Discrete symmetry approach to exact bound-state solutions for a regular hexagon Dirac billiard

Abstract

We consider solving the stationary Dirac equation for a spin-1/2 fermion confined in a two-dimensional quantum billiard with a regular hexagon boundary, using symmetry transformations of the point group C 6v . Closed-form bound-state solutions for this problem are obtained and the non-relativistic limit of our results are clearly discussed. Due to an adequate choice of confining boundary conditions the upper components of the planar Dirac-spinor eigenfunctions are shown to satisfy the corresponding hexagonal Schrödinger billiard, and the Dirac positive energy eigenvalues are proven to reduce directly to their Schrödinger counterparts in the non-relativistic limit. An illustrative application of our group theoretic method to the well-known square billiard problem has been explicitly provided. The success of our approach in solving equilateral-triangle, square and regular hexagon quantum billiards may well imply a possible applicability to other regular polygonal billiards. A quick look on nodal domains of the Schrödinger eigenfunctions for the hexagon billiard is also considered. Moreover, we have determined a number of distinct non-congruent polygonal billiards that have the same eigenvalue spectrum as that of the regular hexagon.