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Physics Letters A

Position operators for systems exhibiting the special relativistic relation between momentum and velocity

Robert F. O’Connell, Eugene Paul Wigner

Published Sep 1, 1978• 25 citations
Publisher
DOI
10.1016/0375-9601(78)90317-1

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The free relativistic particle, by definition, has to move in an inertial reference frame with uniform velocity less than the speed of light. The corresponding movement of a material quantum particle describes a wave packet, composed of matter waves—solutions of the Schr?dinger equation. The maximum of packet, corresponding to the largest probability to find the particle, has to move with the same uniform velocity, defined by the initial condition. It has been shown that the traditional definition of the quantum momentum operator i.e. taking it to correspond to the special relativity theory, relativistic momentum, cannot produce precise description of a relativistic matter particle. Different definitions are investigated and one that solves this issue is found. Obtained original expression of relativistic kinetic energy operator creates new possibilities for relativistic quantum systems theory.
The operators −i∇, the translation generators, are related to mass and velocity by a simple argument that determines the form of the Hamiltonian, in terms of the translation generators, along with the Galilei generators, for a nonrelativistic particle. For a relativistic particle, a similar discussion is given for the analogous problem of determining the position operators for an irreducible unitary representation of the Poincaré group, at least well enough to identify the velocity operators and thus identify the momentum.
For relativistic particles with spin-0 (satisfying the Klein-Gordon equation) and spin-1/2 (satisfying the Dirac equation) in infinitely deep potential well, matrix elements for the coordinate, momentum and the velocity operators are calculated. In the limit of large quantum numbers, these matrix elements give the corresponding classical quantities (nowbeing related quantities in special relativity) and satisfy exact classical relations. These results show that the Heisenberg correspondence principle is applicable to such relativistic systems.