Affiliation:
1. Nonlinear Artificial Intelligence Laboratory, Physics Department, North Carolina State University 1 , Raleigh, North Carolina 27607, USA
2. Physics Department, The College of Wooster 2 , Wooster, Ohio 44691, USA
Abstract
A particle confined to an impassable box is a paradigmatic and exactly solvable one-dimensional quantum system modeled by an infinite square well potential. Here, we explore some of its infinitely many generalizations to two dimensions, including particles confined to rectangle-, ellipse-, triangle-, and cardioid-shaped boxes using physics-informed neural networks. In particular, we generalize an unsupervised learning algorithm to find the particles’ eigenvalues and eigenfunctions, even in cases where the eigenvalues are degenerate. During training, the neural network adjusts its weights and biases, one of which is the energy eigenvalue, so that its output approximately solves the stationary Schrödinger equation with normalized and mutually orthogonal eigenfunctions. The same procedure solves the Helmholtz equation for the harmonics and vibration modes of waves on drumheads or transverse magnetic modes of electromagnetic cavities. Related applications include quantum billiards, quantum chaos, and Laplacian spectra.
Subject
General Physics and Astronomy
Reference29 articles.
1. Physics-enhanced neural networks learn order and chaos;Phys. Rev. E,2020
2. M.
Finzi
, K. A.Wang, and A. G.Wilson, “Simplifying Hamiltonian and Lagrangian neural networks via explicit constraints,” arXiv:2010.13581 (2020).
3. Hamiltonian neural networks for solving equations of motion;Phys. Rev. E,2022
4. R.
Bondesan
and A.Lamacraft, “Learning symmetries of classical integrable systems,” arXiv:1906.04645 (2019).
5. Machine learning conservation laws from trajectories;Phys. Rev. Lett.,2021