Quantum–classical mechanics and the Franck–Condon principle related to quantum mechanics are discussed as two alternative theoretical approaches to molecular optical spectroscopy. The statement of the problem is connected with the singularity of quantum mechanics in describing the joint motion of electrons and nuclei in the transient state of “quantum” transitions. This singularity can be eliminated by introducing chaos into the transient state. Quantum mechanics itself, supplemented by chaos (dozy chaos), is called quantum–classical mechanics. Using the simplest example of quantum transitions, it is shown that the results of quantum–classical mechanics in the case of strong dozy chaos correspond to the physical picture based on quantum mechanics and the Franck–Condon principle. The same chaos can be strong for small molecules in standard molecular spectroscopy and simultaneously weak in the photochemistry and nanophotonics of large molecules, where quantum mechanics no longer works. To describe the chaotic dynamics of the transient state, it is necessary to apply quantum–classical mechanics. Thus, the erroneous Franck–Condon physical picture of molecular “quantum” transitions is workable from a practical point of view as long as we are dealing with sufficiently small molecules, just as the erroneous geocentric picture of the world was workable until we went out into outer space.
Quantum–classical mechanics and the Franck–Condon principle related to quantum mechanics are discussed as two alternative theoretical approaches to molecular optical spectroscopy. The statement of the problem is connected with the singularity of quantum mechanics in describing the joint motion of electrons and nuclei in the transient state of “quantum” transitions. This singularity can be eliminated by introducing chaos into the transient state. Quantum mechanics itself, supplemented by chaos (dozy chaos), is called quantum–classical mechanics. Using the simplest example of quantum transitions, it is shown that the results of quantum–classical mechanics in the case of strong dozy chaos correspond to the physical picture based on quantum mechanics and the Franck–Condon principle. The same chaos can be strong for small molecules in standard molecular spectroscopy and simultaneously weak in the photochemistry and nanophotonics of large molecules, where quantum mechanics no longer works. To describe the chaotic dynamics of the transient state, it is necessary to apply quantum–classical mechanics. Thus, the erroneous Franck–Condon physical picture of molecular “quantum” transitions is workable from a practical point of view as long as we are dealing with sufficiently small molecules, just as the erroneous geocentric picture of the world was workable until we went out into outer space.
In quantum mechanics, the theory of quantum transitions is grounded on the convergence of a series of time-dependent perturbation theory. In nuclear and atomic physics, this series converges because the dynamics of quantum transitions (quantum jumps) are absent by definition. In molecular and chemical physics, on the contrary, the dynamics of "quantum" transitions, being determined by the joint motion of a light electron (or electrons) and very heavy nuclei, are present by definition, and the series of time-dependent perturbation theory becomes singular. An exception is the dynamic problem for stationary states in the Born-Oppenheimer adiabatic approximation, when the electronic subsystem turns out to be "off" from the general dynamic process and therefore is not dynamically full-fledged: it only forms an electric potential in which the nuclei oscillate. Removing the aforementioned singularity can be accomplished in two ways. The first method was consisted of introducing an additional postulate in the form of the Franck-Condon principle into molecular quantum mechanics, in which the adiabatic approximation is used. The second method was proposed by the author and consisted of damping the singular dynamics of the joint motion of an electron and nuclei in the intermediate (transient) state of molecular "quantum" transitions by introducing chaos. This chaos arises only during molecular quantum transitions and is called dozy chaos. Formally, the damping is carried out by replacing an infinitely small imaginary addition in the spectral representation of the complete Green's function of the system with its finite quantity. The damping chaos (dozy chaos) leads to the continuity of the energy spectrum in the molecular transient state, which is a sign of classical mechanics. Meanwhile, the initial and final states of the molecule obey quantum mechanics in the adiabatic approximation. Molecular quantum mechanics, which takes into account the chaotic dynamics of the transient state of molecular "quantum" transitions, can be called quantum-classical (dozy-chaos) mechanics. The efficacy of the damping for the aforementioned singularity was previously shown by dozy-chaos mechanics of elementary electron transfers in condensed matter, which is the simplest case of dozy-chaos mechanics, and its applications to a whole number of problems, especially to the optical spectra in polymethine dyes and their aggregates. This paper provides a regular exposition of this dozy-chaos (quantum-classical) mechanics of the elementary electron transfers. The main results of its applications presented in the introduction are also described.
