Geometrically constrained quantum dynamics: numerical solution of the Schrödinger equation on a comb

Abstract The comb model constitutes a simplified model for investigation of the dynamics in heterogeneous and anisotropic media and its implementation in the Schrödinger equation enables understanding of the wave packet spreading in such media. In this article, the time-dependent Schrödinger equation (TDSE) in the presence of comb-like geometric constraints is numerically solved. The comb-like constraints in the Hamiltonian are tailored by a modified Laplacian operator, which incorporates the Dirac delta function. To devise an appropriate numerical scheme, we regularize the Dirac delta function by its approximation of Gaussian form. Further, the finite difference method for spatial discretization and the fourth order Runge-Kutta method for the time evolution are employed. The obtained results for the wave function are compared with the analytical solutions, previously obtained in an infinite domain and a thorough analysis is carried out. The best comparison is found for short and intermediate time intervals. After validating the numerical scheme, we investigate the influence of the initial conditions on the probability density function (PDF). The TDSE was first solved with the initial conditions of Gaussian form, and then a comb-like shape of the initial wave function was proposed. The latter leads to pronounced localization in the narrow regions along the backbone, which disappears after a long time and resembles the similar behaviour of the PDF, as in the case of Gaussian initial condition. The numerical scheme developed in this work widens the applicability of the comb model for finite domains and nontrivial initial conditions, where analytical solution is not possible. Therefore, this improves the perspectives for modeling of the realistic experimental situations, where the domains are finite.