On the motion of macroscopic bodies in quantum theory

TL;DR

This paper explores the motion of macroscopic bodies within quantum mechanics by linking quantum observables to vector fields, providing new insights into the relationship between quantum and classical dynamics.

Contribution

It introduces a vector field representation of quantum observables to unify the treatment of macroscopic and microscopic bodies and relates Schrödinger and Newton equations beyond Ehrenfest's theorem.

Findings

Derived velocity and acceleration components for quantum state evolution.

Established a connection between Schrödinger and Newton equations.

Presented a formula linking probability distributions to the Born rule.

Abstract

Quantum observables can be identified with vector fields on the sphere of normalized states. The resulting vector representation is used in the paper to undertake a simultaneous treatment of macroscopic and microscopic bodies in quantum mechanics. Components of the velocity and acceleration of state under Schr\"odinger evolution are given for a clear physical interpretation. Solutions to Schr\"odinger and Newton equations are shown to be related beyond the Ehrenfest results on the motion of averages. A formula relating the normal probability distribution and the Born rule is found.