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

Document Type : Research Paper




In this paper, the numerical solution methods of one- particale, one – dimensional time- independent
Schrodinger equation are presented that allows one to obtain accurate bound state eigen values and
eigen 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 in
more details. For this purpose, we first reformulated the Shrodinger equation using dimensionless
variables, the estimating the initial and final values of the reduced variable xr and the value of
intervals sr, and finally making use of Q-Basic or Spread Sheet computer programs to numerically
solved the equation. For each case, we drew the eigen functions versus the related reduced variable
for the corresponding energies. The harmonic oscillator, the Morse potential, and the H-atom radial
Schrodinger equation, … were the examples considered for the method. The paper ended with a
comparison of the result obtained by the numerical solutions with those obtained via the analytical
solutions. The agreement between the results obtained by analytical solution method and numerical
solution for some Potential functions harmonic oscillator̕ Morse was represents the top Numerov
method for numerical solution Schrodinger equation with different potentials energy.