The eigenvalues and eigenfunctions of the toroidal dipole operator

Abstract

Abstract The toroidal dipole represents the lowest order in the family of toroidal multipoles. They appear in physics at all scales, from particle to condensed matter physics, and metamaterials. Nevertheless, in small systems they have to be investigated in the context of quantum mechanics and quantum operators are introduced for the projections of the toroidal dipole on the Cartesian axes. Here we give analytical expressions for the eigenvalues and generalized eigenfunctions of T ˆ 3 –the z-axis projection of the toroidal dipole operator–in a system consisting of a particle confined in a thin film bent into a torus shape. The eigenfunctions are not square integrable, so they do not belong to the Hilbert space of wave functions, but they can be interpreted in the formalism of rigged Hilbert spaces as kernels of distributions. We find the quantization rules for the eigenvalues, which are essential for describing measurements of T ˆ 3 . As an example, we estimate the splitting of the energy levels due to the interaction of the toroidal dipole with an external current.