Numerical solutions of the time-dependent Schrödinger equation: reduction of the error due to space discretization.
We present an improved space-discretization scheme for the numerical solutions of the time-dependent Schrödinger equation. Compared to the scheme of W. van Dijk and F. M. Toyama [Phys. Rev. E 75, 036707 (2007)], the present one, which contains more terms of second-order partial derivatives, greatly reduces the error resulting from the spatial integration. For a (2l+1) -point formula with (2l+1) terms of second-order partial derivatives, the local truncation error can decrease from the order of (Deltax)2l to (Deltax)4l, while the previous one contains only one term of second-order partial derivative. Two well-known numerical examples and the corresponding error analysis demonstrate that the present scheme has an advantage in the precision and efficiency over the previous one.
Shao
Hezhu
H
Department of Physics, Shanghai University, 99 Shangda Road, Shanghai 200444, People's Republic of China. hzshao@hotmail.com
Wang
Zhongcheng
Z