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.
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.
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.
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.
The curvature potential arising from confining a particle initially in three-dimensional space onto a curved surface is normally derived in the hard constraint q → 0 limit, with q the degree of freedom normal to the surface. In this work the hard constraint is relaxed, and eigenvalues and wave functions are numerically determined for a particle confined to a thin layer in the neighborhood of a toroidal surface. The hard constraint and finite layer (or soft constraint) quantities are comparable, but both differ markedly from those of the corresponding two dimensional system, indicating that the curvature potential continues to influence the dynamics when the particle is confined to a finite layer. This effect is potentially of consequence to the modelling of curved nanostructures.
We consider particles constrained to move in the close vicinity of a space curve through a steep quadratic potential in the plane normal to the curve. As is known, the effective 1D Hamiltonian that governs the motion along the curve involves both its curvature and torsion. Thus, the adiabatic cyclic deformations of the curve might give rise to geometric phases being accumulated by the particle. We report that this is indeed, in general, the case, and analyze in detail the origin of the phases, as described in both an inertial Cartesian frame and a noninertial one, adapted to the curve. We extend previous work to include non-locally arclength preserving deformations, and derive general formulas for Berry’s curvature in this case.