Numerical Solution

K. Chadan, Pierre C. Sabatier, Roger G. Newton

Z. Yousefian, N. Shadmani

In this paper, one of the numerical solution method of one- particle, one dimensional timeindependentSchrodinger equation are presented that allows one to obtain accurate bound state eigenvalues and functions for an arbitrary potential energy function V(x).For each case, we draw eigen functions versus the related reduced variable for the correspondingenergies. The paper ended with a comparison of the result obtained by the numerical solutions withthose obtained via the analytical solutions. The agreement between the results obtained by analyticalsolution method and numerical solution is represents the top Numerov method for numerical solutionSchrodinger equation with different potentials energy.

Melinda Hagedorn

Marko Mandarić

This bachelor thesis deals with the time-independent Schrodinger equation and the two ways of solving it: analytical and numerical. The goal is to show whether the equation can be solved precisely enough numerically on the computer, which would allow solutions for complicated cases that don’t have analytical solutions. Specifically, a solution for a particle in a harmonic potential is discussed. Dependence of the accuracy of the numerical solution on the parameters in the method itself is also discussed. This work presents an algorithm for the numerical solution of the Schrodinger equation. With the growth of computer power in the future, we will definitely be able to improve both the speed and precision of solving even more complicated problems than the one depicted here.

Nikolay A. Veretenov, Leonid A. Nesterov, Nikolay N. Rosanov

The motion of the center of a soliton in a trap with oscillating walls is studied analytically and numerically for the case in which the intrinsic frequency of small soliton oscillations in the equilibrium state considerably exceeds the frequency of wall oscillations. this problem can be solved either by applying the gross–pitaevskii equation, which most exactly describes the behavior of the soliton in the trap, or by using the approximate, “mechanical,” equation of motion of the newtonian type for the center of the soliton. an approximate analytical solution of the mechanical equation is obtained and is compared with the numerical solution of the newton equation, while the latter solution is compared with the numerical solution of the gross–pitaevskii equation. good agreement between the first two solutions is revealed. it is also shown that there is a range of parameters in which the numerical solutions of the newton and gross–pitaevskii equations are closest to each other. the frequency-sweeping effect of soliton center oscillations is revealed. an approximate analytical formula for the limiting frequency of these oscillations is obtained and the numerical analysis of this phenomenon is performed.