the Feynman's path integral formula, where Lagrangian is replaced by Lagrangian density. In the case of conservative systems, the modification reduces to standard Feynman's propagator formula. As an example a particle with friction is analyzed in detail.