The Quantum Reduced Action In Higher Dimensions

TL;DR

This paper investigates the quantum reduced action in higher dimensions, revealing that a simple sum of one-variable functions is insufficient to fully describe quantum motion and exploring conditions for microstates.

Contribution

It provides new analytical insights into the construction of the quantum reduced action in higher dimensions and clarifies limitations of the separated variable approach.

Findings

Sum of one-variable functions does not contain complete quantum information.

Conditions for microstates can occur even with complex wave functions.

Analysis of recent quantum trajectory results.

Abstract

The solution with respect to the reduced action of the one-dimensional stationary quantum Hamilton-Jacobi equation is well known in the literature. The extension to higher dimensions in the separated variable case was proposed in contradictory formulations. In this paper we provide new insights into the construction of the reduced action. In particular, contrary to the classical mechanics case, we analytically show that the reduced action constructed as a sum of one variable functions does not contain a complete information about the quantum motion. In the same context, we also make some observations about recent results concerning quantum trajectories. Finally, we will examine the conditions in which microstates appear even in the case where the wave function is complex.