The harmonic oscillator has many different realizations in physics: vibrational and rotational modes in molecules, sound modes in solids, electromagnetic modes in cavities, ... In some cases the harmonic oscillator hamiltonian is exact, and in others it is only approximate near a minimum in the potential. The quantum mechanical harmonic oscillator can be solved exactly. The first step in our exact solution is to re-express the harmonic oscillator hamilonian in terms of the operators, a and a^dag: H = hbar omega (a^dag a + 1/2). The commutation relation of a and a^dag is [a,a^dag] = 1.