A relativistic 'free' particle in a one-dimensional box is studied. The impossibility of the wavefunction vanishing completely at the walls of the box is proven. Various physically acceptable boundary conditions that allow non-trivial solutions for this problem are proposed. The non-relativistic limits of these results are obtained. The problem of a particle in a spherical box, which presents the same type of difficulties as the one-dimensional problem, is also considered. Resumen. Se considera el problema de una partícula 'libre' relativista en una caja unidimensional. Se comprueba la imposibilidad de anular completamente la función de onda en las paredes de la caja. Se proponen diversas condiciones de frontera físicamente aceptables que permiten encontrar soluciones no triviales para este problema. Se discute el límite no relativista de estos resultados. También consideramos el problema de una partícula en una caja esférica, el cual presenta el mismo tipo de dificultades que el problema unidimensional.

The most general relativistic boundary conditions (BCs) for a 'free' Dirac particle in a one-dimensional box are discussed. It is verified that in the Weyl representation there is only one family of BCs, labelled with four parameters. This family splits into three sub-families in the Dirac representation. The energy eigenvalues as well as the corresponding non-relativistic limits of all these results are obtained. The BCs which are symmetric under space inversion P and those which are CP T invariant for a particle confined in a box, are singled out.

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We construct general solutions of the time-dependent Dirac equation in (1+1) dimensions with a Lorentz scalar potential, subject to the so-called Majorana condition, in the Majorana representation. In this situation, these solutions are real-valued and describe a one-dimensional Majorana single particle. We specifically obtain solutions for the following cases: a Majorana particle at rest inside a box, a free (i.e., in a penetrable box with the periodic boundary condition), in an impenetrable box with no potential (here we only have four boundary conditions), and in a linear potential. All these problems are treated in a very detailed and systematic way. In addition, we obtain and discuss various results related to real wave functions. Finally, we also wish to point out that, in choosing the Majorana representation, the solutions of the Dirac equation with a Lorentz scalar potential can be chosen to be real but do not need to be real. In fact, complex solutions for this equation can also be obtained. Thus, a Majorana particle cannot be described only with the Dirac equation in the Majorana representation without explicitly imposing the Majorana condition.

We present formal 1D calculations of the time derivatives of the mean values of the position (x) and momentum (p) operators in the coordinate representation. We call these calculations formal because we do not care for the appropriate class of functions on which the involved (self-adjoint) operators and some of its products must act. Throughout the paper, we examine and discuss in detail the conditions under which two pairs of relations involving these derivatives (which have been previously published) can be formally equivalent. We show that the boundary terms present in d{x}/dt and d{x}/dt can be written so that they only depend on the values taken there by the probability density, its spatial derivative, the probability current density and the external potential V= V9 (x) V = V(x). We also show that d(p)/dt is equal to -dv /dx=(FQ) plus a boundary term (Fq = aQ/ax)is the quantum force and Q is the Bohm's quantum potential). We verify that (fq) is simply obtained by evaluating a certain quantity on each end of the interval containing the particle and by subtracting the two results. That quantity is precisely proportional to the integrand of the so-called Fisher information in some particular cases. We have noted that fQ has a significant role in situations in which the particle is confined to a region, even if V is zero inside that region.

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.