Boson operators

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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

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