We discuss the dynamics of a classical spinless quantum particle carrying electric charge and constrained to move on a nonsingular static surface in ordinary three-dimensional space in the presence of arbitrary configurations of time independent electric currents. Starting from the canonical action in the embedding space we show that a charged particle with charge $q$ couples to a term linear in $q{A}^{3}M$, where ${A}^{3}$ is the transverse component of the electromagnetic vector potential and $M$ is the mean curvature in the surface. This term cancels exactly a curvature contribution to the orbital magnetic moment of the particle. It is shown that particles, independently of the value of the charge, in addition to the known couplings to the geometry also couple to the mean curvature in the surface when a Neumann type of constraint is applied on the transverse fluctuations of the wave function. In contrast to a Dirichlet constraint on the transverse fluctuations a Neumann type of constraint on these degrees of freedom will in general make the equations of motion nonseparable. The exceptions are the equations of motion for electrically neutral particles on surfaces with constant mean curvature. In the presence of electric currents the equation of motion of a charged particle is generally nonseparable independently of the coupling to the geometry and the boundary constraints.
We discuss the dynamics of a classical spinless quantum particle carrying electric charge and constrained to move on a nonsingular static surface in ordinary three-dimensional space in the presence of arbitrary configurations of time independent electric currents. Starting from the canonical action in the embedding space we show that a charged particle with charge q couples to a term linear in qA3M, where A3 is the transverse component of the electromagnetic vector potential and M is the mean curvature in the surface. This term cancels exactly a curvature contribution to the orbital magnetic moment of the particle. It is shown that particles, independently of the value of the charge, in addition to the known couplings to the geometry also couple to the mean curvature in the surface when a Neumann type of constraint is applied on the transverse fluctuations of the wave function. In contrast to a Dirichlet constraint on the transverse fluctuations a Neumann type of constraint on these degrees of freedom will in general make the equations of motion nonseparable. The exceptions are the equations of motion for electrically neutral particles on surfaces with constant mean curvature. In the presence of electric currents the equation of motion of a charged particle is generally nonseparable independently of the coupling to the geometry and the boundary constraints.
We derive the Schrödinger equation for a spinless charged particle constrained to move on a curved surface in the presence of an electric and magnetic field. The particle is confined on the surface using a thin-layer procedure, which gives rise to the well-known geometric potential. The electric and magnetic fields are included via the four potential. We find that there is no coupling between the fields and the surface curvature and that, with a proper choice of the gauge, the surface and transverse dynamics are exactly separable. Finally, we derive an analytic form of the Hamiltonian for spherical, cylindrical, and toroidal surfaces.
The motion of a particle rigidly bounded to a surface is discussed, considering the Schr\"odinger equation of a free particle constrained to move, by the action of an external potential, in an infinitely thin sheet of the ordinary three-dimensional space. Contrary to what seems to be the general belief expressed in the literature, this limiting process gives a perfectly well-defined result, provided that we take some simple precautions in the definition of the potentials and wave functions. It can then be shown that the wave function splits into two parts: the normal part, which contains the infinite energies required by the uncertainty principle, and a tangent part which contains "surface potentials" depending both on the Gaussian and mean curvatures. An immediate consequence of these results is the existence of different quantum mechanical properties for two isometric surfaces, as can be seen from the bound state which appears along the edge of a folded (but not stretched) plane. The fact that this surface potential is not a bending invariant (cannot be expressed as a function of the components of the metric tensor and their derivatives) is also interesting from the more general point of view of the quantum mechanics in curved spaces, since it can never be obtained from the classical Lagrangian of an a priori constrained particle without substantial modifications in the usual quantization procedures. Similar calculations are also presented for the case of a particle bounded to a curve. The properties of the constraining spatial potential, necessary to a meaningful limiting process, are discussed in some detail, and, as expected, the resulting Schr\"odinger equation contains a "linear potential" which is a function of the curvature.
