Abstract
The properties of the eigenvalue problem of the one-dimensional Dirac operator are discussed in terms of the mutual relations between vector, scalar and pseudo-scalar contributions to the potential. Relations to the exact solubility are analyzed.
Highlights
We are concerned with the simplest quantum system: a single particle in a one-dimensional space
Since angular momentum does not exist in one dimension, spin effects, in particular spin-orbit interaction, are absent in one-dimensional Dirac equation
In some cases the reduction of the one-dimensional Dirac equation to the Schrödinger form is associated with the factorization of the Hamiltonian to two mutually Hermite conjugate operators which, in turn, is the first step in an analysis of super-symmetry (SUSY) properties of the equation
Summary
We are concerned with the simplest quantum system: a single particle in a one-dimensional space. From the quantum-chemical perspective the most important are applications of the onedimensional Dirac equation to the description of quantum systems whose spectrum has an energy gap and whose properties may be conveniently simulated by a properly taylored Dirac equation To this category belong works on the edge states, on the Landau levels, on the theoretical modeling of graphene [12,13,14,15], including external electromagnetic fields [16], and advanced studies on radially twisted carbon nanotubes [17, 18], specific calculations relevant to the theoretical simulation of one-dimensional graphene structures performed, among others, in [19, 20]. The paper is dedicated to our old friend, Professor Ramon Carbó-Dorca on the occasion of his 80th birthday
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