The reformed hydrodynamic Schr\"odinger equation and the effective method for eliminating pseudo-quantum potential: a case study on barotropic potential flow

TL;DR

This paper introduces a reformed hydrodynamic Schr"odinger equation (RHSE) for barotropic flows, proposing a correction method to eliminate pseudo-quantum potential without classical limits, and analyzes its conservation properties and limitations.

Contribution

The paper develops the RHSE framework and a novel correction method to remove pseudo-quantum potential, challenging the reliance on classical limits for such elimination.

Findings

The correction method enforces energy and momentum conservation.

Classical limit does not fully remove redundant terms in vorticity cases.

Further investigation is needed for the correction method's universality.

Abstract

Given the discrepancies between the framework of the hydrodynamic Schr\"odinger equation (HSE) and classical fluid dynamics, we propose a reformed framework, termed the reformed hydrodynamic Schr\"odinger equation (RHSE) for clarity. The RHSE for barotropic potential flow is developed; notably, we demonstrate an alternative approach to eliminate the pseudo-quantum potential without resorting to the classical limit. We call this approach the ''correction method.'' We then examine the correction method using Noether's theorem, confirming that it enforces the conservation of energy and momentum. Furthermore, we find that the classical limit fails to completely remove redundant terms when a two-component wave function is used to introduce vorticity. This fact suggests that the classical limit is not a universal fallback, highlighting the importance of further investigating the correction method.