Streuamplitude

The Streuamplitude, or scattering amplitude, is a complex quantity that encodes the probability amplitude for a scattering process to occur between specified initial and final states in quantum mechanics and wave physics. It is the central object that relates the microscopic dynamics of a system to measurable scattering observables.

In nonrelativistic potential scattering, the scattering amplitude is usually denoted f(θ) and depends on the scattering angle θ and the energy. The differential cross section is given by dσ/dΩ = |f(θ)|^2. The total cross section is related to the forward scattering amplitude through the optical theorem: σ_tot = (4π/k) Im f(0), where k is the incident wave number. For central potentials, f(θ) admits a partial wave expansion f(θ) = (1/2ik) ∑_{l=0}^∞ (2l+1) [e^{2iδ_l} − 1] P_l(cos θ), with δ_l the phase shifts and P_l the Legendre polynomials. This expansion expresses how different angular momentum channels contribute to the scattering.

In quantum field theory, the scattering amplitude is the matrix element of the S-matrix between asymptotic states and is often denoted M (the invariant amplitude). For a given process, M depends on the kinematic invariants such as the Mandelstam variables s, t, and u. The observable cross section is proportional to |M|^2 times known kinematic factors, and M is computed from fundamental interactions using Feynman diagrams. Amplitudes can be complex, carrying both magnitude and phase, and are constrained by principles such as unitarity, analyticity, and crossing symmetry. Poles in the amplitude signal resonances or bound states.

In summary, the Streuamplitude is the foundational quantity connecting microscopic dynamics to experimental scattering data across both nonrelativistic and relativistic frameworks.