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:

for

for

in equation:

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 :

thus,

together with the wavefunction:

thus implies,

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,

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