On the classical spin motion of spin-one particles

B. Linet

Kenneth Rafanelli

P. Kyborg

John J. Stachel, Jerzy F Plebanski

This is the first of a series of papers developing the classical theory of a spinning particle. The equations of motion will be derived from a Lagrangian, and solutions for the classical trajectory and spin precession in external fields will be given. In this paper an abstract spin vector is introduced to characterize the spin of a classical particle. Lagrangians for the classical trajectories and for the motion of the abstract spin vector are derived from corresponding quantum-mechanical Lagrangians by the WKBJ approximation method for nonrelativistic and relativistic particles. The equations of motion for the trajectory and the abstract spin vector following from the extremalization of these Lagrangians are given. The equations of motion for the precession in an external electromagnetic field of the spin vector (or tensor) in space–time is derived from the equations of motion for the abstract spin vector. In the relativistic case, they are equivalent to the Bargmann–Michel–Telegdi equations [Phys. Rev. Lett. 2, 435 (1959)]. The relationship between the ensemble and single-particle points of view is also elucidated.

Jędrzej Śniatycki

Francesco R. Ruggeri

Quantum mechanics is often compared with classical mechanics, but it seems that the momentum/spatial treatment of bound quantum mechanics provides for a particle-wave duality with both features having an experimental basis. In bound state quantum mechanics, one has a density pattern often similar to a two slit pattern i.e. with humps, yet at the same time one may knock a particle out of the bound state and obtain a probability distribution a(p)a(p) of momentum states. Describing a particle simply by its momentum even in the bound state seems to be classical as does using pp/2m to describe the kinetic energy of a plane wave in the decomposition of W(x), the wavefunction. Wave aspects are linked to classical particle features like momentum and do not change the particle energy pp/2m. The wave, we argue, may be related to some type of periodic cyclic forward backward motion which creates a spatial probability needed to estables resolution on the order of a wavelength proportional to 1/p. It is often not seen unless there is a potential or other probability waves which may interfere with it In this note, we consider the scenario that momentum (i.e. mov in the nonrelativistic case) is directly linked to a kind of forward-backward cyclic motion of a quantum particle regardless of its spin. The fact that spin exists suggests some kind of internal motion. In the case of a photon, it may be directly linked to angular motion of energy, although for spin ½ matters are more complicated. This motion again does not change classical energy pp/2m unless spin interacts with a magnetic field etc. We argue that a bound state is stochastic and so various free particle momentum states exist without being restricted to an x position. This is the particle scenario. The internal cycling motion, however, gives rise to a resolution of the order of 1/p suggesting a spatial distribution to account for such a resolution. Furthermore, the internal cycling (i.e. forward and backwards motion) even for a particle traveling to the right requires two dimensions to describe overall forwards and backwards motion. The spatial density of the cycling motion may be associated with internal changes in velocity, but overall there is a particle with mass mo moving with v (nonrelativistic case) and kinetic energy pp/2m. This holds for a free particle, but also for a particle in an infinite potential well with a single average |p|. An issue occurs, however, if one wishes to form an average of different p values at the same point x. In such a case the spatial probability density exp(i pj x) differs for each p at x and so the average [p1 exp(i p1 xo) + p2 exp(i p2 xo) ] / [ exp(i p1 xo) + exp(i p2 xo)] becomes relevant. In other words, one takes the average momentum, say p1, and links it to the spatial probability i.e. multiplies two averages and then divides by spatial probability to find a momentum flow at a specific point within a wavelength region. For the particle aspect, such a resolution is non-issue. The particle is the same at each x point in space.