The Fluid Dynamics of the One-Body Stationary States of Quantum Mechanics with Real Valued Wavefunctions

TL;DR

This paper explores the fluid dynamics analogy of quantum stationary states with real wavefunctions, deriving a generalized Bernoulli equation and analyzing flow properties in various quantum systems.

Contribution

It introduces a novel fluid dynamics framework for quantum stationary states, linking probability density to compressible flow equations and analyzing wave-pulse velocities.

Findings

Probability density satisfies a Bernoulli-like equation.

Velocity on a streamline can be opposite to wave-pulse velocity.

Extremums of momentum occur at Mach 1 speed points.

Abstract

It is demonstrated that the probability density function, given by the square of a quantum mechanical wavefunction that is a real-valued eigenvector of a time-independent, one-body Schroedinger equation, satisfies a compressible-flow generalization of the Bernoulli equation, where the mass density is the probability density times the mass of the system; the pressure and velocity fields are defined by functions depending on the probability density, and the gradient and the Laplacian of the probability density, where there are two possible directions of the velocity on a streamline. The velocity given definition implies a generalization of the steady-flow continuity equation where mass is not locally conserved. The gradient of the Bernoullian equation is demonstrated to be equivalent to the steady flow Euler equation for variable mass and irrotational flow. A speed of sound quadratic equation is obtained from a spherical wave-pulse. One of the solutions indicates that the wave-pulse velocity on a streamline is equal in magnitude but opposite in direction of the fluid velocity on the streamline. The other solution is the focus of attention from that point on. It is proven that the extremums of the momentum per volume on a streamline occur at points that are Mach 1 speed. The developed formalism is applied to a particle in a one-dimensional box, the ground and first excited-states of the one-dimensional harmonic oscillator, and the hydrogen 1s and 2s states. Some behavior is repeated in all the applications examined. For example, an antinode, a point of local-maximum density, has zero velocity and zero Mach speed on the streamline, while a node, a point of minimum density, has infinite velocity and Mach 2. In between the node and antinode is an extremum of the momentum, and Mach 1. (This is a short version of the abstract.)