By now the “chaos revolution” has reached nearly every branch of the natural sciences. In fact, chaos is everywhere. To name but a few examples, we talk about chaotic weather patterns, chaotic chemical reactions and the chaotic evolution of insect populations. Atomic and molecular physics are no exceptions. At first glance this is surprising since atoms and molecules are well described by the linear laws of quantum mechanics, while an essential ingredient of chaos is nonlinearity in the dynamic equations. Thus, chaos and atomic physics seem to have little to do with each other. But recently, atomic and molecular physicists have pushed the limits of their experiments to such high quantum numbers that it starts to make sense, in the spirit of Bohr's correspondence principle, to compare the results of atomic physics experiments with the predictions of classical mechanics, which, for the most part, show complexity and chaos. The most striking observation in recent years has been that quantum systems seem to “know” whether their classical counterparts display regular or chaotic motion. This fact can be understood intuitively on the basis of Feynman's version of quantum mechanics. In 1948 Feynman showed that quantum mechanics can be formulated on the basis of classical mechanics with the help of path integrals. Therefore it is expected that the quantum mechanics of an atom or molecule is profoundly influenced, but of course not completely determined, by the qualitative behaviour of its underlying classical mechanics.
Recently, the phenomenon of quantum-classical correspondence breakdown was uncovered in optomechanics, where in the classical regime the system exhibits chaos but in the corresponding quantum regime the motion is regular - there appears to be no signature of classical chaos whatsoever in the corresponding quantum system, generating a paradox. We find that transient chaos, besides being a physically meaningful phenomenon by itself, provides a resolution. Using the method of quantum state diffusion to simulate the system dynamics subject to continuous homodyne detection, we uncover transient chaos associated with quantum trajectories. The transient behavior is consistent with chaos in the classical limit, while the long term evolution of the quantum system is regular. Transient chaos thus serves as a bridge for the quantum-classical transition (QCT). Strikingly, as the system transitions from the quantum to the classical regime, the average chaotic transient lifetime increases dramatically (faster than the Ehrenfest time characterizing the QCT for isolated quantum systems). We develop a physical theory to explain the scaling law.
The application of modern quantum chemical methods to the simulation of chemical reactions makes it obvious that the Copenhagen picture of quantum mechanics cannot be maintained. In ab-initio molecular dynamics methods, any kind of matter is described by applying the Schrödinger equation to the electronic structure and using classical Newton dynamics for the motion of the atomic nuclei. By lots of calculations on this basis, it has become obvious that this concept is superior to a quantum chemical description of nuclear motion. The nuclei move according to Newton's laws in a perfectly local and deterministic way while the surrounding electronic cloud determines bonding and antibonding interactions between the atoms. The resulting theory is a deterministic formulation of quantum mechanics which is free from paradoxa and is able to explain complex phenomena like photochemistry. An obvious consequence is that molecules like ammonia can have a permanent dipole moment in spectroscopic measurements, while consistent application of traditional quantum mechanics predicts a higher symmetry as is outlined in the paper.
Dynamical chaos in quantum systems (quantum chaos) has been studied by a large number of physicists [1-7]. It has been shown that quantum chaos is possible only in the quasiclassical region, for example in the case of quantum states near the dissociation limit in molecules or near the ionization limit in atoms and molecules. However, the quantum equations of motion are linear and hence they cannot contain dynamical chaos. In the present paper we discuss the solution of quantum problems which are nonlinear in the classical limit. It will be show that in solving the quantum problem in this case it is necessary to use the solution of the nonlinear classical problem forming the classical limit of the quantum problem. i. We note first that here we are not interested in the usual field nonlinearities considered, for example, in quantum optics. This is because field nonlinearities cannot determine the bifurcation properties of the quantum equations of motion since the latter are linear in the wave function ~ or density matrix p. In other words, dynamical chaos, which is intimately connected with the bifurcation properties of the corresponding equations, is not determined by nonlinearities in the fields in quantum equations of motion linear in ~ or 0o
In this work the decoherence formalism of quantum mechanics is explored and applied to a number of interesting problems in quantum physics. The boundary between quantum and classical physics is examined, and demonstration made that quantum histories corresponding to classical equations of motion become more probable for a broad class of models, including linear and nonlinear models of Brownian motion. The link between noise, dissipation, and decoherence is studied. This work is then applied to systems which classically exhibit dissipative chaotic dynamics. A theory is explicated for treating these systems, and the ideas are applied to a particular model of the forced, damped Duffing oscillator, which is chaotic for certain parameter values. Differences between classical and quantum chaos are examined, particularly differences arising in the structure of fractal strange attractors, and the conceptual difficulties in framing standard notions of chaos in a quantum system. A brief discussion of previous work on quantum chaos is included, and the differences between Hamiltonian and dissipative chaos pointed out; a somewhat different interpretation of quantum chaos from the standard one is suggested. A class of histories for quantum systems, in phase space rather than configuration space, is studied. Different ways of representing projections in phase space are discussed, and expressions for the probability of phase space histories are derived; conditions for such histories to decohere are also estimated in the semiclassical limit.