Numerical solutions of the time-dependent Schrödinger equation in two dimensions

The generalized Crank-Nicolson method is employed to obtain numerical solutions of the two-dimensional time-dependent Schrödinger equation. An adapted alternating-direction implicit method is used, along with a high-order finite-difference scheme in space. Extra care has to be taken for the needed precision of the time development. The method permits a systematic study of the accuracy and efficiency in terms of powers of the spatial and temporal step sizes. To illustrate its utility the method is applied to several two-dimensional systems.

Figure 1

The potential function in example 1, Eq. (4.5).

Figure 2

Errors $e_2$ of the numerical wave functions for potential (4.2). The parameters are $T=1$ and $N=100$.

Figure 3

CPU time as a function of $r$ and $M$ for calculations with potential (4.5).

Figure 4

Errors of the numerical wave functions as they relate to the order of the Padé approximant for various orders of the spatial expansion. The parameters used are listed in Table 3.

Figure 5

Errors of the numerical wave functions as a function of $r$ for different values of $M$. The parameters used are listed in Table 3.

Figure 6

Oscillating and pulsating wave packet in two dimensions of example 2. The plot in the upper left panel represents the wave function at time $t=0$; in the upper right panel, at $t=\pi /2$; in the lower left panel, at $t=\pi$; and in the lower right panel, at $t=3\pi /2$. See Supplemental Material [30] for an animation.

Figure 7

Movement and dispersion of a free-wave packet in example 3 shown at $t={10}^{-3}$ (left) and at $t=6.3\times {10}^{-3}$ (right). See Supplemental Material [30] for an animation.

Figure 8

Probability density of the wave packet diffracted by a single slit on the plane where $x=x_0=1.25333$ at time $t=0.0057$.

Figure 9

Diffraction of a wave packet passing through a single slit, indicated by framed outlines in the graphs. The time is $t=0.0040$ (left) and $t=0.0079$ (right). The part of the packet to the right of the slit is enhanced by a factor of 10. See Supplemental Material [30] for an animation.