Ladder Operators

#Physics

Types

• Raising Operator
• Lowering Operator

$\displaystyle[\hat{a}{-},\hat{a}{+}]=-[\hat{a}{+},\hat{a}{-}]=1$

• This operator is a commutator

$\displaystyle {\left\langle{\Psi_{n}|\hat{a}{\pm}\Psi{m}}\right\rangle}={\left\langle{\hat{a}{\mp}\Psi{n}|\Psi_{m}}\right\rangle}$

• $\displaystyle \hat{a}{\pm}$ and $\displaystyle \hat{a}{\mp}$ are Hermitian adjoints of each other

$\displaystyle L_{\pm }\equiv L_{x}\pm iL_{y}$

• Ladder operator for angular momentum

$\displaystyle S_{\pm }\equiv S_{x} \pm iS_{y}$