  In this study, potential scatterings are formulated in experimental setups
with Gaussian wave packets in accordance with a probability principle and
associativity of products. A breaking of an associativity is observed in scalar
products with stationary scattering states in a majority of short-range
potentials. Due to the breaking, states of different energies are not
orthogonal and their superposition is not suitable for representing a
normalized isolate state. Free wave packets in perturbative expansions in
coupling strengths keep the associativity, and give a rigorous amplitude that
preserves manifest unitarity and other principles of the quantum mechanics. An
absolute probability is finite and comprises cross sections and new terms of
unique properties. The results also demonstrate an interference term displaying
unique behavior at an extreme forward direction.
