The complete set of eigenstates in one type of N-multiple quantum wells

Abstract

We provide a method to obtain a complete set of eigenstates analytically and the corresponding eigenvalues in one type of N-multiple well potentials. The quantum well profile can be adjusted by many different parameters, which could be applicable to many different physical situations. We demonstrate explicitly the series of eigenstates in double and triple well potentials, which can be used to describe Josephson oscillations and other tunneling dynamics conveniently. The analytic solutions can be used to discuss the topological vector potential hidden in the eigenstates of quantum wells, through performing our recently proposed method of extending Dirac’s monopole theory to a complex plane. Our results suggest that each node of eigenfunctions corresponds to the merging of a pair of magnetic monopoles with inverse charge. The underlying monopoles can be used to determine the phase jump of a pure real wave function with nodes, as observed from an experimental viewpoint.