On some hydrodynamical aspects of quantum mechanics

TL;DR

This paper explores hydrodynamical analogies in quantum mechanics, analyzing spin, vorticity, and velocity fields within a geometric framework, revealing divergence-free flows and their relation to energy levels.

Contribution

It establishes a novel analogy between spin and fluid vorticity, and demonstrates that quantum velocity fields are divergence-free and satisfy stationary Euler equations in finite-dimensional geometric quantum mechanics.

Findings

Schr"odinger velocity field is divergence-free and satisfies Euler equations

Vorticity depends on energy level spacings

Pressure gradient and critical points are explicitly computed

Abstract

In this note we first set up an analogy between spin and vorticity of a perfect 2d-fluid flow, based on the Borel-Weil contruction of the irreducible unitary representations of SU(2), and looking at the Madelung-Bohm velocity attached to the ensuing spin wave functions. We also show that, in the framework of finite dimensional geometric quantum mechanics, the Schr\"odinger velocity field on projective Hilbert space is divergence-free (being Killing with respect to the Fubini-Study metric) and fulfils the stationary Euler equation, with pressure proportional to the Hamiltonian uncertainty (squared). We explicitly compute the pressure gradient of this "Schr\"odinger fluid" and determine its critical points. Its vorticity is also calculated and shown to depend on the spacings of the energy levels. These results follow from hydrodynamical properties of Killing vector fields valid in any (finite dimensional) Riemannian manifold, of possible independent interest.