Perspectives on determinism in quantum mechanics: Born, Bohm, and the “Quantal Newtonian” laws

Abstract

Quantum mechanics has a deterministic Schr\"{o}dinger equation forthe wave function. The G\"{o}ttingen-Copenhagen statisticalinterpretation is based on the Born Rule that interprets the wavefunction as a `probability amplitude'. A precept of thisinterpretation is the lack of determinism in quantum mechanics. TheBohm interpretation is that the wave function is a source of a fieldexperienced by the electrons, thereby attributing determinism toquantum theory. In this paper we present a new perspective on suchdeterminism. The ideas are based on the equations of motion or`Quantal Newtonian' Laws obeyed by \emph{each electron}. TheseLaws, derived from the temporal and stationary-state Schr\"{o}dingerequation are interpreted in terms of `classical' fields whosesources are quantal expectations of Hermitian operators taken withrespect to the wave function. According to the Second Law,\emph{each electron} experiences an \emph{external} field - thequantal Coulomb-Lorentz law. It also experiences an \emph{internal}field representative of properties of the system: correlations dueto Coulomb repulsion and Pauli principle; the density; kineticeffects; and an internal magnetic field component. There is a\emph{response} field. The First Law states that the sum of theexternal and internal fields experienced by each electron vanishes.These fields are akin to those of classical physics: They pervadeall space; their structure is descriptive of the quantum system; theenergy of the system is stored in these fields. It is in theclassical behavior of these fields, which arise from quantal sourcesthat one may then speak of determinismin quantum mechanics.