Newton’s equations from quantum mechanics for a macroscopic body in the vacuum

Abstract

Newton’s force law [Formula: see text] is derived from the Schrödinger equation for isolated macroscopic bodies, composite states of e.g. [Formula: see text] atoms and molecules, at finite body temperatures. We first review three aspects of quantum mechanics (QM) in this context: (i) Heisenberg’s uncertainty relations for their center of mass (CM), (ii) the diffusion of the CM wave packet, and (iii) a finite body temperature which implies a metastable (mixed-) state of the body: photon emissions and self-decoherence. They explain the origin of the classical trajectory for a macroscopic body. The ratio between the range [Formula: see text] over which the quantum fluctuations of its CM are effective, and the body’s (linear) size [Formula: see text], [Formula: see text] or [Formula: see text], tells whether the body’s CM behaves classically or quantum mechanically, respectively. In the first case, Newton’s force law for its CM follows from the Ehrenfest theorem. We illustrate this for weak gravitational forces, a harmonic-oscillator potential, and for constant external electromagnetic fields slowly varying in space. The derivation of the canonical Hamilton equations for many-body systems is also discussed. Effects due to the body’s finite size such as the gravitational tidal forces appear in perturbation theory. Our work is consistent with the well-known idea that the emergence of classical physics in QM is due to the environment-induced decoherence, but complements and completes it, by clarifying the conditions under which Newton’s equations follow from QM, and by deriving them explicitly.