Quantum Hamilton-Jacobi Theory

TL;DR

This paper presents a method to solve the quantum Hamilton-Jacobi equation using the propagator of the Schrödinger equation, enabling practical applications and revealing a link between operator ordering and path density.

Contribution

It introduces a simple prescription to construct solutions to the QHJE from the Schrödinger propagator, overcoming longstanding computational difficulties.

Findings

Solutions to QHJE can be constructed from Schrödinger propagator.

Established a relation between operator ordering and path density.

Enabled practical use of quantum Hamilton-Jacobi theory.

Abstract

Quantum canonical transformations have attracted interest since the beginning of quantum theory. Based on their classical analogues, one would expect them to provide a powerful quantum tool. However, the difficulty of solving a nonlinear operator partial differential equation such as the quantum Hamilton-Jacobi equation (QHJE) has hindered progress along this otherwise promising avenue. We overcome this difficulty. We show that solutions to the QHJE can be constructed by a simple prescription starting from the propagator of the associated Schroedinger equation. Our result opens the possibility of practical use of quantum Hamilton-Jacobi theory. As an application we develop a surprising relation between operator ordering and the density of paths around a semiclassical trajectory.