what is scattering amplitude?

Question

IN QUANTUM MECHANICS ,when we are reading scattering of objects through targets we have this .

Answer 1

In quantum mechanics, scattering theory or quantum field theory, the S-matrix relates the final state in the infinite future (out-channels) and the initial state in the infinite past (in-channels). The "S" stands for "scattering" or "Strahlung" (radiation).

More mathematically, the S-matrix is defined as the unitary matrix connecting asymptotic particle states in the Hilbert space of physical states (scattering channels). While the S-matrix may be defined for any background (spacetime) that is asymptotically solvable and has no horizons, it has a simple form in the case of the Minkowski space. In this special case, the Hilbert space is a space of irreducible unitary representations of the inhomogeneous Lorentz group; the S-matrix is the evolution operator between time equal to minus infinity, and time equal to plus infinity. It can be shown that if a quantum field theory in Minkowski space has a mass gap, the state in the asymptotic past and in the asymptotic future are both described by Fock spaces.

The S-matrix is closely related to the transition probability amplitude in quantum mechanics and to cross sections of various interactions; the elements (individual numerical entries) in the S-matrix are known as scattering amplitudes. Poles of the S-matrix in the complex-energy plane are identified with bound states, virtual states or resonances. Branch cuts of the S-matrix in the complex-energy plane are associated to the opening of a scattering channel.

for more pl. visit:
http://en.wikipedia.org/wiki/Scattering_amplitude

Answer 2

The S-matrix is closely related to the transition probability amplitude in quantum mechanics and to cross sections of various interactions; the elements (individual numerical entries) in the S-matrix are known as scattering amplitudes.