Analisis Energi dan Fungsi Gelombang Persamaan Dirac Potensial Shape Invariant Hulthen, Eckart dan Rosen Morse dengan Menggunakan Metode Polinomial Romanovski

摘要

Umi Khoiriyah. S911208006. 2015. “ Analitical Solution Energy Eigen Value and Wave Function of Shape Invariant Hulthen, Eckart and Rosen Morse Potential With Romanovski Polynomial”. Thesis: Program Pascasarjana Ilmu Fisika Universitas Sebelas Maret Surakarta. Advisor: (1). Prof. Dra. Suparmi, M.A., Ph.D (2). Prof. Drs. Cari, M.Sc., M.A., Ph.D ABSTRACT This research is aimed to determine energy levels and wave functions from Dirac equation for Potential Hulthen, Eckart and Rosen Morse using Polinomial Romanovski method. They are a shaped-invariance potential.Recently developed supersymmetric in field theory has been successfully employed to make a complete mathematical analysis of the reason behind exact solvability of some shaped-invariant potentials in a close form. Then, by operating the lowering operator we get the ground state wave function, and the excited state wave function can be gained by operating raising operator. Non central potential Rosen Morse, Hulthen and Eckart are the potential which separated variable. Wafe function of radial and angular for Hulthen Plus Rosen Morse Potential and Eckart Hulthen Plus Rosen Morse Potential are solved by Romanovski polynomials method. Eigen function that be found can’t be solved by analytical method or approximation value, so that must be solved by numerical method. To solve Dirac equation with Romanovski polynomials we must reduce the two order differential equation to be intermediatery Hypergeometri differential equation with substituting of suitable variable with the Romanovski parameters. To find energy eigen and wave function can be found by subtituting Romanovski’s wave function like into the intermediatery Hypergeometri differential equation and derivating until be obtained the Romanovski’s differential equation. From its Romanovski’s Hypergeometri equation we would determine the energy levels and wave function. So it formed the level of Energy and the wave functions, consist of radial and angular part, are given in Romanovski polynomial form. Energy spectrum, wave functions and probability density graph have been visualized by Matlab 2013. Visualization of radial and polar wave functions might be used to description by the probability of particle position radially and polarly. Key word: Dirac equation, Shape-Invariant Potential, Romanovski polinomial.