Quantum mechanics of a constrained particle and the problem of prescribed geometry-induced potential

Abstract

The experimental techniques have evolved to a stage where various examples of nanostructures with nontrivial shapes have been synthesized, turning the dynamics of a constrained particle and the link with geometry into a realistic and important topic of research. Some decades ago, a formalism to deduce a meaningful Hamiltonian for the confinement was devised, showing that a geometry-induced potential (GIP) acts upon the dynamics. In this work we study the problem of prescribed GIP for curves and surfaces in Euclidean space R 3 , i.e., how to find a curved region with a potential given a priori. The problem for curves is easily solved by integrating Frenet equations, while the problem for surfaces involves a non-linear 2nd order PDE. Here, we explore the GIP for surfaces invariant by a 1-parameter group of isometries of R 3 , which turns the PDE into an ODE and leads to cylindrical, revolution, and helicoidal surfaces. Helicoidal surfaces are particularly important, since they are natural candidates to establish a link between chirality and the GIP. Finally, for the family of helicoidal minimal surfaces, we prove the existence of geometryinduced bound and localized states and the possibility of controlling the change in the distribution of the probability density when the surface is subjected to an extra charge.

Generalization of Wolf effect of light on arbitrary two-dimensional surface of revolution

“…(14) is mathematically equivalent to Eq. (15). Therefore, we can find that when the value of Ξ > t, compared with flat space, longitudinal spectral shift is accelerated, and vice versa.…”

“…It is believed that the curvature of space induces the so-called geometric potential which is determined by both extrinsic and intrinsic curvature. This potential modifies the particles' Hamiltonian and consequently attracts much interest about dynamics of particles on surface in condensed-matter physics [14,15]. Plenty of applications have been presented especially since new technologies enable the synthesis of nanostructures with complex curved geometry [16][17][18][19][20][21][22][23][24][25][26][27][28].…”

Gouy and spatial-curvature-induced phase shifts of light in two-dimensional curved space

“…is generally deemed as geometrical potential and plays a vital role in Hamiltonian of particles constrained on surface [37]. Concretely speaking, H and K are extrinsic and intrinsic curvature defined as average and product of main curvatures κ 1 , κ 2 , respectively.…”

Exact Solutions to Schrödinger Equation for a Charged Particle on a Torus in Uniform Electric and Magnetic Fields

“…In the experimental side of the coin, there are very interesting effects, like condensed matter analogues of Riemann sheets as well as Riemannian geometric and geometrical effects, confirmed [16][17][18]. However, despite there exist analytical solutions for some systems-charged particle along a spherical surface in the presence of magnetic and electric fields as well as circular cylinder with magnetic field off the axis [19], cylinder with a non-circular cross section [20], helicoidal surface [21], some surfaces of revolution [22], and conical space [23]-the equation written in a toroidal surface remains unsolved.…”