The Gaussian wave packets transform for the semi-classical Schrödinger equation with vector potentials
This work provides efficient numerical methods for solving the semi-classical Schrödinger equation in the presence of electromagnetic fields, which is important for quantum mechanics simulations.
The authors reformulate the semi-classical Schrödinger equation with vector potentials using the Gaussian wave packets transform, converting it into a non-oscillatory equation that requires less computational effort. They propose two numerical methods that are unconditionally stable, spectrally accurate in space, and second-order accurate in time.
In this paper, we reformulate the semi-classical Schrödinger equation in the presence of electromagnetic field by the Gaussian wave packets transform. With this approach, the highly oscillatory Schrödinger equation is equivalently transformed into another Schrödinger type wave equation, the $w$ equation, which is essentially not oscillatory and thus requires much less computational effort. We propose two numerical methods to solve the $w$ equation, where the Hamiltonian is either divided into the kinetic, the potential and the convection part, or into the kinetic and the potential-convection part. The convection, or the potential-convection part is treated by a semi-Lagrangian method, while the kinetic part is solved by the Fourier spectral method. The numerical methods are proved to be unconditionally stable, spectrally accurate in space and second order accurate in time, and in principle they can be extended to higher order schemes in time. Various one dimensional and multidimensional numerical tests are provided to justify the properties of the proposed methods.