Relativistic Schroedinger equation for a particle with arbitrary spin. II

Alexander J. Silenko

N. J. Ionesco‐Pallas

Nickola Stefanov Todorov

Hartmut Wachter

This is the second part of a paper about a q-deformed analog of non-relativistic Schroedinger theory. It applies the general ideas of part I and tries to give a description of one-particle states on q-deformed quantum spaces like the braided line or the q-deformed Euclidean space in three dimensions. Hamiltonian operators for the free q-deformed particle in one as well as three dimensions are introduced. Plane waves as solutions to the corresponding Schroedinger equations are considered. Their completeness and orthonormality relations are written down. Expectation values of position and momentum observables are taken with respect to one-particle states and their time-dependence is discussed. A potential is added to the free-particle Hamiltonians and q-analogs of the Ehrenfest theorem are derived from the Heisenberg equations of motion. The conservation of probability is proved.

N. I. Ionescu-Pallas

A. Raiteri

The interpretation proposed in quant-ph/9812011 is extended to the general case of a non-relativistic particle moving in an arbitrary external potential. It is shown that, even in this general case, "particle" solutions exist which do not spread out with time, and remain well localized around their center of mass; it is postulated that these are the only solutions which represent individual physical particles. As a consequence two basic principles of standard QM, namely the superposition principle and the wave-function collapse, are shown to have no ontological meaning. Three simple applications of our approach are then examined: the free particle, the linear harmonic oscillator and the delta barrier potential; the corresponding "particle" solutions are explicitly shown. Finally, it is argued that the persisting confusion about the meaning of the wave-function (does it represent an individual particle or a statistical ensemble?) calls for a non-linear extension of the Schroedinger equation.