Some typical features of relativistic boundary conditions across δ-function potential

嘉三 中沢

Chandana Roy

Chandana Roy

We have developed a unified perspective for boundary conditions concerned with (I) a direct relativistic treatment of a \ensuremath{\delta}-function potential and (II) the \ensuremath{\delta}-function limit of solutions for relativistic treatment of peaked potential. We have presented a critical discussion of our findings, especially in the context of (i) certain conditions related very importantly to this unified perspective and (ii) the difficulty in making a choice between the approaches (I) and (II) on the basis of mathematical analyses alone.

I. Yanetka

The solution of the one-dimensional Dirac equation is examined for the symmetrical double delta-function potential. A formula for the relativistic transmission coefficient is derived from the solution. The maximum condition as well as the minimum condition for the transmission coefficient is determined. The transmission coefficient is shown to be equal to unity in its maximum. In its minimum, it is smaller than the transmission coefficient for the single delta-function potential of the twofold strength. Further, some differences between the relativistic and non-relativistic transmission coefficient are presented. It is found that the wave numbers, at which the maxima and minima occur according to the non-relativistic treatment, are shifted by the relativistic treatment. Unlike the non-relativistic transmission coefficient, the relativistic transmission coefficient does not approach unity in the range of high energies. It oscillates between unity and a constant. The value of this constant depends on the strength of the delta-interaction. Thus, the relativistic corrections to the transmission coefficient for the symmetrical double delta-function potential are found to represent significant shifts in its characteristics.

G. K. Artimovich, V. I. Ritus, P. J. Shepherd

Ch. Grosche, Hamburg (Germany). Projekttraeger ''Erforschung der Kondensierten Materie: Synchrotronstrahlung'' Deutsches Elektronen-Synchrotron (DESY)

#delta#'-function perturbations and Neumann boundary conditions are incorporated into the path integral formalism. The starting point is the consideration of the path integral representation for the one dimensional Dirac particle together with a relativistic point interaction. The non-relativistic limit yields either a usual #delta#-function or a #delta#'-function perturbation; making their strengths infinitely repulsive one obtains Dirichlet, respectively Neumann boundary conditions in the path integral. (orig.)