Solving the Time Dependent Schr¨odinger Equation using Machine Learning

Abstract

The time-dependent Schr¨odinger equation plays a central role in quantum physics, yet the methods used to solve it are typically computationally expensive. In this work, we use a Physics-Informed Neural Network approach to learn the dynamics of the quantum harmonic oscillator. Our model successfully reproduces the expected oscillatory motion of the coherent state and conserves energy with only very small deviations, with relative energy errors below 10−3. The method achieves extremely low infidelities with respect to the analytical results, of the order of 10−5. We also test the model on breathing mode dynamics, obtaining a low average infidelity of the order of 10−2 and a modest relative energy error around 10−2. These results show that Physics-Informed Neural Networks can accurately learn and generalise solutions to the time-dependent Schr¨odinger equation, providing an efficient alternative to traditional solvers.