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This chapter shows how scattering problems are formulated in the framework of path integrals. In quantum mechanics, the state of an isolated system evolves under the action of a unitary operator, as a consequence of the conservation of probabilities and, thus, of the norm of vectors in Hilbert space. Quantum evolution (that is, in real time) is introduced, after which a path integral representation of the scattering matrix is constructed. From this *S* matrix, the standard perturbative expansion in powers of the potential is recovered. Even the evolution of a free quantum particle is slightly non-trivial; in general, one observes a spreading of wave packets. Scattering is then characterized by the asymptotic deviations at infinite time from this free evolution and this leads to the definition of a scattering or *S*-matrix. An S-matrix is defined in the example of bosons and fermions. Various other semi-classical approximation schemes are then discussed.

*Keywords: *
quantum evolution;
scattering matrix;
path integrals;
perturbative expansion;
bosons;
fermions;
semi-classical approximation;
free quantum particle

*Chapter.*
*5189 words.*

*Subjects: *
Astronomy and Astrophysics
;
Mathematical and Statistical Physics

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