Abstract
A high-order finite difference discretisation of the Schrödinger equation on domains that deform in time is presented. For the domain deformation, a time-dependent coordinate transformation is used. The utilisation of summation-by-parts finite difference operators, combined with boundary conditions imposed weakly by a penalty technique, leads to a provably stable scheme. The convergence properties of the scheme are verified by numerical computations. The capabilities of the numerical scheme are demonstrated by simulations of Berry phases in deforming quantum billiards.
Highlights
Particles localised in two-dimensional (2D) spatial domains surrounded by infinite potential walls is a much used model in the study of quantum chaos [1]
The corresponding wave functions show complex structures, such as ‘scars’ resembling classical periodic orbits [3] and non-trivial Berry phases when adiabatically deforming the domain around a crossing point [4]
As a test of our method, we demonstrate how these Berry phases can be obtained in a fully quantum-mechanical setting by solving the time-dependent Schrödinger equation in this system
Summary
Particles localised in two-dimensional (2D) spatial domains surrounded by infinite potential walls is a much used model in the study of quantum chaos [1]. The corresponding wave functions show complex structures, such as ‘scars’ resembling classical periodic orbits [3] and non-trivial Berry phases when adiabatically deforming the domain around a crossing point [4]. To study such systems there is a need for numerical methods that can efficiently solve the Schrödinger equation on time-dependent domains. We derive a stable and accurate high-order summation-by-parts (SBP) finite difference scheme for the 2D Schrödinger equation on deforming domains that can be used to efficiently model this type of complex quantum systems.
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