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Physical Review A

More stringent confinement limit of the Dirac particle in one dimension

Jian-Yuan Cheng

Published May 26, 2011• 3 citations
Publisher

Abstract

Unanyan, Otterbach, and Fleischhauer [Phys. Rev. A 79, 044101 (2009)] found that the confinement limit of a one-dimensional Dirac particle in a symmetric potential is half its corresponding Compton length and can be derived from the Dirac equation. Then, Toyama and Nogami [Phys. Rev. A 81, 044106 (2010)] conjectured that a more stringent limit holds for any symmetric potential. I would like to show, in this Brief Report, that the confinement limit conjectured by Toyama and Nogami can be derived from the Dirac equation.

DOI
10.1103/physreva.83.054103

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A comment on the analogous confinement of a neutral particle to a quantum dot via noninertial effects

K. Bakke

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Unanyan, Otterbach, and Fleischhauer [Phys. Rev. A 79, 044101 (2009)] found that the confinement limit of a one-dimensional Dirac particle in a symmetric potential is half its corresponding Compton length and can be derived from the Dirac equation. Then, Toyama and Nogami [Phys. Rev. A 81, 044106 (2010)] conjectured that a more stringent limit holds for any symmetric potential. I would like to show, in this Brief Report, that the confinement limit conjectured by Toyama and Nogami can be derived from the Dirac equation.
For a particle of mass mmm that obeys the time-independent Dirac equation in one dimension with a symmetric potential, Unanyan et al. [Phys. Rev. A 79, 044101 (2009)] recently pointed out that the inequality $\ensuremath{\Delta}x=\sqrt{\ensuremath{\langle}{x}^{2}}\ensuremath{\rangle}>\ensuremath{\lambda}/2$ can be derived simply from the Dirac equation. Here $\ensuremath{\lambda}=\ensuremath{\hbar}/(\mathit{mc})$ is the Compton wavelength, xxx is the particle coordinate, and $\ensuremath{\langle}{x}^{2}\ensuremath{\rangle}$ is the expectation value of x2{x}^{2}x2. We conjecture that a new, more stringent limit $\ensuremath{\Delta}x\ensuremath{\geqslant}\ensuremath{\lambda}/\sqrt{2}$ holds for any symmetric potential. We present a model analysis on which the conjecture is based.
Unanyan, Otterbach, and Fleischhauer [Phys. Rev. A 79, 044101 (2009)] found that the confinement limit of a one-dimensional Dirac particle can be derived from the Dirac equation. The generalization of this problem to three dimensions is discussed in this paper. It shows that the three-dimensional Dirac particle in vector and scalar potentials has a confinement limit proportional to the modulus of expectation value of spin. This result obtained in Dirac equation is applicable for any Dirac particle confined in a finite region of space, even when vector and scalar potentials of quite general character are present. In addition, a Dirac particle confined in Lorentz scalar potential is discussed.
We show that the first order form of the Schrodinger equation proposed in [1] can be obtained from the Dirac equation in the non-relativistic limit. We also show that the Pauli Hamiltonian is obtained from this equation by requiring local gauge invariance. In addition, we study the problem of a spin up particle incident on a finite potential barrier and show that the known quantum mechanical results are obtained. Finally, we consider the symmetric potential well and show that the quantum mechanical expression for the quantized energy levels of a particle is obtained with periodic boundary conditions. Based on these conclusions, we propose that the equation introduced in [1] is the non-relativistic limit of the Dirac equation and more appropriately describes spin 1/2 particles in the non-relativistic limit.
We discuss the properties of 1D stationary pulses of light in an atomic ensemble with electromagnetically induced transparency in the limit of tight spatial confinement. When the size of the wave packet becomes comparable or smaller than the absorption length of the medium, it must be described by a two-component vector which obeys the one-dimensional two-component Dirac equation with an effective mass m;{*} and effective speed of light c;{*}. Then a fundamental lower limit to the spatial width in an external potential arises from Klein tunneling and is given by the effective Compton length lambda_{C}=variant Planck's over 2pi/(m;{*}c;{*}). Since c;{*} and m;{*} can be externally controlled and can be made small, it is possible to observe effects of the relativistic dispersion for rather low energies or correspondingly on macroscopic length scales.
Abstract In this contribution, we discuss the nonrelativistic limit of the Dirac equation for a neutral particle with a permanent electric dipole moment interacting with external fields in a noninertial frame. We show a case where the geometry of the manifold can play the role of a hard-wall confining potential due to noninertial effects, and can yield bound states analogous to a confinement of the spin-half neutral particle interacting with external fields to a quantum dot described by a hard-wall confining potential [33].