ABSTRACT
In this chapter, the authors introduce the Hamilton-Jacobi equation showing that classical mechanics can be described with a wave equation for the classical action. In order to find the equation showing that classical mechanics already contains a wave behavior in matter particles, the people need to go back to the Lagrange/Hamilton formalism. Davisson-Germer experiment has shown without any doubt that electrons behave as massive particles whose dynamics is governed by some sort of wave. A free particle is apparently the conceptually simplest system to study because the TISE will reduce to its simplest form. The simple (linear) harmonic oscillator is of paramount importance in classical as well as quantum mechanics. A proper solution to the Harmonic oscillator problem can be found by solving Schrodinger equation analytically. Charles Hermite was a French mathematician who gave important contributions in many fields of mathematics used in physics like, for example, Hermitian operators and Hermite polynomials.
