Lowering Operator

${\hat{a}}_{-}=i\frac{\hat{p}}{p_0}+\frac{\hat{x}}{x_0}=\frac{1}{\sqrt{2m\hslash \omega }}\left(\hslash \frac{\mathrm{d}}{\mathrm{d}x}+m\omega x\right)$

• Applying this operator on $\Psi$ decreases its energy by $\hslash \omega$
• $\hat{p}$ is the momentum operator
• $p_0$ is the characteristic momentum
• $\hat{x}$ is the position operator
• $x_0$ is the characteristic length

$\hat{H}({\hat{a}}_{-}{\Psi }_n)=E_n-\hslash \omega$

• Each application of ${\hat{a}}_{-}$ decreases the energy by $\hslash \omega$

${\hat{a}}_{-}{\psi }_n=\sqrt{n}{\psi }_{n-1}$

${\hat{a}}_{-}{\psi }_0=0$