Annihilation and creation operators for the quantum harmonic oscillator

The annihilation (a) and creation (a^†) (aka lowering and raising) operators enable you to go back and forth between the adjacent eigenfunctions of a given particle's wave function.

However, the real fun of these operators comes into play when they are used together. The two operators are derived from the factorization of the Hamiltonian (H) for the quantum harmonic oscillator problem where

H = kinetic energy + potential energy
    = p²/2m + √(w/m)x²/2
    = w (√(mw/2)x - ip/√(2mw))((√(mw/2)x + ip/√(2mw)
    = hw/2π( ½ + a^†a)

(w represents the angular frequency of the particle, m is the particle's mass, and x and p denote the position and momentum, respectively)

One can then use the annihilation and creation operators to represent both the position and momentum functions for the particle:

x = (1/2)√(h/πmw)(a + a^†)

p = (1/2)√(mwh/π)(a - a^†)