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arXiv:0901.3446 · Physical Review A

Confinement limit of Dirac particles in scalar one-dimensional potentials

Razmik G. Unanyan, Johannes Otterbach, Michael Fleischhauer

Published Apr 10, 2009• 8 citations
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Abstract

We present a general proof that Dirac particles cannot be localized below their Compton length by symmetric but otherwise arbitrary scalar potentials. This proof does not invoke the Heisenberg uncertainty relation and thus does not rely on the nonrelativistic linear momentum relation. Further it is argued that the result is also applicable for more general potentials, as, e.g., generated by nonlinear interactions. Finally a possible realization of such a system is proposed.

DOI
10.1103/physreva.79.044101

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We present a general proof that Dirac particles cannot be localized below their Compton length by symmetric but otherwise arbitrary scalar potentials. This proof does not invoke the Heisenberg uncertainty relation and thus does not rely on the nonrelativistic linear momentum relation. Further it is argued that the result is also applicable for more general potentials, as, e.g., generated by nonlinear interactions. Finally a possible realization of such a system is proposed.
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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 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.
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.
The Dirac equation with vector plus scalar surface delta interactions supported by a sphere is exactly solved for all partial waves and parities. Unlike the relativistic one-dimensional delta potentials, surface delta interactions with finite values of the vector and scalar coupling constants require a minimum strength to bind particles. Strong scalar couplings confine relativistic particles at high energies inside the sphere, while the confinement is no longer possible for strong vector potentials. It is also found that the potential causes no effects on the scattering phase shift in the zero-range limit. Therefore, the well known Dirac-Kronig-Penney model in one dimension cannot be generalised to three-dimensional crystal lattices.
For the Dirac equation in one space dimension with a potential of the Lorentz scalar type, we present a complete solution for the problem of constructing a transparent potential. This is a relativistic extension of the Kay-Moses method which was developed for the nonrelativistic Schr\"odinger equation. There is an infinite family of transparent potentials. The potentials are all related to solutions of a class of coupled, nonlinear Dirac equations. In addition, it is argued that an admixture of a Lorentz vector component in the potential impairs perfect transparency.
We show that care is required in formulating the nonrelativistic limit of generalized Dirac Hamiltonians which describe particles and antiparticles interacting with static electric and/or gravitational fields. The Dirac-Coulomb and the Dirac-Schwarzschild Hamiltonians, and the corrections to the Dirac equation in a non-inertial frame, according to general relativity, are used as example cases in order to investigate the unitarity of the standard and "chiral" approaches to the Foldy-Wouthuysen transformation, and spurious parity-breaking terms. Indeed, we find that parity-violating terms can be generated by unitary pseudo-scalar transformations ("chiral" Foldy-Wouthuysen transformations). Despite their interesting algebraic properties, we find that "chiral" Foldy-Wouthuysen transformations change fundamental symmetry properties of the Hamiltonian and do not conserve the physical interpretation of the operators. Supplementing the discussion, we calculate the leading terms in the Foldy-Wouthuysen transformation of the Dirac Hamiltonian with a scalar potential (of the (1/r)-form and of the confining radially symmetric linear form), and obtain compact expressions for the leading higher-order corrections to the Dirac Hamiltonian in a non-inertial rotating reference frame "Mashhoon term").