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:

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

${\displaystyle {\hat {b}}}$

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

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