Quantum mechanics of a constrained particle and the problem of prescribed geometric potential

Research paper by L. C. B. da Silva, C. C. Bastos

Indexed on: 01 Feb '16Published on: 01 Feb '16Published in: Quantum Physics


Nowadays the experimental techniques in nanoscience have evolved to a stage where various examples of nanostructures with non-trivial shapes have been synthesized and analyzed, turning the study of the quantum dynamics of a constrained particle and the relation with geometry into a realistic and important topic of research. Some decades ago, a formalism capable of giving a meaningful Hamiltonian for the confined dynamics was devised. These results showed that a quantum scalar potential, which depends explicitly on the geometry of the curved region, acts upon the particle. In this work, we study the confinement on curves and the problem of prescribed geometric potential, i.e., finding a curve or a surface with a quantum geometric potential given {\it a priori}. For the confinement on curves we show that in an intrinsic scheme, i.e., in the absence of the quantum geometric potential, the spectrum of a curve on any manifold only depends on the fixed length and imposed boundary conditions. After, we investigate the one-dimensional confinement in a surface (then, in a non-euclidean ambient space), where we show that the geometric potential can give rise to both negative and positive potentials, a result which is not possible if the ambient surface is flat. In addition, we solve the problem of prescribed geometric potential for planar curves with an emphasis on the case of power-law curvature functions, which allows us to model the Hydrogen atom through a confinement in a curve. In the two-dimensional case, we solve the prescribed geometric potential problem for surfaces with rotation and translation symmetry, where in the last case we also show that the problem for curves is equivalent to the problem for cylindrical surfaces. Finally, we discuss on the nature of the quantum geometric potential and the influence that different embeddings may have on the constrained dynamics.