A Wave-Mechanical Treatment of the Simple Harmonic Oscillator and the Hydrogen Atom

Joseph H. Simmons, Kelly S. Potter

H. Beker

Shabnam Siddiqui

In this chapter, the authors solve the Schrodinger equation for a harmonic oscillator using two methods, analytical method and algebraic method. They discuss the quantum properties of the harmonic oscillator. A harmonic oscillator is a system that exhibits simple harmonic motion or periodic motion, a motion that repeats itself after equal intervals of time. Many systems such as spring, simple pendulum, vibrating string and molecular vibration can be approximated as simple harmonic oscillators. The authors consider a simple harmonic oscillator, a spring whose spring constant is "k" attached to a mass "m" as a simple harmonic oscillator. They describe the Schrodinger equation in spherical coordinates. The authors also solve the three-dimensional Schrodinger equation for the hydrogen atom. For the hydrogen atom, the potential energy function is dependent only on radial distance and is independent of angular variables. The hydrogen atom consists of a positively charged proton surrounded by an electron that orbits around it.

Zhao Su-qin

Through the counterchanges of two methodologies of coordinates and special equations,the eigen equations of the three-dimensional harmonic oscillator and hydrogen atom are converted into the same equations in form.Then the relationship between the energy and wave function of the three-dimensional harmonic oscillator and hydrogen atom is given.

M. Kibler, Tidjani Négadi