Solving the Schrödinger equation using the finite difference time domain method

Abstract

In this paper, we solve the Schrödinger equation using the finite difference time domain (FDTD) method to determine energies and eigenfunctions. In order to apply the FDTD method, the Schrödinger equation is first transformed into a diffusion equation by the imaginary time transformation. The resulting timedomain diffusion equation is then solved numerically by the FDTD method. The theory and an algorithm are provided for the procedure. Numerical results are given for illustrative examples in one, two and three dimensions. It is shown that the FDTD method accurately determines eigenfunctions and energies of these systems.

Heavy quarkonia in strong magnetic fields

“…Further improvement on present results could also be obtained once lattice simulations provide information about the exact angular dependence of the static potential (which in the present study was partially based on an ansatz) and, possibly, about the spin-dependent part of the potential. The algorithm used to numerically compute eigenvalues and eigenfunctions of the reduced Hamiltonian (7) is the Finite Difference Time Domain method (FDTD) described, e.g., in [41,42]. The main idea of this approach is the following: once the Schrödinger equation is Wick rotated to imaginary time…”

The binding energies of a bulged GaAs nanowire

“…Therefore, the Finite Difference (FD) approximation [16][17][18] has been hired in order to solve such Schrödinger equation and obtain the eigenvalues and corresponding eigenvectors. The Schrödinger equation in cylindrical coordinate with zero azimuthal quantum number is given as:…”

“…For discretized space on different boundaries, different FD schemes are used (see Refs. [16][17][18][19]) as: At the bottom of the wire for boundary (q, z = À100 nm) the forward FD scheme is used. At the top of the wire for boundary (q, z = +100 nm) the backward FD scheme is used.…”

The linear and nonlinear optical properties of an off-center hydrogenic donor impurity in nanowire superlattices: Comparison between arrays of spherical and cylindrical quantum dots

“…On the other hand, based on selection rules, those transitions are allowed that take place between states with the same azimuthal quantum numbers, i.e., Δ = m 0. Therefore, the Schrodinger equation can be rewritten as: The finite difference (FD) approximation is employed [30][31][32] in order to solve the Schrödinger equation and determine the energy levels and wave functions. In this regards we use the same procedure which has been applied in Ref.…”