Lowering Operator
#Physics
$\displaystyle \hat{a}{-}=i \frac{\hat{p}}{p{0}}+ \frac{\hat{x}}{x_{0}}=\frac{1}{\sqrt{ 2m\hbar \omega }}\left( \hbar \frac{ \mathrm{d} }{ \mathrm{d}x }+m\omega x \right)$
- Applying this operator on $\displaystyle \Psi$ decreases its energy by $\displaystyle \hbar \omega$
- $\displaystyle \hat{p}$ is the momentum operator
- $\displaystyle p_{0}$ is the characteristic momentum
- $\displaystyle \hat{x}$ is the position operator
- $\displaystyle x_{0}$ is the characteristic length
$\displaystyle \hat{H}(\hat{a}{-}\Psi{n})=E_{n}-\hbar \omega$
- Each application of $\displaystyle \hat{a}_{-}$ decreases the energy by $\displaystyle \hbar \omega$