Exact solution of the problem of an electron in a magnetic field consisting of a uniform field and arbitrarily arranged magnetic strings parallel to the uniform field

Abstract

It is shown that the requirements of finiteness, single-valuedness, and definiteness of the wave function and probability current density necessarily lead to the fact that as a magnetic string is approached the wave functions of the electron must decrease in modulus more rapidly than the square root of the distance from the string (a magnetic string is an infinitely thin solenoid carrying a finite magnetic flux). The energy spectrum of an electron is obtained. In general the spectrum is identical to the spectrum in the absence of strings. The general form of the eigenfunctions of the ground state and an operator whose powers give the eigenfunctions of the excited states are found. When there is only one string with magnetic flux which is not a multiple of twice the flux quantum another equidistant sequence of eigenvalues appears in the energy spectrum. This sequence is shifted with respect to the main sequence by a fraction equal to the positive fractional part of the quotient obtained by dividing the magnetic flux by twice the flux quantum. This sequence starts from a level whose number equals the number of the remaining magnetic strings. The wave functions for these special states are also obtained.