Solving the time-dependent Schrödinger equation numerically

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Abstract
We introduce an explicit scheme to solve the time-dependent Schrödinger equation. The scheme is a straightforward extension of the second order differencing scheme to the fourth, sixth, and higher order accuracy. The accuracy is remarkably improved with minor changes in the second order differencing program. This method is conditionally stable. There is a trade-off between the higher order accuracy and the condition of stability. The performance is evaluated and compared to the standard methods, such as the Crank-Nicholson scheme (CN) and the Chebyshev scheme (CH). The new scheme is much more accurate than CN, almost equal to CH.