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Physica Scripta

Ehrenfest-Type Theorems for a One-Dimensional Dirac Particle

Vidal Alonso Secades, Salvatore De Vincenzo

Published Apr 1, 2000• 9 citations
Publisher

Abstract

The time evolution of the mean values of the standard position, velocity and momentum operators, for a relativistic Dirac particle in a one-dimensional box, as well as for a free Dirac particle on a line with a hole, are studied. By considering the cases of “free” particle, confined particle and particle with a delta function interaction, it is shown that the Ehrenfest-type theorems for these operators are not always valid.

DOI
10.1238/physica.regular.061a00396

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The time evolution of the mean values of the standard position, velocity and momentum operators, for a relativistic Dirac particle in a one-dimensional box, as well as for a free Dirac particle on a line with a hole, are studied. By considering the cases of “free” particle, confined particle and particle with a delta function interaction, it is shown that the Ehrenfest-type theorems for these operators are not always valid.
We calculate formal time derivatives of the mean values of the standard position, velocity, and mechanical momentum operators, i. e., the Ehrenfest theorem for a one-dimensional Dirac particle in the coordinate representation.We show that these derivatives contain boundary terms that essentially depend on the values taken there by characteristic bilinear densities.We do not automatically take the boundary terms to vanish (as is usually done); nevertheless, we relate the boundary terms to similar terms that must be zero if one requires the hermiticity of certain specific unbounded operators.Throughout the article, we thoroughly discuss and illustrate all these aspects, which include the relations to certain boundary conditions.To clarify, we call our approach formal because all operations involving operators (for example, some operators products) are performed without respecting the restrictions imposed by the sets of functions on which the self-adjoint operators can act.Moreover, the Dirac Hamiltonian that we consider in our calculations contains a potential that is the time component of a Lorentz two-vector; nevertheless, we also obtain and concisely discuss the Ehrenfest theorem for a Hamiltonian with the most general Lorentz potential in (1+1) dimensions.
The canonical commutation relation between the quantum mechanical operators of momentum and coordinate and the Ehrenfest theorem in the free particle case are derived solely from the rule of differentiation of the delta function [xδ′(x)=−δ(x)x\delta'(x) = - \delta(x)xδ′(x)=−δ(x)].
Ehrenfest’s theorem states that as a quantum state evolves in time, the rate of change of the expectation value of momentum is equal to the expectation value of the force. In the familiar ‘‘particle-in-a-box,’’ however, the probability of finding the particle at a wall—the only place where forces act—is zero, so at first glance it appears that the expectation value of the force should vanish for any state, which would violate Ehrenfest‘s theorem. This argument is flawed, however, since the forces at the walls of the box are infinite. We consider the box as a limit of a very deep square well and confirm that Ehrenfest’s theorem emerges safe and sound.
"We study the issues of average forces and the Ehrenfest theorem for a particle restricted to a semi-infinite interval by an impenetrable wall. We consider and discuss two specific cases: (i) a free particle in an infinite step potential, and (ii) a free particle on a half-line. In each situation, we show that the mean values of the position, momentum and force, as functions of time, verify the Ehrenfest theorem (the state of the particle being a general wave packet that is a continuous superposition of the energy eigenstates for the Hamiltonian). However, the involved force is not the same in each case. In fact, we have the usual external classical force in the first case and a type of nonlocal boundary quantum force in the second case. In spite of these different forces, the corresponding mean values of these quantities give the same results. Accordingly, the Ehrenfest equations in the two situations are equivalent. We believe that a careful and clear consideration of how the two cases differ but, in the end, agree, is pertinent, and has not been included in the literature."
In this article, we investigate Ehrenfest's theorem from the Dirac equation in a noncommutative phase-space where we calculate the time derivative of the position and the kinetic momentum operators for Dirac particles in interaction with electromagnetic field and within a noncommutative setting. This allows examining the effect of the phase-space noncommutativity on Ehrenfest's theorem. Knowing that with both the linear Bopp-Shift and Moyal-Weyl product, the noncommutativity is inserted.