Integrated Nanophotonic Devices, 2nd Edition

(1.104)

and the creator:

${\hat{a}}^{+}={(2m\hslash \mathrm{\Omega })}^{-1/2}(m\mathrm{\Omega }\hat{q}-i\hat{p})$ (1.105)

(1.105)

From its definition it is clear that:

$\left[\hat{a},{\hat{a}}^{+}\right]=1$ (1.106)

(1.106)

Note that the commutator for any operator with a constant (here denoted as c) is always zero:

$\left[\hat{\theta },c\right]=0$ (1.107)

(1.107)

The Hamiltonian may also be defined according to the two new operators as:

$\hat{H}(\hat{p},\hat{q})=\hslash \mathrm{\Omega }\left({\hat{a}}^{+}\hat{a}+\frac{1}{2}\right)$ (1.108) ...