Abstract
In this work we approach the Schrödinger equation in quantum wells with arbitrary potentials, using the machine learning technique. Two neural networks with different architectures are proposed and trained using a set of potentials, energies, and wave functions previously generated with the classical finite element method. Three accuracy indicators have been proposed for testing the estimates given by the neural networks. The networks are trained by the gradient descent method and the training validation is done with respect to a large training data set. The two networks are then tested for two different potential data sets and the results are compared. Several cases with analytical potential have also been solved.
Highlights
In this work we approach the Schrödinger equation in quantum wells with arbitrary potentials, using the machine learning technique
The neural network (NN) are differently trained using a set of potentials, energies, and wave functions (WFs) previously generated by the finite element method (FEM)
Two NNs with different architectures have been proposed and trained using a set of potentials, energies, and WFs previously generated with the classical FEM
Summary
In this work we approach the Schrödinger equation in quantum wells with arbitrary potentials, using the machine learning technique. Two neural networks with different architectures are proposed and trained using a set of potentials, energies, and wave functions previously generated with the classical finite element method. Mills et al proposed a deep learning method for solving the two-dimensional (2D) SE3 They trained a NN to predict the ground and the first excited state energies of an electron in different classes of 2D confining potentials. By the present work we intend to approach the Schrödinger problem in quantum wells (QWs) with finite walls and arbitrary potentials, using rather simple NNs. When SE is solved by conventional numerical methods, the computation of the energy eigenvalues and eigenfunctions cannot be conceived as separate, independent problems. Several cases with analytical potential will be solved and analyzed
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