Expected Value of an Observable

To determine the mean (EV) of an operator A^\hat{A}, we must know the state ∣ψ⟩\left\lvert \psi \right\rangle you’re in.

E(A)=∑nAnP(n)=∑An∣⟨n∣ψ⟩∣2=∑An⟨ψ∣n⟩⟨n∣ψ⟩=⟨ψ∣∑nAn∣n⟩⟨n∣∣ψ⟩=⟨ψ∣A^∣ψ⟩≜⟨A^⟩\begin{gather*} \mathbb{E}(A)=\sum_nA_nP(n)\\ =\sum A_n|\left\langle n|\psi \right\rangle|^2\\ =\sum A_n\left\langle \psi|n \right\rangle\left\langle n|\psi \right\rangle\\ =\left\langle \psi \right\rvert\sum_n A_n\left\lvert n \right\rangle\left\langle n \right\rvert\left\lvert \psi \right\rangle\\ =\left\langle \psi \right\rvert\hat{A}\left\lvert \psi \right\rangle\\ \triangleq\left\langle \hat{A} \right\rangle \end{gather*}

Another convientn reprsentation is

ρ^≜∣ψ⟩⟨ψ∣⇒⟨A^⟩=tr(ρ^A^)=tr(∣ψ⟩⟨ψ∣A^)tr(⟨ψ∣A^∣ψ⟩)\begin{gather*} \hat{\rho}\triangleq\left\lvert \psi \right\rangle\left\langle \psi \right\rvert\\ \Rightarrow\left\langle \hat{A} \right\rangle=tr(\hat{\rho}\hat{A})\\ =tr(\left\lvert \psi \right\rangle\left\langle \psi \right\rvert\hat{A})\\ tr(\left\langle \psi \right\rvert\hat{A}\left\lvert \psi \right\rangle) \end{gather*}

where trtr is trace