Speaker
Description
Background: The time-independent Schrödinger equation is central to the description of
bound states and quantum tunneling in double-well potential systems. Such potentials are
important in molecular inversion phenomena and can also arise as effective quantum
potentials in plasma environments due to external fields, confinement effects, and plasma-
surface interactions. Conventional numerical approaches, such as finite-difference matrix
methods, rely on spatial discretization and can become computationally demanding for
accurate solutions.
Purpose: The purpose of this study is to investigate the applicability of Physics-Informed
Neural Networks (PINNs) for solving the one-dimensional time-independent Schrödinger
equation with a symmetric double-well potential and to obtain accurate ground and low-lying
excited state solutions relevant to quantum plasma systems.
Methods: A Physics-Informed Neural Network framework is employed in which the
wavefunction is represented using fully connected feedforward neural networks, while the
energy eigenvalues are treated as trainable parameters. The Schrödinger equation is enforced
through a residual-based loss function, supplemented by boundary condition, normalization,
and orthogonality constraints. Even- and odd-parity neural network architectures are used to
directly capture symmetric and antisymmetric eigenstates. The PINN solutions are validated
by comparison with reference results obtained from a finite-difference-based matrix
diagonalization method.
Results: The PINN approach successfully reproduces the ground and low-lying excited states
of the double-well potential. In atomic units (ℏ = 1, 𝑚 = 1), the ground-state energy is
obtained as 𝐸0 ≈ 1.973, compared to the finite-difference value 𝐸0 ≈ 1.971. The first
excited-state energy is predicted as 𝐸1 ≈ 2.220, while the corresponding finite-difference
result is 𝐸1 ≈ 2.012. The second excited-state energy is also well captured, with PINN and
finite-difference values of 𝐸2 ≈ 4.911and 𝐸2 ≈ 4.908, respectively. The small energy
separation between the lowest two states reflects the near-degeneracy characteristic of
symmetric double-well potentials, arising from quantum tunneling between the wells.
Conclusions: This study demonstrates that Physics-Informed Neural Networks provide an
accurate and flexible alternative to traditional numerical techniques for solving the
Schrödinger equation with double-well potentials. The approach is suitable for modeling
tunneling phenomena and effective quantum potentials in plasma-related systems.
