行列表示
A ^ = 1 ^ A ^ 1 ^ = ∑ m , n | m ⟩ ⟨ m | A ^ | n ⟩ ⟨ n | = ∑ m , n | m ⟩ A m n ⟨ n | | ψ ⟩ = 1 ^ | ψ ⟩ = ∑ n | n ⟩ ⟨ n | ψ ⟩ = ∑ n ψ n | n ⟩ {\displaystyle {\begin{aligned}{\hat {A}}&={\hat {1}}{\hat {A}}{\hat {1}}=\sum _{m,n}|m\rangle \langle m|{\hat {A}}|n\rangle \langle n|=\sum _{m,n}|m\rangle A_{mn}\langle n|\\|\psi \rangle &={\hat {1}}|\psi \rangle =\sum _{n}|n\rangle \langle n|\psi \rangle =\sum _{n}\psi _{n}|n\rangle \end{aligned}}}
この ⟨ m | A ^ | n ⟩ = A m n {\displaystyle \langle m|{\hat {A}}|n\rangle \ =A_{mn}\ } を、「演算子 A ^ {\displaystyle {\hat {A}}\ } の行列要素」と呼ぶ。