Derivation of eq. (N.10) on page 791 of Appendix N in Ashcroft's SSP.

[MathematicalPhysicist]

Homework 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.

 

[MathematicalPhysicist]

I think I got it, $\sum_f A \phi_f^* \phi_f = A$ since $\sum_f \phi_f^*\phi_f = 1$.

Am I correct?