Neural network approaches for solving Schrödinger equation in arbitrary quantum wells

Area of Science:

• Quantum mechanics
• Computational physics
• Machine learning

Background:

• The Schrödinger equation is fundamental to quantum mechanics.
• Solving it for arbitrary potentials in quantum wells is computationally challenging.
• Machine learning offers a novel approach to accelerate these calculations.

Purpose of the Study:

• To apply machine learning techniques to solve the Schrödinger equation in quantum wells.
• To develop and compare two distinct neural network architectures for this task.
• To establish accuracy indicators for evaluating the machine learning model's performance.

Main Methods:

• Utilizing two neural network architectures.
• Training networks with data generated via the finite element method (FEM).
• Employing gradient descent for network training and validation against extensive datasets.

Main Results:

• Developed and validated two neural networks for solving the Schrödinger equation.
• Proposed three accuracy indicators to assess network performance.
• Achieved accurate predictions for energies and wave functions across diverse potentials.

Conclusions:

• Machine learning, specifically neural networks, provides an effective method for solving the Schrödinger equation in quantum wells.
• The proposed models demonstrate high accuracy and generalizability for arbitrary potentials.
• This approach offers a promising alternative to traditional numerical methods.