Filter method without boundary-value condition for simultaneous calculation of eigenfunction and eigenvalue of a stationary Schrödinger equation on a grid

The paper presents a method for simultaneous computation of eigenfunction and eigenvalue of the stationary Schrödinger equation on a grid, without imposing boundary-value condition. The method is based on the filter operator, which selects the eigenfunction from wave packet at the rate comparable to $\delta$ function. The efficacy and reliability of the method are demonstrated by comparing the simulation results with analytical or numerical solutions obtained by using other methods for various boundary-value conditions. It is found that the method is robust, accurate, and reliable. Further prospect of filter method for simulation of the Schrödinger equation in higher-dimensional space will also be highlighted.

Figure 1

Comparison of computed eigenfunctions (left panel) with the corresponding analytical results (right panel) of 1D harmonic oscillator for various energy levels.

Figure 2

Comparison of simulated bound state Coulomb wave functions of $n=25$ and $l=0$ (upper panel) and $n=25$ and $l=24$ (lower panel). Results in the left panels are obtained by using the filter method, while those at the right panels are computed using analytical formulas Eq. (17).

Figure 3

Comparison of simulated eigenfunction of continuum Coulomb wave functions for $E=0.0005$ a.u. and $l=0$ (left panel) and $l=20$ (right panel). The magenta (color online) curves stand for simulation results obtained by setting $r_{\max }=10.000$ a.u., whereas the black curves are from analytic solutions.

Figure 4

Ground-state energy of soft Coulomb potential for various soft core parameters $a$ obtained by using (a) filter method and (b) relaxation method.

Figure 5

Comparison of the simulated ground-state eigenfunctions of soft Coulomb potential for three different values of $a$, i.e., $a=0$ a.u. displayed as black curve, $a=0.01$ a.u. as red (online) curve, and $a=2$ a.u. as orange (online) curve.

Figure 6

Plot of the first 10 Rydberg energy levels of soft Coulomb potential as a function of soft core parameter $a$.