Novel Numerical Solution of Schrodinger Equation for Hydrogen-like Atoms

Maysoon A. Mahmood

Abstract: An improved numerical approach is used to find electron probability distribution around the nucleus and the energy levels for all subshells in Hydrogen-like atoms. This approach exploit the built-in capability of MATLAB to find Eigen values and Eigen vectors and its capability to draw the electron distribution in three dimensions. The space around the nucleus is divided into a large number of finite elements in three dimensions. The size of element is increased with the shell number to cover the space of existence of electron without increasing the number of calculations. Hamiltonian operator matrix was constructed according to Schrodinger equation for the space around the nucleus. The Eigen values for the Hamiltonian operator matrix are found to be the electron energies for each electron subshell around the nucleus. The corresponding Eigen vectors give the value of the wave function of the electron for the corresponding subshell in each spatial element. Three dimensional plots for the square of the magnitude of the wave functions for several subshells are plotted using MATLAB to show the distribution of electron around the nucleus for these subshells. The results of this method shows fabulous coincidence with analytical results for both, energy levels and electron distribution around nucleus

Keywords: Quantum chemistry, quantum mechanics, Hamiltonian operator matrix, schrodinger equation, numerical solution

Edition: Volume 6 Issue 3, March 2017,

Pages: 653 - 657