The quantum operator
T^3, corresponding to the projection of the toroidal moment on the
z axis, admits several self-adjoint extensions, when defined on the whole
R3 space.
T^3 commutes with
L^3 (the projection of the angular momentum operator on the
z axis) and they have a \textit{natural set of coordinates}
(k,u,ϕ) where
ϕ is the azimuthal angle. The second set of \textit{natural coordinates} is
(k1,k2,u), where
k1=kcosϕ,
k2=ksinϕ. In both sets,
T^3=−iℏ∂/∂u, so any operator that is a function of
k and the partial derivatives with respect to the \textit{natural variables}
(k,u,ϕ) commute with
T^3 and
L^3. Similarly, operators that are functions of
k1,
k2, and the partial derivatives with respect to
k1,
k2, and
u commute with
T^3. Therefore, we introduce here the operators
p^k≡−iℏ∂/∂k,
p^(k1)≡−iℏ∂/∂k1, and
p^(k2)≡−iℏ∂/∂k2 and express them in the
(x,y,z) coordinates. One may also invert the relations and write the typical operators, like the momentum
p^≡−iℏ∇ or the kinetic energy
H^0≡−ℏ2Δ/(2m) in terms of the "toroidal" operators
T^3,
p^(k),
p^(k1),
p^(k2), and, eventually,
L^3. The formalism may be applied to specific physical systems, like nuclei, condensed matter systems, or metamaterials. We exemplify it by calculating the momentum operator and the free particle Hamiltonian in terms of \textit{natural coordinates} in a thin torus, where the general relations get considerably simplified.
The orthogonality requirement usually specified for the eigenfunctions φ of a quantum-mechanical operator P which has a continuous spectrum of eigenvalues λ, is ∫φ*(λ)φ(λ′) dλ=δ(λ−λ′). The derivation of this result is examined; it is seen to be merely a sufficient, but not a necessary condition for the consistent expansion of a bound-state wavefunction ψ in the form ∫ a(λ)φ(λ) dλ. The situation when P is a generalized momentum operator is investigated in detail. Such operators were discussed in the preceding paper. It is shown that any two of the corresponding momentum eigenfunctions are not usually orthogonal. Nevertheless, with the help of Fourier analysis, a consistent expansion ψ=∫ a(λ)φ(λ) dλ is established. The theory is illustrated with the familiar examples of a ``particle in a box'' and a ground-state hydrogen atom. When a space coordinate has a finite range, a quantum condition can be imposed on the generalized momentum eigenvalues so that the momentum eigenfunctions form a complete discrete orthogonal set. We attempt to justify the belief that such quantization is not essential unless it is necessary to ensure single-valued eigenfunctions.
In quantum mechanics textbooks the momentum operator is defined in the Cartesian coordinates and rarely the form of the momentum operator in spherical polar coordinates is discussed. Consequently one always generalizes the Cartesian prescription to other coordinates and falls in a trap. In this work we introduce the difficulties one faces when the question of the momentum operator in general curvilinear coordinates arises. We have tried to elucidate the points related to the definition of the momentum operator taking spherical polar coordinates as our specimen coordinate system and proposed an elementary method in which we can ascertain the form of the momentum operator in general coordinate systems.
In addition to electric and magnetic multipoles, the expansion of current density also yields toroidal terms, a lesser-known family of multipoles. A recent proposal, I. Kuprov
et al., Science Adv. 8 abq6751 (2022), explores the possibility of a direct observation of optical toroidal transitions in hydrogen and alkali atoms in the presence of a large magnetic field that decouples the spin and the angular momentum of the electron. However, the difficulty of observing these transitions against the nearby electric dipole (E1) transitions were underestimated because of extra admixture coming from diamagnetic coupling. Here, we revisit the toroidal coupling in atoms, taking diamagnetic contribution into account, and discuss the technical challenges of observing toroidal coupling in atomic physics. We show that toroidal transition should be searched in transitions with low principal quantum numbers. The remaining strong electric-dipole contribution could be removed using an appropriate differential measurement.