The classical limit of quantum partition functions

Bernhard Baumgartner

Bingyu Cui

Andrew Jackson, Theodoros Kapourniotis, Animesh Datta

Pere Seglar, Enric Pérez Canals

We analyze the so-called classical limit of the quantum-mechanical canonical partition function. In order to do that, we define accurately the density matrix for symmetrized and antisymmetrized wave functions only (Bose-Einstein and Fermi-Dirac), and find an exact relation between them and the density matrix for non symmetrized functions (Maxwell-Boltzmann). Our results differ from the generally assumed in a numerical factor N!, for which we suggest a physical interpretation. We derive as well the reverse (and also exact) relation, to find the canonical partition function for non-symmetrized wave functions in terms of the corresponding function for fermions and bosons.

Filippo Cesi

The author considers a class of spin systems whose single-site configuration space is an orbit of a representation of a compact Lie group G. For these models the author gets upper and lower bounds to the quantum partition function in terms of two classical partition functions. If a certain group-theoretic condition is satisfied, then these bounds allow one to prove the convergence of a suitable sequence of quantum partition functions to the 'corresponding' classical one. This condition is shown to be satisfied, in particular, for the D-component rotators when D is odd. The result could be useful for the extension of the Lee-Yang theorem to such models.

H. Hogreve

Spin- 1/2 particles subject to external fields will often exhibit paramagnetism. This effect can be formalised in terms of an inequality involving appropriate partition functions constructed with spin Hamiltonians which are chosen to be of Pauli type. Whereas in quantum mechanics it is known that a paramagnetic inequality does not hold true in general, the paramagnetism of spin in the classical limit to 0 is investigated. The author applies previous results to derive a classical partition function which, in satisfying the relevant inequality, gives rise to general paramagnetism.