Radiation boundary conditions for the numerical solution of the three-dimensional time-dependent Schrödinger equation with a localized interaction

Exact radiation boundary conditions on the surface of a sphere are presented for the single-particle time-dependent Schrödinger equation with a localized interaction. With these boundary conditions, numerical computations of spatially unbounded outgoing wave solutions can be restricted to the finite volume of a sphere. The boundary conditions are expressed in terms of the free-particle Green’s function for the outside region. The Green’s function is analytically calculated by an expansion in spherical harmonics and by the method of Laplace transformation. For each harmonic number a discrete boundary condition between the function values at adjacent radial grid points is obtained. The numerical method is applied to quantum tunneling through a spherically symmetric potential barrier with different angular-momentum quantum numbers $l$. Calculations for $l=0$ are compared to exact theoretical results.

Figure 1

Real (solid) and imaginary (dashed) parts of the Green’s function (28) and the one-dimensional ($l$–independent) Green’s function (dotted) at various values of $l$ for $R=50$. The real part of the one-dimensional Green’s function is equal to its imaginary part.

Figure 2

Real (solid) and imaginary (dashed) parts of the integral kernel ${\mu }_l\left(R,\tau \right)$ for $l=15$ and $R=50$.

Figure 3

Quantum tunneling of an initially localized isotropic wave function through a spherically symmetric delta-function potential as given by Eq. (35). At the times shown, the initial state is already strongly depleted and a standing-wave structure between the outgoing wave packet and the potential well has been established. The numerical result compares very well to the analytical solution from [30], Fig. 3.

Figure 4

Comparison of the reference solution to a boundary at $R=2000$ (dotted) to solutions with radiation boundary conditions at $R=50$ (solid) and $R=100$ (dashed) and to the solution with absorbing boundary conditions in a region between $R=50$ and $R=60$ (gray) after ${10}^4$ time steps. Shaded bar indicates the position of the potential well. Simulation parameters: $E_0=0.5$, $r_0=25$, $R_1=36.25$, $R_2=38.75$, $V_0=0.1$, $\Delta t=0.05$, $\Delta r=0.1$.

Figure 5

The deviation in the scaled probability densities after ${10}^4$ time steps for radiation boundary conditions at $R=50$. All simulation parameters are the same as in Fig. 4.

Figure 6

A snapshot of the wave function for $l=15$, $E_0=0.5$, $R_1=36.25$, $R_2=38.75$, $V_0=0.1$, $\Delta t=0.05$, and $\Delta r=0.1$ at time $t\approx 2500$. Shaded bar shows the position of the finite potential. Dashed: real part, dotted: imaginary part, solid line: modulus.

Figure 7

Static shape functions $S_l$ for calculations with the parameters $E_0=0.5$, $R_1=36.25$, $R_2=38.75$, $V_0=0.1$, $\Delta t=0.05$, and $\Delta r=0.1$ at different values of $l$.

Figure 8

Time history of the probability $P$ to find the particle in the computational volume. Simulation parameters: $E_0=0.5$, $R_1=36.25$, $R_2=38.75$, $V_0=0.1$, $\Delta t=0.05$, $\Delta r=0.1$. The decay rate rises with the parameter $l$.

Figure 9

Static decay rate ${\gamma }_l$ as a function of the quantum number $l$ for various heights $V_0$ of the finite barrier. Simulation parameters: $E_0=0.5$, $R_1=36.25$, $R_2=38.75$, $\Delta t=0.05$, $\Delta r=0.1$.