Numerical solutions of the Schrödinger equation with source terms or time-dependent potentials

We develop an approach to solving numerically the time-dependent Schrödinger equation when it includes source terms and time-dependent potentials. The approach is based on the generalized Crank-Nicolson method supplemented with an Euler-MacLaurin expansion for the time-integrated nonhomogeneous term. By comparing the numerical results with exact solutions of analytically solvable models, we find that the method leads to precision comparable to that of the generalized Crank-Nicolson method applied to homogeneous equations. Furthermore, the systematic increase in precision generally permits making estimates of the error.

Figure 1

The $\Delta {\tau }_s^{\left(M\right)}$ for each $s=1,\cdots ,M$ from left to right plotted as dots on the complex plane. In this graph $M=20$, 10, and 4. The individual contributions for $M=20$, namely $-2/z_s^{\left(20\right)}$, are plotted as green dots.

Figure 2

The errors (exact and estimated) for the calculations with the parameters of Table 2 including $\Delta t=2\nu$ and $J=200$. The $\eta$ in the legend of the graph refers to ${\eta }_{M,r}$.

Figure 3

The wave functions of example 2 at (a) $t=0$ and at (b) $t=2.5\pi$. The units used are such that $\hslash =m=1$.

Figure 4

The potential of Eq. (4.16) as a function of $x$ at $t=0$ and $t=2$. Since $V(x,2)$ is relative small, the curve plotted is magnified by a factor of ten. The units used are such that $\hslash =2m=1$.

Figure 5

The wave functions Eq. (4.17) as a function of $x$ at $t=0$ and $t=2$. The units used are such that $\hslash =2m=1$.

Figure 6

The errors (exact and estimated) for the calculations with the parameters $x_0=-15,x_J=15,J=200,\Delta t=0.0075$. The units used are such that $\hslash =1,m=1/2$. For the $M=3$ case we terminate iteration for $i$ when $e^{\left(i\right)}<{10}^{-19}$. The $\eta$ in the legend of the graph refers to ${\eta }_{M,r}$.