Long time dynamical behavior of the confined quantum time of arrival eigenfunctions

Abstract

Dynamical phenomena unique to the confined quantum time of arrival (CTOA) operator's eigenfunctions for vanishing potentials are observed when these eigenfunctions are evolved over a long interval of time. The first is the emergence of pseudo fractional quantum revival behavior, which is similar to standard fractional quantum revival behavior but which occur at instants of time which are numerically equal to the sum of the eigenvalue corresponding to the eigenfunction and a fraction of the standard quantum revival time. The second is the emergence of quantum fractals in the position static probability densities and, in the case of the nonnodal CTOA eigenfunctions, in the time static probability densities as well as in the quantum carpets generated by the time evolved free particle CTOA eigenfunctions. The results show that there is a need to determine the analytic form of the CTOA operator's eigenfunctions for an arbitrary potential if one is to generalize the results obtained.