# QUANTUM EVOLUTION AND SCATTERING MATRIX

# QUANTUM EVOLUTION AND SCATTERING MATRIX

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

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