Calculation of Expectation Values and Uncertainty for Linear Potential by Using Numerov Method

K. Hagino

A standard way to solve a Schrödinger equation is to discreteize the radial coordinates and apply a numerical method for a differential equation, such as the Runge-Kutta method or the Numerov method. Here I employ a discrete basis formalism based on a finite mesh method as a simpler alternative, with which the numerical computation can be easily implemented by ordinary linear algebra operations. I compare the numerical convergence of the Numerov integration method to the finite mesh method for calculating penetrabilities of a one-dimensional potential barrier, and show that the latter approach has better convergence properties.

Bruce Cameron Reed

Zidan Afidah, Bambang Supriadi, Rif’ati Dina Handayani

The positive helium ion consists of two protons and one orbital electron to be viewed as a hydrogenic atom. The use of ions has very potential, especially in the development of spintronic devices and plasma-based sterilization. This study aims to determine the expected value and position uncertainty of He^+ ion electrons. Determination of the expected value and position uncertainty is done by analytical calculation through the Schrodinger method approach, especially the radial part at quantum number n≤4, and numerical calculation with the help of a simulation program. The results showed that the main quantum number and orbital n, l influence the expectation value and position uncertainty. The greater the main quantum number n, the expectation value of electron position decreases and the uncertainty of electron position increases. At the same time, the increase in the orbital quantum number l has an impact on the value, which tends to go down. This research is in accordance with de Broglie's hypothesis about the nature of dualism and the principle of uncertainty in the review of position space.

Scott H. Werden, Ernest R. Davidson

FendaRizky Pratama

The eigenfunctions as the solutions of Schrodinger equation for particle in the box, simple harmonic and perturbed oscillators, as well as in the case of Coloumb, linear, Morse, Yukawa, and modified Woods-Saxon potentials are simulated with fourth-order Numerov algorithm. The simulation techniques for initialization, which is aimed to acquire qualitative behaviour of eigenfunctions are derived by solving Schrodinger equation in particular region directly or by WKB approximation and Bohr-Sommerfeld quatization rule. The simulated eigenfunctions possess good accuracy, with maximum relative error in the order 10─3 compared to the analytical eigenfunctions or are in the good agreements with the analytical eigenfunctions calculated in the previous works. This work also provides some illustrations of how approximation methods are able to improve the accuracy in computational simulations.