  The parity violation in nuclear reactions led to the discovery of the new
class of toroidal multipoles. Since then, it was observed that toroidal
multipoles are present in the electromagnetic structure of systems at all
scales, from elementary particles, to solid state systems and metamaterials.
The toroidal dipole ${\bf T}$ (the lowest order multipole) is the most common.
In quantum systems, this corresponds to the toroidal dipole operator $\hat{\bf
T}$, with the projections $\hat{T}_i$ ($i=1,2,3$) on the coordinate axes. Here
we analyze a quantum particle in a system with cylindrical symmetry, which is a
typical system in which toroidal moments appear. We find the expressions for
the Hamiltonian, momenta, and toroidal dipole operators in adequate curvilinear
coordinates, which allow us to find analytical expressions for the
eigenfunctions of the momentum operators. While the toroidal dipole is
hermitian, it is not self-adjoint, but in the new set of coordinates the
operator $\hat{T}_3$ splits into two components, one of which is (only)
hermitian, whereas the other one is self-adjoint. The self-adjoint component is
the one that is physically significant and represents an observable.
Furthermore, we numerically diagonalize the Hamiltonian and the toroidal dipole
operator and find their eigenfunctions and eigenvalues. We write the partition
function and calculate the thermodynamic quantities for a system of ideal
particles on a torus. Besides proving that the toroidal dipole is self-adjoint
and therefore an observable (a finding of fundamental relevance) such systems
open up the possibility of making metamaterials that exploit the quantization
and the quantum properties of the toroidal dipoles.
