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

TL;DR

This paper investigates how to design curved nanostructures with specific geometry-induced potentials (GIP) to control quantum states, focusing on surfaces with symmetries like helicoids, and demonstrates the existence of bound states on minimal helicoidal surfaces.

Contribution

It provides a method to solve the prescribed GIP problem for symmetric surfaces and explores the quantum effects on helicoidal minimal surfaces, linking geometry with quantum localization.

Findings

Solutions for prescribed GIP on symmetric surfaces using ODEs.

Existence of bound and localized quantum states on helicoidal minimal surfaces.

Control of probability density distribution through surface charge modifications.

Abstract

The experimental techniques have evolved to a stage where various examples of nanostructures with non-trivial 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 $\mathbb{R}^3$, i.e., how to find a curved region with a potential given {\it a priori}. The problem for curves is easily solved by integrating Frenet equations, while the problem for surfaces involves a non-linear 2nd order partial differential equation (PDE). Here, we explore the GIP for surfaces invariant by a 1-parameter group of isometries of $\mathbb{R}^3$, which turns the PDE into an ordinary differential equation (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 geometry-induced 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.