Confinement limit of the Dirac particle in one dimension

Abstract

For a particle of mass $m$ 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 $\ensuremath{\Delta}x=\sqrt{\ensuremath{\langle}{x}^{2}}\ensuremath{\rangle}g\ensuremath{\lambda}/2$ can be derived simply from the Dirac equation. Here $\ensuremath{\lambda}=\ensuremath{\hbar}/(\mathit{mc})$ is the Compton wavelength, $x$ is the particle coordinate, and $\ensuremath{\langle}{x}^{2}\ensuremath{\rangle}$ is the expectation value of ${x}^{2}$. 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.