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Spectrum of the relativistic energy ratio ${\epsilon }_{{n}_{1},{n}_{2}}^{(\pm )}={E}_{{n}_{1},{n}_{2}}^{(\pm )}/{m}_{0}{c}^{2}$ . The upper and lower curves represent ${\epsilon }_{{n}_{1},{n}_{2}}^{(+)}$ and ${\epsilon }_{{n}_{1},{n}_{2}}^{(-)}$ , respectively, as a function of α for any arbitrary ordered pair of quantum numbers (n 1, n 2).
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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 C6v.
Closed-form bound-state solutions for this problem are obtained and the nonrelativistic limit of our results are clearly discussed. Due to an...
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Citations
... The BC imposes a phase relation on the wave-function components ψ 1,2 (s) at ∂Ω and provides a quantization condition whose solutions are the eigenstates of the HamiltonianĤ NB associated with the NB. Alternative BCs for the confinement of relativistic particles to a bounded domain are proposed in [70,72] and based on the 'MIT' bag model [73]. ...
Rectangular billiards have two mirror symmetries with respect to perpendicular axes and a twofold (fourfold) rotational symmetry for differing (equal) side lengths. The eigenstates of rectangular neutrino billiards (NBs), which consist of a spin-1/2 particle confined through boundary conditions to a planar domain, can be classified according to their transformation properties under rotation by π (π/2) but not under reflection at mirror-symmetry axes. We analyze the properties of these symmetry-projected eigenstates and of the corresponding symmetry-reduced NBs which are obtained by cutting them along their diagonal, yielding right-triangle NBs. Independently of the ratio of their side lengths, the spectral properties of the symmetry-projected eigenstates of the rectangular NBs follow semi-Poisson statistics, whereas those of the complete eigenvalue sequence exhibit Poissonian statistics. Thus, in distinction to their nonrelativistic counterpart, they behave like typical quantum systems with an integrable classical limit whose eigenstates are non-degenerate and have alternating symmetry properties with increasing state number. In addition, we found out that for right triangles which exhibit semi-Poisson statistics in the nonrelativistic limit, the spectral properties of the corresponding ultrarelativistic NB follow quarter-Poisson statistics. Furthermore, we analyzed wave-function properties and discovered for the right-triangle NBs the same scarred wave functions as for the nonrelativistic ones.
