Quantization of the Magnetic Moment of a Magnetic Multipole

Abstract

Based on the electric multipole theory, which has been extensively studied in electrostatics, and by exploiting the symmetry in the vacuum field effects of electric and magnetic fields in Maxwell’s equations, this work presents a comprehensive classical study of magnetic multipole moments.

We first define the magnetic monopole according to the Dirac–Schwinger quantization condition, and then extend the analysis to include quantum effects from the perspective of quantum mechanics principles. To carry out this study, we employ the generating function method for spherical harmonics (Legendre functions).

It is shown, through the expansion of the generating function, that the terms of the series correspond to multipole moments: the constant term represents the monopole, the second term the dipole, and so on.

Our results indicate that the magnetic monopole can exist in nature under the quantization condition proposed by Dirac and Schwinger. Furthermore, we find that the quantization conditions can be determined mathematically from the recurrence relations that connect different spherical harmonics of various orders, where both the functions and their derivatives are quantized. This is expressed through the dependence of the orbital and magnetic angular momentum quantum numbers (l,m)(l, m)(l,m), respectively.