Quantum harmonic oscillator/Boson operators

The single mode boson operators come in two flavours, the creation operator, and the annihilation operator. They are denoted like the following:


 * $$\hat{b}^\dagger$$
 * $$\hat{b}$$

The pair are called the "ladder operators". They arise from consideration of the quantum harmonic oscillator. The commutation relation that they satisfy is


 * $$[\hat{b}, \hat{b}^\dagger] = 1$$