Coherent State

It is a minimum uncertainty state. If represented in the basis of Fock state:

\begin{align} \ket{\alpha}&=e^{-\frac{|\alpha|^2}{2}}\sum^\infty_{n=0}\frac{\alpha^n}{\sqrt{n!}}\ket{n}\tag{1}\\ &=e^{-\frac{|\alpha|^2}{2}}e^{\alpha \hat{a}\dagger}e^{-\alpha^*\hat{a}}\ket{0}\tag{2}\\ &=e^{\alpha \hat{a}\dagger-\alpha^*\hat{a}}\ket{0}\tag{3} \end{align} $$From (2) to (3), we use [[Baker–Campbell–Hausdorff Formula|Baker–Campbell–Hausdorff]] (BCH) formula.