In an exercise, we are supposed to show that the scattering matrix on the right of $$S_1(E)= \begin{pmatrix}t_1 & r_1' \\ r_1& t_1'\end{pmatrix}\delta(E_f-E_i)$$ is unitary. We are explicitly told, that the left side is, i.e. $S_1(E)^{-1}=S_1(E)^\dagger$ Now, this means the entire right side must be unitary. However, this silly Delta function is still in our way. We tried to write the delta function as a Fourier transform of a constant like in this unfortunately wrong answer: wrong answer on similar problem. The real problem is, that I do not see, what the delta function even has to do with the unitarity from a mathematical point of view it is a distribution that does not really make sense without an integral anyway. And, I certainly do not know how to get its adjoint. Pls, clarify for me.
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- 2$\begingroup$A tensor product is unitary iff its factors are unitary. The Dirac delta (i.e., the unit operator) is obviously unitary, so you only care about the unitarity of the matrix.$\endgroup$– AccidentalFourierTransformCommentedJan 21, 2018 at 18:17
- $\begingroup$How is the Dirac delta the unit operator? The delta for me is rather $\delta(E_f-E_i)=\langle E_f | E_i \rangle $. For a Kronecker delta, I see how it is connected to the identity but not for the Dirac delta.$\endgroup$– MarslCommentedJan 21, 2018 at 18:32
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