Investigation of analytical and numerical solutions for one-dimensional independent-oftime Schrödinger Equation

Subject Areas : Journal of Physical & Theoretical Chemistry

Z. Yousefian Molla ¹ , Gh. Islampour ²

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Received: 2014-04-25 Accepted : 2014-04-25 Published : 2013-01-01

Keywords: Independent-of-time Schrödinge, Numerical Solution, analytical solution, Numero method, Spreadsheet,

Abstract :

In this paper, the numerical solution methods of one- particale, one dimensional time- independentSchrodinger equation are presented that allows one to obtain accurate bound state eigen values andeigen functions for an arbitrary potential energy function V(x). These methods included the FEM(Finite Element Method), Cooly, Numerov and others. Here we considered the Numerov method inmore details. For this purpose, we first reformulated the Shrodinger equation using dimensionlessvariables, the estimating the initial and final values of the reduced variable xr and the value ofintervals sr, and finally making use of Q-Basic or Spread Sheet computer programs to numericallysolved the equation. For each case, we drew the eigen functions versus the related reduced variablefor the corresponding energies. The harmonic oscillator, the Morse potential, and the H-atom radialSchrodinger equation, … were the examples considered for the method. The paper ended with acomparison of the result obtained by the numerical solutions with those obtained via the analyticalsolutions. The agreement between the results obtained by analytical solution method and numericalsolution for some Potential functions harmonic oscillator̕ Morse was represents the top Numerovmethod for numerical solution Schrodinger equation with different potentials energy.