Speaker
Description
Numerical Simulation of Hydrogen Atom using Physics-Informed Neural Networks
Kanika1, Ishwar Kant1, Ayushi Awasthi1, and O. S. K. S. Sastri1
Department of Physics and Astronomical Sciences Central University of Himachal Pradesh, ?
Dharamshala - 176215, India
Abstract
Background: The hydrogen atom is the simplest bound quantum system and serves as a ?
standard benchmark for testing numerical methods in quantum mechanics. Traditional ?
numerical techniques, such as matrix-based approaches with predefined basis sets, often ?
require domain truncation and careful treatment of singular potentials and boundary
conditions.
Purpose: This study aims to apply Physics-Informed Neural Networks (PINNs) to solve the ?
time-independent Schrödinger equation for the hydrogen atom, accurately determining ?
energy eigenvalues and radial wavefunctions, and to assess the effectiveness of PINNs as an ?
alternative computational approach for quantum systems.
Methods: The radial Schrödinger equation is solved using PINNs by approximating the ?
radial wavefunction with a neural network. A physics-informed loss function is formulated ?
by incorporating the residual of the Schrödinger equation along with the imposed boundary ?
conditions. The energy eigenvalue is treated as a trainable parameter and optimized ?
simultaneously with the network weights using gradient-based learning.
Results: The PINNs model accurately reproduces the ground and low-lying excited states of ?
the hydrogen atom in atomic units. The predicted ground-state energy is -0.499 Hartree,
showing an absolute error of 2.3 × 10−5relative to the exact value -0.500 Hartree. For the
first excited states (𝑛= 2, 𝑙= 0) and (𝑛= 2, 𝑙= 1), the PINN yields energies of -0.136 ?
Hartree and -0.140 Hartree, respectively, demonstrating close agreement with the analytical
value -0.125 Hartree. The learned radial wavefunctions exhibit correct nodal structure and
asymptotic behavior.
Conclusion: Physics-Informed Neural Networks provide an accurate and basis-free approach ?
for solving the hydrogen atom Schrödinger equation. The quantitative agreement between ?
PINN-predicted and exact energies, along with physically consistent wavefunctions, ?
highlights the potential of PINNs as a robust computational framework for quantum ?
mechanical systems.