In the course of one's studies, it may be desired to approximate a particle constrained to a curved surface, such as a sheet of graphene or a fullerene, but without modelling the lattice of Coulomb potentials. This can be done using the machinery of differential geometry to formulate a Schr?dinger equation for a particle constrained so by applying an infinitely narrow infinite square well potential normal to the surface. The purpose of this paper is to clarify two mistakes that might be made when attempting to formulate this equation. These mistakes are: (1) incorrectly assuming that the Laplacian of the surface is the same as the Laplacian of the space, with the constrained term removed; and (2) neglecting to include the effective potential energy due to the curvature of the surface. It is also noted that the constrained approach produces results that, if the resulting Schr?dinger equation is separable, will produce one constant of separation instead of the traditional two. The emergence of the second quantum number is demonstrated and, hence, the apparent inconsistency is resolved. The prolate spheroid is given as a non-trivial example because it highlights the issues presented.
In the course of one's studies, it may be desired to approximate a particle constrained to a curved surface, such as a sheet of graphene or a fullerene, but without modelling the lattice of Coulomb potentials. This can be done using the machinery of differential geometry to formulate a Schr?dinger equation for a particle constrained so by applying an infinitely narrow infinite square well potential normal to the surface. The purpose of this paper is to clarify two mistakes that might be made when attempting to formulate this equation. These mistakes are: (1) incorrectly assuming that the Laplacian of the surface is the same as the Laplacian of the space, with the constrained term removed; and (2) neglecting to include the effective potential energy due to the curvature of the surface. It is also noted that the constrained approach produces results that, if the resulting Schr?dinger equation is separable, will produce one constant of separation instead of the traditional two. The emergence of the second quantum number is demonstrated and, hence, the apparent inconsistency is resolved. The prolate spheroid is given as a non-trivial example because it highlights the issues presented.
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 discuss the introduction of constraints in a system of n noninteracting particles which obey the laws of nonrelativistic quantum mechanics. In this paper the particles are first thought of as being unconstrained (described by the 3n Cartesian coordinates of a flat space R), but subject to an external potential V which, in a certian suitable limit, forces the system to remain in a curved subspace V of R. This idea was already employed in a previous work where we have discussed the motion of one constrained particle. It was then shown that in order to obtain a meaningful result the particle wave function should be "uniformly compressed" into a surface (or curve), avoiding, in this way, the tangential forces which correspond to the dissipative constraints of classical mechanics. The resulting Schr\"odinger equation could then be separated in such a way that the part which contained the surface (or curve) variables was independent of the potential V employed in the constraining process. In the present paper we show that this procedure cannot be carried out for all subspaces V of R in the case of a many-particle system, unless V satisfies a certain geometrical condition. The many-particle Schr\"odinger equation (as was already the case of the one-particle systems) contains, besides the kinetic energy term, potentials which are not bending invariants, that is, cannot be obtained from the metric tensor gij of V or its derivatives. This gives rise to different Schr\"odinger equations for isometric subspaces, in striking contrast with the usual quantization procedures which start from the classical Lagrangian of the constrained system.
We derive the Schrödinger equation of a particle constrained to move on a rotating curved surface S. Using the thin-layer quantization scheme to confine the particle on S, and with a proper choice of gauge transformation for the wave function, we obtain the well-known geometric potential V g and an additive Coriolis-induced geometric potential in the co-rotational curvilinear coordinates. This novel effective potential, which is included in the surface Schrödinger equation and is coupled with the mean curvature of S, contains an imaginary part in the general case which gives rise to a non-Hermitian surface Hamiltonian. We find that the non-Hermitian term vanishes when S is a minimal surface or a revolution surface which is axially symmetric around the rolling axis.
Since electrons in a ballistic regime perceive a carbon nanotube or a graphene layer structure as a continuous medium, we can use the study of the quantum dynamics of one electron constrained to a curve or surface to obtain a qualitative description of the conduction electrons' behavior. The confinement process of a quantum particle to a curve or surface leads us, in the so-called da Costa's approach, to a geometry-induced potential (GIP) in the effective Schr\odinger equation. With these considerations, this work aims to study in detail the consequences of constraining a quantum particle to a helix, catenary, helicoid, or catenoid, exploring the relations between these curves and surfaces using differential geometry. Initially, we use the variational method to estimate the energy of the particle in its ground state, and then, we obtain better approximations with the use of the confluent Heun function through numerical calculations. Thus, we conclude that a quantum particle constrained to an infinite helix has its angular momentum quantized due to the geometry of the curve, while in the cases of the catenary, helicoid, and catenoid the particle can be found either in a single bound state or in excited states which constitute a continuous energy band.
When quantum mechanical motion of a particle constrained to a curved surface in the Euclidean space is considered, three different approaches may be adopted to describe the system. One is the confining approach which enforces a particle to stay in the surface by a physical confining potential. Both of the others follow Dirac's general prescription to treat a constrained system, but employ different constraints, namely the usual and the conservative constraints, respectively. The three approaches lead to results which do not in general agree with each other, though they give the same classical equation of motion. It is emphasized that the arbitrariness in the ordering of operators cannot be avoided when the latter two approaches are adopted. In the approach with the conservative constraint, however, there is a physical ordering, which leads to the result in agreement with that of the confining approach. We interpret these discrepancy and agreement in simple physical terms.