PX262 - C5 - expectation value

when measurements of a dynamical variable represented by the operator, $\hat{Q}$ , are repeated on identical copies of a system, the average value obtained is called the expectation value
the prediction for the expectation value is:

\begin{aligned} ⟨ \hat{Q} ⟩ & = \int ψ^{*} \hat{Q} ψ d τ \\ = \int (\sum_{m} a_{m}^{*} ϕ_{m}^{*}) \hat{Q} (\sum_{n} a_{n} ϕ_{n}) d τ \\ = \sum_{m, n} a_{m}^{*} a_{n} q_{n} \int ϕ_{m}^{*} ϕ_{n} d τ \\ = \sum_{m, n} a_{m}^{*} a_{n} q_{n} δ_{m n} \\ = \sum_{n} a_{n}^{*} a_{n} q_{n} \\ ∴ ⟨ \hat{Q} ⟩ & = \sum_{n} | a_{n} |^{2} q_{n} \end{aligned}

the standard deviation can be defined as: $\sqrt{δ Q} = \sqrt{⟨ {\hat{Q}}^{2} ⟩ - ⟨ \hat{Q} ⟩^{2}}$