### Abstract:

This report presents a characterization of the quantum mechanical
analog of the Gibbs canonical density. The approach is
based on a method developed by D.S. Carter for the case of classical
statistical mechanics, which considers composite mechanical systems
composed of mechanically and statistically independent components.
After a brief introductory chapter, Chapter II outlines how
the case of classical mechanics may be described in terms of the
usual measure theoretic treatment of probability. The necessary
statistical background of quantum mechanics is then discussed in
Chapter III, relying on the classic treatment of J. von Neumann and
the more recent work of G. W. Mackey. The basic idea of probability
measure in quantum mechanics differs from that in classical measure
theory, for the measure is defined on a non-Boolean lattice consisting
of all closed linear subspaces of a Hilbert space. Because of this
difference, the classical theory of product measures does not apply. Chapter IV presents a detailed treatment of probability measures for
composite quantum systems.
The analog of the Gibbs canonical density is characterized in
Chapter V, by considering a large collection Q of noninteracting
quantum systems, each of which is in an equilibrium statistical state.
The set Q, the Hamiltonian operator for each system, and the
equilibrium states are assumed to have certain properties which are
given as axioms.
The axioms require each Hamiltonian operator to have a pure
point spectrum. It is assumed, without loss of generality, that the
lowest characteristic value of each Hamiltonian is zero. The set Q
is assumed to be closed under the formation of pairwise mechanically
independent composite systems. This implies that the set D of all
Hamiltonian spectra is closed under addition. It is further assumed
that D is closed under positive differences. The final requirement
on the set Q is that it contain certain "harmonic oscillators".
More precisely, for each positive λεD, Q must contain a system
whose Hamiltonian has the spectrum {nλ : n=0,1,2,[superscript ...]}. The usual
assumption is made that each density operator is a function of the
system Hamiltonian. Finally, it is assumed that for each composite
system in Q, with two mechanically independent components, the
component systems are statistically independent.
It is shown that these assumptions imply that each member of Q is in a canonical state at a temperature which is the same for all
systems. The possibility of zero absolute temperature is included.