Hamiltonian Operator

#Physics

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

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

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