Wave Functions of Positrons Channeling in [111] Direction of a Silicon Crystal

For a positively charged particle, the repulsive uniform potentials of the three neighboring [111] chains of the silicon crystal form a small potential well with the symmetry of an equilateral triangle is described by the C_3v group. The motion of a quantum particle in such a well is of interest in terms of manifestations of quantum chaos. A previously developed procedure for numerically finding the energy levels and wave functions of stationary states, taking into account the symmetry of this problem, is used to study the transverse motion of the channeling positrons with energies of 5, 6 and 20 GeV. A classification of stationary states of transverse motion of a positron is given based on the theory of group representations. The wave functions of the stationary states in an axially symmetric potential well are also found, and it is shown how these functions are modified under the influence of a perturbation with the symmetry of an equilateral triangle. In the upper part of the triangular potential well, the classical motion is chaotic for the majority of initial conditions. The structure of the wave functions in this domain has the features predicted by the quantum chaos theory.