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Berichte der Bunsengesellschaft für physikalische Chemie

Time Operator, D‐Fluctuations, and Microscopic Irreversibility in Spectroscopic Processes

C. A. Chatzidimitriou‐Dreismann

Published Jun 1, 1985• 1 citation
Publisher

Abstract

Abstract Prigogine has emphasized that there is a “degeneracy” in quantum theory, where the Hamiltonian plays a dual role as (a) the generator of time evolution and (b) the energy observable. This prevents the exact treatment of irreversible phenomena on the microscopic quantum mechanical level. The removal of the aforementioned degeneracy can be achieved with the aid of a star‐unitary similarity transformation depending explicitly on the star‐Hermitian operator of the “intrinsic time”. Its physical significance is shown in the context of luminescence spectroscopy, where a novel kind of (dynamically induced or D‐) fluctuations around steady photoemission states is predicted. A photon counting/spectroscopic experiment has been performed which permits to detect D‐fluctuations and, at the same time, allows us to resolve broad spectroscopic bands by detection of the underlying finer “pure” vibronic transitions. The experimental findings clearly show the physical significance of Prigogine's Microscopic Theory.

DOI
10.1002/bbpc.19850890607

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New quantum rules for dissipative systems

Christophe George, Françoise Hénin, F. Mayné +1 more

It is difficult to incorporate the concept of interacting physical particles in the frame of conventional quantum mechanics. It comes from the fact that in this formalism the Hamiltonian plays a dual role as the generator of motion and as the observable associated with the energy. As a result it is easy to define either a representation in which particles interact but are not well defined (a representation in which the Hamiltonian is not diagonal) or a representation in which the units are well defined but do not interact. We expect that in the search for a representation in which we may speak of both well defined and still interacting particles, the distinction between reversible and irreversible processes should play a basic role. Certainly we do not want to eliminate irreversible processes such as scattering, production or decay which are basically related to the mechanisms through which we observe particles. To distinguish between reversible and irreversible processes we have to work in a representation such that a Liapounov function (the entropy) may be ascribed to the time evolution of the system. The introduction of such a representation through a specific class of non-unitary transforms is described. We then propose newmore » quantum rules directly applicable to dissipative systems with a Liapounov function. For systems with a discrete spectrum these new quantum rules reduce trivially to the usual quantum rules as formulated by Born, Jordan and Heisenberg. The theory is applied to a simple field theoretical model - the so-called Friedrichs model-for which all calculations can be performed exactly.« less

