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Manuscript Title: A Mathematica program for the two-step twelfth-order method with multi-derivative for the numerical solution of a one-dimensional Schrödinger equation
Authors: Z. Wang, Y. Ge, Y. Dai, D. Zhao
Program title: ShdEq.nb
Catalogue identifier: ADTT
Journal reference: Comput. Phys. Commun. 160(2004)23
Programming language: Mathematica 4.2.
Computer: The program has been designed for the microcomputer and been tested on the microcomputer.
Operating system: Windows XP.
RAM: 51 712 bytes
Keywords: Multi-derivative method, High-order linear two-step methods, Schrödinger equation, Eigenvalue problems, High precision methods, Numerov's method.
PACS: 02.60.Cb, 02.70.Bf.
Classification: 4.3.

Nature of problem:
Numerical integration of one-dimensional or radial Schrödinger equation to find the eigenvalues for a bound states and phase shift for a continuum state.

Solution method:
Using a two-step method twelfth-order method to integrate a Schrödinger equation numerically from both two ends and the connecting conditions at the matching point, an eigenvalue for a bound state or a resonant state with a given phase shift can be found.

The analytic form of the potential function and its high-order derivatives must be known.

Unusual features:
Take advantage of the high-order derivatives of the potential function and efficient algorithm, the program can provide all the numerical solution of a given Schrödinger equation, either a bound or a resonant state, with a very high precision and within a very short CPU time. The program can apply to a very broad range of problems because the method has a very large interval of periodicity.

Running time:
Less than one second.

[1] T.E. Simos, Proc. Roy. Soc. London A 441 (1993) 283.
[2] Z. Wang, Y. Dai, An eighth-order two-step formula for the numerical integration of the one-dimensional Schrödinger equation, Numer. Math. J. Chinese Univ. 12 (2003) 146.
[3] Z. Wang, Y. Dai, An twelfth-order four-step formula for the numerical integration of the one-dimensional Schrödinger equation, Internat. J. Modern Phys. C 14 (2003) 1087.