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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The time evolution of the mean values of the standard position, velocity and momentum operators, for a relativistic Dirac particle in a one-dimensional box, as well as for a free Dirac particle on a line with a hole, are studied. By considering the cases of “free” particle, confined particle and particle with a delta function interaction, it is shown that the Ehrenfest-type theorems for these operators are not always valid.

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