Hadronic J.; (United States)•Jun 1, 1978•4 citations
Abstract Prigogine has emphasized that there is a “degeneracy” in quantum theory, where the Hamiltonian plays a dual role as (a) the generator of time evolution and (b) the energy observable. This prevents the exact treatment of irreversible phenomena on the microscopic quantum mechanical level. The removal of the aforementioned degeneracy can be achieved with the aid of a star‐unitary similarity transformation depending explicitly on the star‐Hermitian operator of the “intrinsic time”. Its physical significance is shown in the context of luminescence spectroscopy, where a novel kind of (dynamically induced or D‐) fluctuations around steady photoemission states is predicted. A photon counting/spectroscopic experiment has been performed which permits to detect D‐fluctuations and, at the same time, allows us to resolve broad spectroscopic bands by detection of the underlying finer “pure” vibronic transitions. The experimental findings clearly show the physical significance of Prigogine's Microscopic Theory.
Some basic concepts of the Prigogine microscopic theory of irreversibility are discussed in the physical context of molecular dynamical processes in condensed systems. The removal of the “dual role” that the Hamiltonian plays within conventional quantum theory corresponds to the introduction of a star-Hermitian superoperator called “intrinsic time.” The existence of dynamically induced fluctuations around steady states is predicted. With the aid of a new photon-counting experiment these fluctuations have been detected. The experiment allows us to resolve broad luminescence bands through the detection of the underlying vibrational structure.
Abstract Some basic concepts of the Prigogine quantum statistical theory of irreversible processes are discussed in connection with a previous paper by Chatzidimitriou‐Dreismann and Lippert. The removal of the “dual role” that the Hamiltonian plays within conventional quantum theory corresponds to the introduction of a star‐unitary operator 𝒯 called “intrinsic time.” The formalism of the previous paper is extended to systems that interact with external fields. A formal specification of 𝒯, furthermore, follows from the Lt invariance of the basic equations of motion and its connection with the intrinsic time. As an application to fluorescence spectroscopy, a photon counting experiment is presented. It allows us to detect certain predicted “dynamical” fluctuations in emission spectra. Some current experimental results are reported concerning the resolution of the broad fluorescence band of fluoranthene (dissolved in ethanol) through detection of the underlying finer vibrational structure.
In many problems of molecular physics of condensed matter the coupling V of the system of interest with its environment is strong. During a time interval of the order of the characteristic relaxation time ΓTy due to V, both the energy spectrum and the time evolution of the system cannot be determined exactly. A new representation is introduced in which the coupling V does not affect the energy spectrum, but it determines the time evolution. This causes the appearance of a new quantity ΓT, called “intrinsic time.” Some results of this procedure are (a) invariance of the basic physical equations under the new trnsformation, (b) noncommutation of ΓT with the Liouvillian, and (c) existence of a nonvanishing “entropy production” with respect to the intrinsic time.
Explicitly time-dependent Hamiltonians are ubiquitous in applications of quantum theory. It is therefore necessary to solve the time-dependent Schrödinger equation directly. The system’s dynamics is associated with a unitary time-evolution operator (a propagator), formally given as an infinite Dyson series. Time-dependent observables are invariant under unitary time-dependent transformations, where it is sometimes useful to transform the time-evolution from the states into the corresponding operators. This is carried out in part (in full) by transforming to the interaction (Heisenberg) picture. The corresponding equations of motion for the time-dependent operators are introduced. For quadratic potential energy functions, the time evolution of quantum expectation values coincides with the corresponding classical dynamics. This is demonstrated and analyzed in detail for Gaussian wave packets and a coherent state. Finally, we derive exact and approximate expressions for time-dependent transition probabilities and transition rates between quantum states. The validity of time-dependent perturbation theory is analyzed by comparison to exact dynamics.
Abstract The aim of our work was to find an unambiguous connection between irreversible macroscopic dynamics and reversible microdynamics that makes it possible to manifest irreversibility on a submacroscopic level without the use of coarse graining arguments. On this level the state of a physical system can be described by a field amplitude Ψ and the time evolution of this system is determined by a field equation for Ψ. For conservative systems, this field equation is formally identical with the linear Schrödinger equation, which can be constructed with the help of the classic Hamiltonian function for the corresponding problem. Regarding irreversible phenomena like damping due to a frictional force, for those dissipative systems no classic Hamiltonian function exists. Therefore the corresponding field equation cannot be obtained in the usual way. Nevertheless, also for dissipative systems it is possible to obtain a field equation in an unambiguous way using only Newton's form of classic mechanics. The result of our method is a nonlinear Schrödinger‐type field equation with a logarithmic nonlinearity. We discuss in more detail the properties of our logarithmic nonlinearity that corresponds to a macroscopic frictional force in a unique way. A figurative interpretation in terms of environment and interaction can be given. From a more formal point of view, the compatibility of our nonlinear operator with principles known from the theory of linear operators is investigated. One of the surprising results is the fact that although our nonlinear Hamiltonian HNL is “Hermitean” in the usual sense, in contrast to the linear theory an operator exp(i · HNL) is not unitary. Furthermore, in our theory the time‐derivative of the mean value of an operator is not only essentially determined by (the mean value of) its commutator with the Hamiltonian. There also occurs an additional term that causes irreversibility of the evolution and is connected with the feature of our theory that (in general) time derivative and construction of the mean value are no longer commuting operations. This fact shows some similarity with coarse graining theories, but in our theory the reason can be traced back unambiguously to an irreversible physical phenomenon.
It is difficult to incorporate the concept of interacting physical particles in the frame of conventional quantum mechanics. It comes from the fact that in this formalism the Hamiltonian plays a dual role as the generator of motion and as the observable associated with the energy. As a result it is easy to define either a representation in which particles interact but are not well defined (a representation in which the Hamiltonian is not diagonal) or a representation in which the units are well defined but do not interact. We expect that in the search for a representation in which we may speak of both well defined and still interacting particles, the distinction between reversible and irreversible processes should play a basic role. Certainly we do not want to eliminate irreversible processes such as scattering, production or decay which are basically related to the mechanisms through which we observe particles. To distinguish between reversible and irreversible processes we have to work in a representation such that a Liapounov function (the entropy) may be ascribed to the time evolution of the system. The introduction of such a representation through a specific class of non-unitary transforms is described. We then propose newmore » quantum rules directly applicable to dissipative systems with a Liapounov function. For systems with a discrete spectrum these new quantum rules reduce trivially to the usual quantum rules as formulated by Born, Jordan and Heisenberg. The theory is applied to a simple field theoretical model - the so-called Friedrichs model-for which all calculations can be performed exactly.« less