ANALYTICAL SOLUTION OF ATOMIC HELIUM USING ALGEBRAIC COMPUTATION METHOD - ROSYID MAHMUDI (NIM:20210020); Pembimbing : Prof. Toto Winata, Ph. D, MOH.
The solution of Schroedinger equations of many electrons atom is usually done by many researchers using numerical approximation, perturbations method, and variational calculus. In this thesis the solution of Schroedinger equation will be described by using the algebraic computational method. The algebraic computational method is an analytical method which uses the algebraic software. Matlab was used throughout this tesis. The solution which obtained by Matlab is Whittaker’s functions. Whittaker’s functions is hypergeometric functions which confluent with Kummer’s function. The general solution by this method is in the form of Kummer’s functions. The general solution was then associated with the boundary conditions of the wave functions. The energy levels were obtained by diagonalization of Hamiltonian using the general solution. The solution was then called the algebraic solution. This method was successfully validation for hydrogen atom case. The obtained energy levels of atomic hydrogen are exactly the same as the result of the analytical method using Laguerre’s functions. This method was further applied to solve the atomic helium Schroedinger equation case which is a simplest many electron atom. The most difficult problem is in the interaction of two electrons, which is called Coulomb potential interactions. This interactions depend of the distance between two electrons. Therefore we used Legendre’s series to approximate this interaction. First order term in the Legendre’s series, is used in algerbraic computation calculation because only depend on one electron. And second order will be calculated later due to depend on position of both electron and can’t be separately. The obtained energy levels of helium is almost the same as references with the error of under 1%. The benefit this method is more simple compared other methods.