从费马原理到波动方程

费马原理在经典和量子系统中的推广

A Generalization of Fermat's Principle for Classical and Quantum Systems

动力学和光学之间的类比对经典力学和量子力学基础的发展产生了巨大影响。我们把这个类比更进一步，研究费马原理在描述动力系统的多维空间中的有效性（即，量子希尔伯特空间和经典相位和配置空间）。我们建议，如果度量距离的概念是很好地定义在该空间和系统的代表点的速度是一个不变量的运动，然后一个广义版本的费马原理将举行。我们证实了这个猜想的时间无关的量子系统和一个经典的系统组成的耦合谐振子。这个原理的一个例外是带电粒子在恒定磁场中的位形空间;在这种情况下，这个原理在旋转拉莫尔频率一半的坐标系中是有效的，而不是静止的实验室坐标系。

Photonic Rutherford Scattering：光线光学和波动光学中的经典和量子力学类比

Photonic Rutherford Scattering: A Classical and Quantum Mechanical Analogy in Ray- and Wave-Optics

本文利用费马最小光程原理，解析地求解了通过一种特殊而又常见的梯度折射率透镜n（r）=n_0+Δ nR/r的光线轨迹族。该解，即射线方程r（phi），被证明是密切相关的著名的卢瑟福散射，因此被称为光子卢瑟福散射。结果表明，不仅这些经典的限制对应，但也波动力学图片一致：时间无关的薛定谔方程和非齐次亥姆霍兹方程允许大质量粒子散射和衍射光学标量波之间的映射相同。窄波包的散射最终恢复了经典轨迹。分析表明，光热单粒子显微镜实际上测量光子卢瑟福散射在特定的限制。

Critique of Quantum Optical Experimental Refutations of Bohr's Principle of Complementarity, of the Wootters-Zurek Principle of Complementarity, and of the Particle-Wave Duality Relation

I argue that quantum optical experiments that purport to refute Bohr's principle of complementarity (BPC) fail in their aim. Some of these experiments try to refute complementarity by refuting the so called particle-wave duality relations, which evolved from the Wootters-Zureck reformulation of BPC (WZPC). I therefore consider it important for my forgoing arguments to first recall the essential tenets of BPC, and to clearly separate BPC from WZPC, which I will argue is a direct contradiction of BPC. This leads to a need to consider the meaning of particle-wave duality relations and to question…

Pilot-wave quantum theory in discrete space and time and the principle of least action

The idea of obtaining a pilot-wave quantum theory on a lattice with discrete time is presented. The motion of quantum particles is described by a $|Ψ|^2$-distributed Markov chain. Stochastic matrices of the process are found by the discrete version of the least-action principle. Probability currents are the consequence of Hamilton's principle and the stochasticity of the Markov process is minimized. As an example, stochastic motion of single particles in a double-slit experiment is examined.

变分原理、波粒二象性与薛定谔方程

A variational principle, wave-particle duality, and the Schrödinger equation

本文提出一个基于两个泛函变分相等的变分原理，分别刻画量子粒子的微粒性与波动性，作为波粒二象性的数学表述，并由此导出薛定谔方程。

Explaining Retrocausality Phenomena in Quantum Mechanics using a Modified Variational Principle

A modified lagrangian with causal and retrocausal momenta was used to derive a first causal wave equation and a second retrocausal wave equation using the principle of least action. The retrocausal wave function obtained through this method was found to be equivalent to the complex conjugate of the causal wave function, thus leading to the conclusion that a retrocausal effect is already implicit in quantum mechanics through the means of complex conjugation of the wave function when computing the probability density for a particle. Lastly, the same variational principle was employed with a frac…