Quantum Harmonic Oscillator Wave Function

The wave function of a quantum harmonic oscillator varies depending on the energy level of the particle being described. Using the number operator, the wave function of a ground state harmonic oscillator can be found. Repetitively applying the raising operator to the ground state wave function then allows the derivation of the general formula describing wave functions of higher energy levels.

Equivalency of Hermite Polynomial Definitions

Hermite polynomials are a set of polynomials which arise frequently in probability calculations and in quantum physics when finding the wave functions of harmonic oscillators. Often, Hermite polynomials are defined by two seemingly different mathematical definitions. Here, it is proved that these definitions in fact yield equivalent results.