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 (Δx)^2l to (Δx)^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.