Toward a thermo-hydrodynamic like description of Schrodinger equation via the Madelung formulation and Fisher information

TL;DR

This paper explores the hydrodynamic and thermodynamic analogies of the Schrödinger equation using Madelung's formulation, linking Fisher information to internal energy and disorder in quantum systems.

Contribution

It introduces a thermodynamic perspective to Madelung's fluid model, connecting Fisher information with internal energy and providing insights into quantum disorder measures.

Findings

Pressure-like term derived from Bohm potential does not violate Ehrenfest's theorem.

Incompressibility leads to non-spreading wave packets in 1D.

Fisher information is proportional to the expectation of Bohm potential, linking quantum disorder and internal energy.

Abstract

We revisit the analogy suggested by Madelung between a non-relativistic time-dependent quantum particle, to a fluid system which is pseudo-barotropic, irrotational and inviscid. We first discuss the hydrodynamical properties of the Madelung description in general, and extract a pressure like term from the Bohm potential. We show that the existence of a pressure gradient force in the fluid description does not violate Ehrenfest's theorem since its expectation value is zero. We also point out that incompressibility of the fluid implies conservation of density along a fluid parcel trajectory and in 1D this immediately results in the non-spreading property of wave packets, as the sum of Bohm potential and an exterior potential must be either constant or linear in space. Next we relate to the hydrodynamic description a thermodynamic counterpart, taking the classical behavior of an adiabatic barotopric flow as a reference. We show that while the Bohm potential is not a positive definite quantity, as is expected from internal energy, its expectation value is proportional to the Fisher information whose integrand is positive definite. Moreover, this integrand is exactly equal to half of the square of the imaginary part of the momentum, as the integrand of the kinetic energy is equal to half of the square of the real part of the momentum. This suggests a relation between the Fisher information and the thermodynamic like internal energy of the Madelung fluid. Furthermore, it provides a physical linkage between the inverse of the Fisher information and the measure of disorder in quantum systems - in spontaneous adiabatic gas expansion the amount of disorder increases while the internal energy decreases.