Creation / Annihilation operators

Definitions§

$$\hat{a}\ket{n}=\sqrt{n}\ket{n-1}\\ \hat{a}^\dagger\ket{n}=\sqrt{n+1}\ket{n+1}\\$$

Where, $\hat{a}$ is the annihilation operator and it's conjugate is the creation oprator.

Properties§

Arbitrary states§

By direct recursion, we easily find that,

$$\left(\hat{a}^\dagger\right)^n\ket{0}=\sqrt{n!}\ket{n}$$

Thus

$$\ket{n} = \frac{1}{\sqrt{n!}} \left( \hat{a}^\dagger\right)^n\ket{0}$$

Commutation§

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

The number operator§

$$\hat{a}^\dagger\hat{a} = n$$

And by the commutation relation,

$$\hat{a}\hat{a}^\dagger = n + 1$$

Anti-commutation§

$$\{\hat{a}, \hat{a}^\dagger\}=\hat{a}\hat{a}^\dagger + \hat{a}^\dagger\hat{a} = 2n+1$$

Modes§

For different modes ( $k$ or $\omega$ ),

$$[\hat{a}_k, \hat{a}^\dagger_{k'}]=\delta_{kk'}$$

General single photon§

$$\ket{\psi;f}=\sum_k f(k)\hat{a}^\dagger_k \ket{0}$$

Other Commutation relation§

$$[\hat{a}^\dagger \hat{a}, \hat{a}^\dagger]=\hat{a}^\dagger[\hat{a}, \hat{a}^\dagger]=\hat{a}^\dagger$$ $$[\hat{a}^\dagger \hat{a}, \hat{a}]=[\hat{a}^\dagger, \hat{a}]\hat{a}=-\hat{a}$$