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STATISTICAL QUANTUM MECHANICS. DENSITY OPERATORS AND THERMODYNAMIC VALUES

Jacques Yvon

Published Jul 1, 1960• 0 citations
OpenAlex

Abstract

The simple and double density operators, as functions of temperature and activity, are computed for a system in equilibrium. The scattering operator allows, in some cases, separating the thermodnamic variables. In particular, these variables do not appear in the expression of the double density operator, as a function of the simple density operator; which explains the fact that this expression is still valid out of equilibrium. (auth)

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The simple and double density operators, as functions of temperature and activity, are computed for a system in equilibrium. The scattering operator allows, in some cases, separating the thermodnamic variables. In particular, these variables do not appear in the expression of the double density operator, as a function of the simple density operator; which explains the fact that this expression is still valid out of equilibrium. (auth)
General relationship among the density matrix formalism, the double-space formulation and thermo field dynamics is discussed in non-equilibrium dissipative quantum systems. The concept of weakly equivalent operators in the double space formulation is introduced to review many well-known results and it is shown to be useful in mapping between the density matrix formalism and the double space formulation. A new non-equilibrium thermo field dynamics is formulated to discuss dissipative quantum systems. A simple example of damped harmonic oscillators is discussed in the present representation.
In contrast to classical physics, the language of quantum mechanics involves operators and wave functions (or, more generally, density operators). However, in 1932, Wigner formulated quantum mechanics in terms of a distribution function W(q,p)W(q,p)W(q,p), the marginals of which yield the correct quantum probabilities for qqq and ppp separately \cite{wigner}. Its usefulness stems from the fact that it provides a re-expression of quantum mechanics in terms of classical concepts so that quantum mechanical expectation values are now expressed as averages over phase-space distribution functions. In other words, statistical information is transferred from the density operator to a quasi-classical (distribution) function.
In quantum physics, we often need to consider ensembles that are not pure cases, that is, that cannot be described by state vectors. This is so, for instance, when an ensemble is composed of several subensembles that are described by nonequivalent state vectors. It also happens in most cases in which we consider ensembles of definite subsystems of some larger systems. The density matrix—or "statistical operator"—formalism is one that makes it possible to deal with such situations swiftly and efficiently. The rules apply only to ensembles that can be described by state vectors. These rules entail as consequences new expressions for quantum probabilities and mean values, which can be called the—derived— "statistical operator rules." However, it is also possible to formulate the statistical operator rules directly instead. The Heisenberg "uncertainty" relations imply that in general f(x,p) cannot be interpreted as really being a joint probability distribution.