MathematicalPhysicist
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 Problem Statement

I want to derive equation (N.10) on page 791, i.e.:
$$\sum_f(\Phi_i, A\Phi_f)(\Phi_f,B\Phi_i)=(\Phi_i,AB\Phi_i)$$
where ##A,B## are operators, and ##\Phi_i, \Phi_f## are the initial and final wavefunctions before and after the scattering.
 Relevant Equations
 $$(\Phi_i,\Phi_f) = \int \Phi_i^* \Phi_f$$
Well as always start with the definition of scalar product:
$$\sum_f (\Phi_i,A\Phi_f)(\Phi_f ,B\phi_i) = \sum_f \int \Phi_i^*A\Phi_f \int \Phi_f^* B\Phi_i=\int \int \Phi_i^* \sum_f A\Phi_f \Phi_f^* B\Phi_i$$
How to continue from the last equality?
Thanks.
$$\sum_f (\Phi_i,A\Phi_f)(\Phi_f ,B\phi_i) = \sum_f \int \Phi_i^*A\Phi_f \int \Phi_f^* B\Phi_i=\int \int \Phi_i^* \sum_f A\Phi_f \Phi_f^* B\Phi_i$$
How to continue from the last equality?
Thanks.