Using the thin-layer quantization, we formulate the problem of a Schrödinger particle constrained to move along a coordinate surface of the bi-spherical coordinate system. In three-dimensional space, the free Schrödinger equation is not separable in this coordinate system. However, when we consider the equation for a particle constrained to a given surface, there are only two degrees of freedom. One has to introduce a geometrical potential to attach the particle to the surface. This well-known potential has two contributions: one from Gauss’ curvature and the other from the mean curvature. The Schrödinger equation leads to a general Heun equation. We solve it exactly and present the eigenfunctions and plots of the probability densities, and, as an application of this methodology, we study the problem of an electric charge propagating along these coordinate surfaces in the presence of a uniform magnetic field.
We consider the quantum dynamics of a charged particle in Euclidean space subjected to electric and magnetic fields under the presence of a potential that forces the particle to stay close to a compact surface. We prove that, as the strength of this constraining potential tends to infinity, the motion of this particle converges to a motion generated by a Hamiltonian over the surface superimposed by an oscillatory motion in the normal directions. Our result extends previous results by allowing magnetic potentials and more general constraining potentials.
Essentially, we solve the particle-field equations obtained in the matter distribution quantum mechanics. In the framework of this theory, a particle is described by a density amplitude function, we call wave function, as a packet of waves with the time dependent phases proportional to the relativistic Lagrangian. The mass quantization of such a particle, as a distribution of matter, is obtained from the equality of the spatial integral of the density with the mass as a characteristic of the relativistic dynamics of this matter. The particle dynamics, as a motion of a mass spatial distribution, is obtained from the equality of the wave propagation velocities with the distribution coordinate velocities. This description is a physical one, compared to the Schrödinger-Heisenberg description leading to unphysical results as a null velocity for the kinetic energy as a constant of motion, or Zitterbewegung. The description of a particle interaction with an electromagnetic field by additional Lagrangian terms, with a vector potential conjugated to coordinates, as in the Aharonov-Bohm effect, and an electric potential conjugated to time, leads to the Lorentz force, and the Maxwell equations. In the two conjugated spaces, of the coordinates and of the momentum, the dynamics of a quantum particle is described by propagation operators, depending on the coordinate-momentum product, applied to time dependent wave functions satisfying matter-field equations. Unlike the standard Schrödinger or Schrödinger-Dirac equations, depending only on the Hamiltonian operator, instead of the Hamiltonian these equations contain the Lagrangian, which, besides the Hamiltonian, include additional terms with an explicit dependence on velocity, as it is expected in a relativistic theory. We obtain the wave function of a particle-antiparticle system interacting with an electromagnetic field, as a wave packet of spinors propagating in space with finite dimensions, given by the finite definition domain of the density function, in the space of the matter-field momentum, as a constant of motion. We show that an electrically charged quantum particle, rotating in a central electric field, generates a vector potential with a magnetic field perpendicular to the rotational plane and an electric field in this plane. By a quantum transition, a photon is emitted in the particle rotational plane, with the magnetic field perpendicular to this plane, and the electric field contained in this plane. In this framework, the vector potential rotation is correlated to a particle momentum rotation, for the canonical matter-field momentum as a constant of motion, without any electromagnetic field radiated outside.
We derive the Pauli equation for a charged spin particle confined to move on a spatially curved surface S in an electromagnetic field. Using the thin-layer quantization scheme to constrain the particle on S, and in the transformed spinor representations, we obtain the well-known geometric potential Vg and the presence of ${e}^{\ensuremath{-}i\ensuremath{\varphi}}$, which can generate additive spin connection geometric potentials by the curvilinear coordinates derivatives, and we find that the two fundamental evidences in the literature [Giulio Ferrari and Giampaolo Cuoghi, Phys. Rev. Lett. 100, 230403 (2008)] are still valid in the present system without source current perpendicular to S. Finally, we apply the surface Pauli equation to spherical, cylindrical, and toroidal surfaces, in which we obtain expectantly the geometric potentials and new spin connection geometric potentials, and find that only the normal Pauli matrix appears in these equations.