Quantization of the canonical ensemble

The task of quantization consists of forming a quantum mechanical system from a given classical system. Thereby, classical observables are replaced by quantum mechanical observables. We carry out this procedure with Gibbs’ canonical ensemble. Therefrom, we obtain a density matrix which represents a quantum mechanical concept of equipartition. The most important justification for this principle is provided by the virial theorem. Thereafter, the implications of the density matrix are evaluated. For this purpose, we define a virtual free energy. Thus, the equilibrium condition of the quantum canonical ensemble is derived by minimizing the virtual free energy. We demonstrate our method for the case of a mean field like order–disorder phase transition. 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.