Canonical states in quantum statistical mechanics Public Deposited

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  • 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.
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