Solving the Time-Dependent Schr¨odinger Equation for a Harmonic Oscillator and Extracting the Bohm Quantum Potential Using PINNs

Abstract

Physics-Informed Neural Networks (PINNs) offer an effective framework for solving partial differential equations by incorporating governing physical laws into the learning process. When applied to quantum mechanical differential equations such as the Schr¨odinger equation, this approach produces smooth wavefunctions suitable for computing both conventional observables and Bohmian quantities. In this work, a PINNs based framework is developed to solve the one-dimensional time-dependent Schr¨odinger equation for the Quantum Har monic Oscillator, a fundamental model with broad relevance in quantum and plasma physics. A fully connected neural network is used to represent the real and imaginary parts of the wavefunction, with the Schr¨odinger equation enforced through the loss function together with appropriate initial and boundary conditions. Automatic differentiation is employed to compute the  required derivatives and to extract physical quantities from the learned solution. The method accurately reproduces the ground-state wavefunction and yields a ground-state energy in agreement with the analytical result. The Bohm quantum potential obtained from the network also matches the corresponding analytical expression, demon strating that the essential spatial structure of the quantum amplitude is well captured. These results demonstrate that PINNs constitute a reliable and conceptually transparent framework for solving the time-dependent Schr¨odinger equation. Within this differential equation based setting, PINNs also enable systematic exploration of Bohmian aspects of quantum systems, where Bohmian quantities such as the quantum potential provide physical insight into confinement effects and the emergence of effective potentials.