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

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.