at the point of scattering ( t = 0 ), the wave function and the incoming and out-going wave function can be related as:
where the incoming and out-going wavefunction is very far away from the field of the scatter, and thus, they are free and we say they are asymptotic.
Let a time propagator with a full Hamiltonian be U(t), and a time propagator with free Hamiltonian be . thus the time behavior of the scattering wave functions can be related as:
Thus, we have the scattering matrix or the S-matrix.
Now assume for a particular state generated by an accelerator is Φ, and a particular out-going asymptotic state is χ. we have:
thus, the probability amplitude for the scattering between these 2 states is:
if we expand a wave function in momentum basis:
now, we are going to show the energy conservation of the scattering operator or matrix, by showing that the scattering operator S commutes with the Hamiltonian. from
differential it then we have :
together with the wavefunction:
since, at the forward direction, the change of momentum is zero, we can write S = 1 + R, then,
the is called on-shell T-matrix. since the energy must be equal, required by the delta function, thus, the momentum magnitude must be equal, therefore, the 2 momentums s on a shell. The T-matrix also related to the scattering amplitude by:
then the S-matrix becomes,