The Quantum Mechanical Simple Harmonic Oscillator

Joseph H. Simmons, Kelly S. Potter

Murilo B. Alves

The harmonic oscillator is one of the most studied systems in Physics with a myriad of applications. One of the first problems solved in a Quantum Mechanics course is calculating the energy spectrum of the simple harmonic oscillator with analytic and algebraic approaches. In the algebraic solution, creation and annihilation operators are introduced to factorize the Hamiltonian. This work presents an algebraic solution for the simple harmonic oscillator in the context of Classical Mechanics, exploring the Hamiltonian formalism. In this solution, similarities between the canonical coordinates in a convenient basis for the classical problem and the corresponding operators in Quantum Mechanics are highlighted. Moreover, the presented algebraic solution provides a straightforward procedure for the quantization of the classical harmonic oscillator, motivating and justifying some operator definitions commonly used to solve the correspondent problem in Quantum Mechanics.

Gerardo Muñoz

The energy eigenstates for the simple harmonic oscillator in quantum mechanics are found through the solution of an integral equation.

Erich W. Schmid, Gerhard Spitz, Wolfgang Lösch

David K. Ferry

One of the most commonly used quantum mechanical systems is that of the simple harmonic oscillator . The prototype system is the simple pendulum. Classically, the motion of the system exhibits harmonic, or sinusoidal oscillations. The sinusoidal motion of classical mechanics is the simplest system that produces oscillatory behavior, and therefore is found in almost an infinity of physical systems. Thus, the properties of the harmonic oscillator become important for their use in describing quite diverse physical systems. In a solid, the motion of the atoms is much like the motion of free particles, with the important exception that the atoms are forced on average to retain positions within the solid that specify a particular crystal structure. The motion remains that of a harmonic oscillator, but now the shift of the wave function and the energy levels are hybrids of the confinement harmonic oscillator and the magnetic harmonic oscillator.