Quantization of the canonical ensemble

The task of quantization consists of forming a quantum mechanical system fr om a given classical system. Thereby, classical observables are replaced by quantum mechanical observables. We carry out this procedure with Gibbs' ca nonical ensemble. Therefrom, we obtain a density matrix which represents a quantum mechanical concept of equipartition. The most important justificati on for this principle is provided by the virial theorem. Thereafter, the im plications of the density matrix are evaluated. For this purpose, we define a virtual free energy. Thus, the equilibrium condition of the quantum cano nical ensemble is derived by minimizing the virtual free energy. We demonst rate our method for the case of a mean field like order-disorder phase tran sition. This theory leads to new predictions for solids at low temperatures . The model may, for instance, be applied to potassium dihydrogen phosphate or its isomorphous compounds. (C) 2000 Elsevier Science B.V. All rights re served.