The Quantum Canonical Ensemble

The phase space of quantum mechanics can be viewed as the complex projective space CP endowed with a Kählerian structure given by the Fubini-Study metric and an associated symplectic form. Therefore, we can interpret the Schrödinger equation as generating a natural Hamiltonian dynamics on CP. Based upon the geometric structure of the quantum phase space we introduce the corresponding natural microcanonical and canonical ensembles. The resulting distribution implied by the quantum canonical ensemble differs from distribution of the conventional density matrix approach, which in some respects can be viewed as semiclassical. The results are applied in detail to the case of a spin one-half particle in a heat bath with an applied magnetic field.