How is the Schrödinguer equation actually solved? Three different methods to address the Quantum Harmonic Oscillator problem

Abstract

Quantum chemistry is a central subject in Chemistry degree because it provides ultimate answers to explain reactivity, structural, stability and spectroscopy of atoms molecules and solids. However, to derive practical conclusions, one need to be familiar with several specific mathematical concepts and techniques. Undergraduate courses on Quantum Chemistry often omit much of this mathematical background in order to highlight the physical concepts, as doing otherwise would likely confuse the majority of students. Who would “not see the forest because of the leaves”. As a result, the solutions of the Schrödinguer equation are usually provided right after presenting the Hamiltonian. Obviously, a student with interest in the matter realizes he is missing an important step, for he is then unable to obtain solutions autonomously. The goal of this work is to provide an introductory view on some of the mathematical tools used in Quantum Chemistry to obtain eigenvalues and eigenstates of the time-independent Schrödinguer equation. Namely, we study polynomial method, the factorization method and the method of finite differences. All three techniques are applied to solve the Quantum Harmonic oscillator problem. We purpose fully fully choose this problem because it is pervasive in most areas of Physical Chemistry, and susceptible of being solved through any of the techniques, which will allow us to draw comparisons and conclusions in the end.