Hamiltonian Operator

$\hat{H}=\hat{K}+\hat{U}=-\frac{{\hslash }^2}{2m}\nabla +U$

$\hat{H}=\left({\hat{a}}_{+}{\hat{a}}_{-}+\frac{1}{2}\right)\hslash \omega =\left({\hat{a}}_{-}{\hat{a}}_{+}-\frac{1}{2}\right)\hslash \omega$

• ${\hat{a}}_{+}$ is the raising operator
• ${\hat{a}}_{-}$ is the lowering operator
• A mnemonic here is to apply +-+ or -+- wrt to the operators and $\frac{1}{2}$ factor