Statistical Ensemble (mathematical Physics) - Ensembles in Quantum Statistical Mechanics

Ensembles in Quantum Statistical Mechanics

Putting aside for the moment the question of how statistical ensembles are generated operationally, we should be able to perform the following two operations on ensembles A, B of the same system:

• Test whether A, B are statistically equivalent.

• If p is a real number such that 0 < p < 1, then produce a new ensemble by probabilistic sampling from A with probability p and from B with probability 1 – p.

Under certain conditions therefore, equivalence classes of statistical ensembles have the structure of a convex set. In quantum physics, a general model for this convex set is the set of density operators on a Hilbert space. Accordingly, there are two types of ensembles:

• Pure ensembles cannot be decomposed as a convex combination of different ensembles. In quantum mechanics, a pure density matrix is one of the form . Accordingly, a ray in a Hilbert space can be used to represent such an ensemble in quantum mechanics. A pure ensemble corresponds to having many copies of the same (up to a global phase) quantum state.
• Mixed ensembles are decomposable into a convex combination of different ensembles. In general, an infinite number of distinct decompositions will be possible.

Thus a quantum mechanical ensemble is specified by a mixed state in general. For example, one can specify the density operators describing microcanonical, canonical, and grand canonical ensembles of quantum mechanical systems, in a mathematically rigorous fashion.

The normalization factor required for the density operator to have trace 1 is the quantum mechanical version of the partition function.

We note here that ensembles of quantum mechanical system are sometimes treated by physicists in a semi-classical fashion. Namely, one considers the phase space of the corresponding classical system (e.g. for an ensemble of quantum harmonic oscillators, the phase space of a classical harmonic oscillator is considered). Then, using physical arguments, one derives a suitable "fundamental volume" for the particular system to reflect the fact that quantum mechanical microstates are discretely distributed on the phase space. From the uncertainty principle, it is expected this fundamental volume to be related to the Planck constant, in some way.