A Mathematica program for the two-step twelfth-order method with multi-derivative for the numerical solution of a one-dimensional Schrödinger equation

In this paper, we present the detailed Mathematica symbolic derivation and the program which is used to integrate a one-dimensional Schrödinger equation by a new two-step numerical method. We add the fourth- and sixth-order derivatives to raise the precision of the traditional Numerov's method from fourth order to twelfth order, and to expand the interval of periodicity from (0,6) to the one of (0,9.7954) and (9.94792,55.6062). In the program we use an efficient algorithm to calculate the first-order derivative and avoid unnecessarily repeated calculation resulting from the multi-derivatives. We use the well-known Woods-Saxon's potential to test our method. The numerical test shows that the new method is not only superior to the previous lower order ones in accuracy, but also in the efficiency. This program is specially applied to the problem where a high accuracy or a larger step size is required.

Program summary

Title of program: ShdEq.nbCatalogue number: ADTTProgram summary URL:http://cpc.cs.qub.ac.uk/summaries/ADTTProgram obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandLicensing provisions: noneComputer for which the program is designed and others on which it has been tested: The program has been designed for the microcomputer and been tested on the microcomputer.Computers: IBM PCOperating systems under which the program has been tested: Windows XPProgramming language used: Mathematica 4.2Memory required to execute with typical data: 51 712 bytesNo. of bytes in distributed program, including test data, etc.: 45 381No. of lines in distributed program, including test data, etc.: 7311Distribution format: tar gzip fileCPC Program Library subprograms used: noNature of physical 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.Method of solution: 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.Restrictions on the complexity of the problem: The analytic form of the potential function and its high-order derivatives must be known.Typical running time: Less than one second.Unusual features of the program: 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.References: 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.