Introducing S matrix in scattering theory What was the need to introduce the concept of S matrix in scattering theory?while we were studying scattering with partial wave analysis.
 A: The $S$ matrix is what permits you to do predictions on the final state of some process whenever the initial state is known. You build it up from wave-operators which give you the possibility to find some interacting asymptotic states which approach the free evolution for $t\to\pm\infty$.
Using Dirac braket notation, if you have an initial state $|i\rangle$ and you want to know what is the probability that the initial state goes to a specific final state $\langle f|$, then you have to compute the following matrix element $$\langle f|S|i\rangle$$
The beauty of the scattering matrix is that it enables you to do perturbative calculations since in most of the cases, you won't be able to compute practically the full matrix element, due to the impossibility to evaluate precisely the eigenfunctions of an interacting hamiltonian. 
Moreover, the $S$ matrix can be built up beginning from any given hamiltonian, without special restrictions. To do perturbative calculations, obviously, you need the interacting term to be small. 
