Newton's law from quantum mechanics: macroscopic bodies in the vacuum

Kenichi Konishi

Published 2022-09-15Version 1

Newton's force law $\frac{d {\bf P}}{dt} = {\bf F}$ is derived from the Schr\"odinger equation for isolated macroscopic bodies in the vacuum. First we identify three elements that ensure well-defined unique center-of-mass position and momentum for a macroscopic body at each instant of time, i.e., a classical trajectory. They are (i) Heisenberg's uncertainty relations, (ii) absence of the diffusion of the wave packet for the center of mass of a macroscopic body, due to its large mass, and (iii) a finite body-temperature which implies a radiating, metastable state of the body - a mixed state, with decoherence caused by entanglement with the photons it emits and which carry away information. Newton's equation follows from the Ehrenfest theorem, as we explicitly verify for a macroscopic body in weak gravitational forces, in a harmonic potential, and under constant external electromagnetic fields slowly varying in space. Corrections due to its finite size such as the gravitational tidal forces known in classical physics, also appear correctly, as can be checked by application of perturbation theory to the Ehrenfest theorem. The present work in several ways complements and strengthens the well-known view that the emergence of classical physics in quantum mechanics is due to environment-induced decoherence, but sharpens further our understanding of the problem, by emphasizing the (body) temperature as a key element and demonstrating an explicit derivation of classical equations of motion for the CM of a macroscopic body in the vacuum.