I am reading Landau's quantum mechanics textbook for interests. I try to rewrite it by Dirac notation but this is not as obvious as I thought. e.g. , eq(3.8) in Landau, defines the mean value of an operator: \begin{equation} \bar{f} = \int\Psi^*(q)(\hat{f}\Psi(q))\,dq \end{equation} In Dirac notation: \begin{equation} \bar{f} = \langle\Psi|\hat{f}|\Psi\rangle = \iint\Psi^*(q)f_{qq'}\Psi(q')\,dqdq' \end{equation} where $\Psi(q)=\mathinner{\langle q}|{\Psi\rangle}; f_{qq'} = \mathinner{\langle q|}\hat{f}\mathinner{|q'\rangle}$. I am wondering how to show the equivalence between these two formulas?
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