受约束粒子的量子力学与预定几何诱导势问题
Quantum mechanics of a constrained particle and the problem of prescribed geometry-induced potential
摘要：
实验技术已经发展到这样一个阶段，其中合成了具有非平凡形状的纳米结构的各种例子，将约束粒子的动力学和与几何形状的联系转变为现实而重要的研究课题。几十年前，人们设计了一种推导约束有意义的汉密尔顿函数的形式，表明几何诱导势（GIP）作用于动力学。本文研究欧氏空间$\mathbb{R}^3$中曲线曲面的指定GIP问题，即，如何找到具有给定{ \it a priori}的势的弯曲区域。曲线的问题很容易通过积分Frenet方程来解决，而曲面的问题涉及非线性二阶偏微分方程（PDE）。在这里，我们探讨的GIP不变的表面由一个1参数组的等距的$\mathbb{R}^3$，它把PDE成一个常微分方程（ODE），并导致圆柱形，革命，螺旋面。螺旋面特别重要，因为它们是建立手征性和GIP之间联系的天然候选者。最后，对于螺旋极小曲面族，我们证明了几何诱导的束缚态和局域态的存在性，以及当表面受到额外电荷时控制概率密度分布变化的可能性。
英文摘要：
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.
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