Traditionally, finite differences and finite element methods have been by many regarded as the basic tools for obtaining numerical solutions in a variety of quantum mechanical problems emerging in atomic, nuclear and particle physics, astrophysics, quantum chemistry, etc. In recent years, however, an alternative technique based on the semi-spectral methods has focused considerable attention. The purpose of this work is first to provide the necessary tools and subsequently examine the efficiency of this method in quantum mechanical applications. Restricting our interest to time independent two-body problems, we obtained the continuous and discrete spectrum solutions of the underlying Schroedinger or Lippmann-Schwinger equations in both, the coordinate and momentum space. In all of the numerically studied examples we had no difficulty in achieving the machine accuracy and the semi-spectral method showed exponential convergence combined with excellent numerical stability.
Physics-Informed Support Vector Kernels via Green-Function Analogies and Jackson-Chebyshev Spectral Design
Department of Physics, National Taiwan University, Taipei, Taiwan（物理系，国立台湾大学）；Department of Artificial Intelligence, Chang Gung University, Taoyuan, Taiwan（人工智能系，长庚大学）；Department of Neurology, Chang Gung Memorial Hospital, Keelung, Taiwan（神经科，长庚纪念医院）
Semiclassical trajectory-coherent approximation in quantum mechanics: II. High order corrections to the Dirac operators in external electromagnetic field
Signatures of exciton orbits in quantum mechanical recurrence spectra of Cu$_2$O
T. Kazimierczuk等人[Nature 514，343（2014）]已经表明在对应原理适用并且量子力学转变为经典力学的情况下存在高度激发的激子态，然而，仍然缺少基于激子的经典方法的激子谱的任何解释。在这里，我们通过计算和比较氧化亚铜的量子力学和半经典递归光谱来缩小这一差距。我们表明，量子力学递归谱表现出的峰值，其中，通过应用半经典理论和标度变换，可以直接与经典的周期激子轨道。半经典理论在激子物理中的应用需要详细分析经典激子动力学，包括三维轨道，这些轨道强烈偏离类氢开普勒轨道。我们的研究结果阐明了激子在半导体中的重要方面，直接相关的激子的量子力学能带结构分裂相应的经典激子动力学。
