Solving the Hydrogen Atom using Physics-Informed Neural Network for ground state

The hydrogen atom is a standard benchmark for testing and validating numerical techniques in quantum mechanics. Although high-precision studies in the literature include small quantum electrodynamic correc-tions such as the Lamb shift, the present work focuses on the numerical solution of the radial Schr¨odinger equation without incorporating these corrections. In this study, an unsupervised learning framework is em-ployed to model radial wavefunctions at fixed energy values. A neural network is trained to learn physically acceptable solutions that satisfy boundary conditions and normalisation behaviour. The obtained wavefunctions show strong agreement with expected theoretical profiles, reproducing correct nodal structures and asymptotic decay. The results demonstrate that unsupervised neural networks can serve as an effective numerical tool for approximating hydrogenic wavefunctions with good accuracy and precision. This approach provides a flexible  foundation for future extensions toward more complex atomic systems and higher-order corrections.