Ladder Operators

Types

• Raising Operator
• Lowering Operator

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

• This operator is a commutator

$⟨{\Psi }_n|{\hat{a}}_{\pm }{\Psi }_m⟩=⟨{\hat{a}}_{\mp }{\Psi }_n|{\Psi }_m⟩$

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

$L_{\pm }\equiv L_x\pm i{L}_y$

• Ladder operator for angular momentum

$S_{\pm }\equiv S_x\pm i{S}_y$