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

For a positively charged particle, the repulsive continuous potentials of three adjacent [111] string in a silicon crystal create a shallow potential well with the symmetry of an equilateral triangle, which is described by the \({{C}_{{3{v}}}}\) group. The quantum motion of a particle in such a well is of interest in the context of quantum chaos phenomena. A previously developed numerical procedure for determining energy levels and wave functions of stationary states, which takes into account the symmetry of this problem, is applied to investigate the transverse motion of channeled positrons with energies of 5, 6, and 20 GeV. A classification of stationary states in transverse motion is proposed based on the theory of group representations. The wave functions of stationary states in an axially symmetric potential well are also determined, and the modification of these functions under the influence of a perturbation with equilateral triangle symmetry is demonstrated. In the upper region of the triangular potential well, classical motion is chaotic for the vast majority of initial conditions. The structure of the obtained wave functions in this region exhibits characteristic features predicted by the theory of quantum chaos.