Some concepts and fundamental mathematical tools of the quantum theory of many particle systems, indispensable to the study of solid state physics are presented. Among concepts, we highlight those of collective stimuli and quasi-particles. As far as mathematical tools are concerned, quantum mechanics is presented in its general formulation, independent of representation, and the problem of many particles is considered in the representation of occupancy number or second quantization. Finally, as an application, the interaction between electrons and the vibrations of a crystalline lattice, descited in terms of elementary excitations of collective waves, or phonons, is expressed and discussed within this scheme.
