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American Journal of Undergraduate Research

Optimization of the Split-Operator Method for Modeling of Quantum System Time Evolution

D Vlasov

Published Sep 17, 2005• 0 citations
Publisher

Abstract

A continuing and important problem in modeling physical processes deals with creating optimum calculation algorithms. These algorithms should provide the necessary precision yet not demand a lot of storage and time for evaluations. In the present paper the numerical integration of the non-stationary Schrödinger equation (NSSE) is considered. There are many numerical methods of integration of this equation to the present moment. In this paper, we explore the split-operator method (SOM) and consider a way of lessening time computing expenses in view of specificity of the given method. We also consider the choice of optimum discretization steps in coordinate and momentum space. The computer application that we outline is applicable, in method and preparation, to the solution more difficult problems, and it has been applied to simple chemical reactions. We use standard atomic units to simplify equations in the analysis that follows.

DOI
10.33697/ajur.2005.016

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A continuing and important problem in modeling physical processes deals with creating optimum calculation algorithms. These algorithms should provide the necessary precision yet not demand a lot of storage and time for evaluations. In the present paper the numerical integration of the non-stationary Schrödinger equation (NSSE) is considered. There are many numerical methods of integration of this equation to the present moment. In this paper, we explore the split-operator method (SOM) and consider a way of lessening time computing expenses in view of specificity of the given method. We also consider the choice of optimum discretization steps in coordinate and momentum space. The computer application that we outline is applicable, in method and preparation, to the solution more difficult problems, and it has been applied to simple chemical reactions. We use standard atomic units to simplify equations in the analysis that follows.
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In recent decades a lot of research has been done on the numerical solution of the time-dependent Schrodinger equation. On the one hand, some of the proposed numerical methods do not need any kind of matrix inversion, but source terms cannot be easily implemented into this schemes; on the other, some methods involving matrix inversion can implement source terms in a natural way, but are not easy to implement into some computational software programs widely used by non-experts in programming (e.g. Mathematica). We present a simple method to solve the time-dependent Schrodinger equation by using a standard Crank-Nicholson method together with a Cayley's form for the finite-difference representation of evolution operator. Here, such standard numerical scheme has been simplified by inverting analytically the matrix of the evolution operator in position representation. The analytical inversion of the N x N matrix let us easily and fully implement the numerical method, with or without source terms, into Mathematica or even into any numerical computing language or computational software used for scientific computing.
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Abstract The application of the numerical functional integration method to the solution of differential equations in quantum physics is discussed. We have developed a method of numerical evaluation of functional integrals in abstract complete separable metric spaces, which proves to have important advantages over the conventional Monte Carlo method of path integration. One of the considered applications is the investigation of open quantum systems (OQS), i.e., systems interacting with their environment. The density operator of OQS satisfies the known Lindblad differential equation. We have obtained the expression for matrix elements of this operator in the form of the double conditional Wiener integral and considered its application to some problems of nuclear physics. Another application is the solution of the Scr¨odinger equation with imaginary time and anticommuting variables for studying many-fermion systems. We have developed a numerical method based on functional integration over ordered subspaces. The binding energies of some nuclei are computed using this method. Comparison of the results with those obtained by other authors and with experimental values is presented.