Time-dependent Aharonov-Bohm Hamiltonian and admissibility criteria of quantum wave functions

Abstract

Self-adjointness of the time-independent Aharonov-Bohm Hamiltonian is shown to allow a continuous family of different dynamics including those following from Pauli's criterion of rotational invariance, Aharonov-Bohm criterion of single valuedness and a version of Pauli's criterion appropriate to cylindrical symmetry suggested by Henneberger. A time-dependent flux F(t) linking the Aharonov-Bohm solenoid leads to the time-dependent AB Hamiltonian. Explicit solutions in cases with and without inaccessible regions for the charged particle rule out applicability of both versions of the Pauli criterion. The solutions contain one time-independent parameter α, integer values of which correspond to single-valued wave functions. Any real (integer or noninteger) value of α is allowed. Charge and current densities depend on α and F(t) only through the combination α-eF(t)/2π and lead to the conclusion that physical effects of changing the flux during an experiment can be understood as local effects of the electric field inevitably associated with changing magnetic flux.