Free-particle model

All models of this type have become known colloquially by the misnomer free-particle model. Diverse objects with formal resemblance to chemical systems are included here, such as an electron in an impenetrable sphere to model activated atoms particle on a line segment to model delocalized systems particle interacting with finite barriers to simulate tunnel effects particle interacting with periodic potentials to simulate electrons in solids, and combinations of these. 

To conclude, after the canonical transformation we have two equivalent models (1) the initial model (145) with the eigenstates (160) and (2) the fictional free-particle model (154) with the eigenstates (158). We shall call this second model polaron representation. The relation between the models is established by (155)-(157). It is also clear from the Hamiltonian (148), that the operators < , d. ad. and a describe the initial electrons and vibrons in the fictional model. 

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