Velocity field and operator in (non relativistic) quantum mechanics

TL;DR

This paper introduces a new tensorial definition of velocity fields in non-relativistic quantum mechanics, including a novel velocity operator for spin 1/2 particles that accounts for internal zitterbewegung motion.

Contribution

It proposes a new tensor-based framework for quantum velocity fields and introduces a non-relativistic velocity operator incorporating internal spin motion, extending previous models.

Findings

Decomposition of velocity field into mean motion and internal zitterbewegung.

Velocity field associated with zitterbewegung is solenoidal.

Local angular velocity aligns with the spin vector.

Abstract

Starting from the formal expressions of the hydrodynamical (or ``local'') quantities employed in the applications of Clifford Algebras to quantum mechanics, we introduce --in terms of the ordinary tensorial framework-- a new definition for the field of a generic quantity. By translating from Clifford into tensor algebra, we also propose a new (non-relativistic) velocity operator for a spin 1/2 particle. This operator is the sum of the ordinary part p/m describing the mean motion (the motion of the center-of-mass), and of a second part associated with the so-called zitterbewegung, which is the spin ``internal'' motion observed in the center-of-mass frame. This spin component of the velocity operator is non-zero not only in the Pauli theoretical framework, i.e. in presence of external magnetic fields and spin precession, but also in the Schroedinger case, when the wave-function is a spin eigenstate. In the latter case, one gets a decomposition of the velocity field for the Madelung fluid into two distinct parts: which constitutes the non-relativistic analogue of the Gordon decomposition for the Dirac current. We find furthermore that the zitterbewegung motion involves a velocity field which is solenoidal, and that the local angular velocity is parallel to the spin vector. In presence of a non-constant spin vector (Pauli case) we have, besides the component normal to spin present even in the Schroedinger theory, also a component of the local velocity which is parallel to the rotor of the spin vector.