Abstract
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.
Highlights
The fundamental and at the first glance the simplest problem of quantum mechanics is description of moving in vacuum free particle
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
The quantum mechanical momentum operator −i ∇ is associated with the classical relativistic momentum p = γ mv, present in the energy expression
Summary
The fundamental and at the first glance the simplest problem of quantum mechanics is description of moving in vacuum free particle. The main argument for this association is the photon energy definition E = pc , producing after Schrödinger equation solution the electromagnetic waves, whose packet moves with the speed of light and defines the probability distribution of a photon This definition, applied to a free particle, whose m ≠ 0, leads to a result not consistent with fundamental conditions, following from the inertial reference system definition, stating that a free particle in an arbitrary, such system has to move with uniform velocity, defined by the initial condition and less than speed of light. From our considerations it follows that both definitions cannot produce packets, satisfying the mentioned conditions.
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