Coupled anharmonic oscillators: the Rayleigh-Ritz approach versus the collocation approach

Variational collocation for systems of coupled anharmonic oscillators

We have applied a collocation approach to obtain the numerical solution to the stationary Schrödinger equation for systems of coupled oscillators. The dependence of the discretized Hamiltonian on scale and angle parameters is exploited to obtain optimal convergence to the exact results. A careful comparison with results taken from the literature is performed, showing the advantages of the present approach.

Group theoretical analysis of a quantum-mechanical three-dimensional quartic anharmonic oscillator

This paper illustrates the application of group theory to a quantum-mechanical three-dimensional quartic anharmonic oscillator with $O_{h}$ symmetry. It is shown that group theory predicts the degeneracy of the energy levels and facilitates the application of perturbation theory and the Rayleigh-Ritz variational method as well as the interpretation of the results in terms of the symmetry of the solutions . We show how to obtain suitable symmetry-adapted basis sets.

复平面中的瑞利-里茨变分法

Rayleigh-Ritz Variational Method in The Complex Plane

本文研究了量子振子在Segal-Bargmann空间中瑞利-里茨变分法的应用，推导了广义高斯试函数的可积条件，并展示了四次非谐振子中自适应高斯近似方法的高阶能量展开。

The optimized Rayleigh-Ritz scheme for determining the quantum-mechanical spectrum

The convergence of the Rayleigh-Ritz method with nonlinear parameters optimized through minimization of the trace of the truncated matrix is demonstrated by a comparison with analytically known eigenstates of various quasi-solvable systems. We show that the basis of the harmonic oscillator eigenfunctions with optimized frequency ? enables determination of boundstate energies of one-dimensional oscillators to an arbitrary accuracy, even in the case of highly anharmonic multi-well potentials. The same is true in the spherically symmetric case of V (r) = ω2r2 2 + λrk, if k > 0. For spiked oscilla…

一种用于物理引导机器学习的教学框架：从经典单摆到量子非谐振子使用PyTorch在现代GPU硬件上

A Pedagogical Framework for Physics-Informed Machine Learning: From Classical Pendulum to Quantum Anharmonic Oscillator Using PyTorch on Modern GPU Hardware

本文提出一个五模块教学框架，通过两个递进的物理系统展示物理引导机器学习，比较了五种模型架构，展示了GPU加速在不同模型中的应用效果。