A velocity field and operator for Spinning Particles in (nonrelativistic) quantum mechanics

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 language-- 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 appears as 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 (CMF). This spin component of the velocity operator is non-zero not only in the Pauli theoretical framework, i.e., in the presence of external electromagnetic fields with a non-constant spin function, but also in the Schroedinger case, when the wave-function is a spin eigenstate. Thus, one gets even in the latter case 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. Explicit calculations are presented for the velocity field in the particular cases of the hydrogen atom, of the spherical well potential, and of an electron in a uniform magnetic field. 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 the presence of a non-uniform spin vector (Pauli case) we have, besides the component of the local velocity normal to the spin (present even in the Schroedinger theory), also a component which is parallel to the rotor of the spin vector.