Introduction We now move to considering the dynamics of a system of nuclei and electrons. Of course, both electrons and nuclei are subject to the laws of quantum mechanics, but since nuclei are 2000 to 200,000 times heavier than electrons, we expect that classical mechanics will be a much better approximation for the motion of nuclei than for the motion of electrons. This means that we expect a level of approximation to be valid, where some of the degrees of freedom (d.o.f.) of a system behave essentially classically and others behave essentially quantum-mechanically. The system then is of a mixed quantum/classical nature. Most often the quantum subsystem consists of system of electrons in a dynamical field of classical nuclei, but the quantum subsystem may also be a selection of generalized nuclear coordinates (e.g., corresponding to high-frequency vibrations) while other generalized coordinates are supposed to behave classically, or describe the motion of a proton in a classical environment. So, in this chapter we consider the dynamics of a quantum system in a non-stationary potential. In Section 5.2 we consider the time-dependent potential as externally given, without taking notice of the fact that the sources of the time-dependent potential are moving nuclei, which are quantum particles themselves, feeling the interaction with the quantum d.o.f. Thus we consider the time evolution of the quantum system, which now involves mixing-in of excited states, but we completely ignore the back reaction of the quantum system onto the d.o.f. that cause the time-dependent potential, i.e., the moving nuclei.
Introduction We now move to considering the dynamics of a system of nuclei and electrons. Of course, both electrons and nuclei are subject to the laws of quantum mechanics, but since nuclei are 2000 to 200,000 times heavier than electrons, we expect that classical mechanics will be a much better approximation for the motion of nuclei than for the motion of electrons. This means that we expect a level of approximation to be valid, where some of the degrees of freedom (d.o.f.) of a system behave essentially classically and others behave essentially quantum-mechanically. The system then is of a mixed quantum/classical nature. Most often the quantum subsystem consists of system of electrons in a dynamical field of classical nuclei, but the quantum subsystem may also be a selection of generalized nuclear coordinates (e.g., corresponding to high-frequency vibrations) while other generalized coordinates are supposed to behave classically, or describe the motion of a proton in a classical environment. So, in this chapter we consider the dynamics of a quantum system in a non-stationary potential. In Section 5.2 we consider the time-dependent potential as externally given, without taking notice of the fact that the sources of the time-dependent potential are moving nuclei, which are quantum particles themselves, feeling the interaction with the quantum d.o.f. Thus we consider the time evolution of the quantum system, which now involves mixing-in of excited states, but we completely ignore the back reaction of the quantum system onto the d.o.f. that cause the time-dependent potential, i.e., the moving nuclei.
A classical limit of quantum dynamics can be defined by compensation of the quantum potential in the time-dependent Schrödinger equation. The quantum potential is a non-local quantity, defined in the trajectory-based form of the Schrödinger equation, due to Madelung, de Broglie, and Bohm, which formally generates the quantum-mechanical features in dynamics. Selective inclusion of the quantum potential for the degrees of freedom deemed "quantum," defines a hybrid quantum/classical dynamics, appropriate for molecular systems comprised of light and heavy nuclei. The wavefunction is associated with all of the nuclei, and the Ehrenfest, or mean-field, averaging of the force acting on the classical degrees of freedom, typical of the mixed quantum/classical methods, is avoided. The hybrid approach is used to examine evolution of light/heavy systems in the harmonic and double-well potentials, using conventional grid-based and approximate quantum-trajectory time propagation. The approximate quantum force is defined on spatial domains, which removes unphysical coupling of the wavefunction fragments corresponding to distinct classical channels or configurations. The quantum potential, associated with the quantum particle, generates forces acting on both quantum and classical particles to describe the backreaction.
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.
Abstract This chapter the reviews essential concepts and methods of time-dependent quantum mechanics. (1) Formal solutions of the time-dependent Schrödinger equation. (2) The two-level system. (3) Interacting systems: The time-dependent self-consistent field approximation and the Ehrenfest approximation. (4) The Born–Oppenheimer approximation and nuclear potential surfaces. (5) Time-dependent Hamiltonians: A two-level system in a time-dependent field. (6) Non-adiabatic dynamics: The surface-hopping method. (7) Expressing the time evolution in terms of the Green operator. (8) Representations (the Schrödinger and Heisenberg representations, the interaction representation). (9) Time-dependent perturbation theory. (10) Quantum dynamics of the free particles. (11) Quantum dynamics of the harmonic oscillator. (12) The raising/lowering operators formalism. (13) Tunneling.
Chemical dynamics, in principle, should be understood by solving the time-dependent Schrödinger equation for a molecular system, describing motion of the nuclei and electrons. However, the computational efforts to solve this partial second-order differential equation scales exponentially with the system size, which prevents us from getting exact numerical solutions for systems larger than 4-5 atoms. Thus, approximations simplifying the picture are necessary. The so-called Born-Oppenheimer approximation, separating motion of the electrons and nuclei is the central one: solution to the electronic Schrödinger equation defines the potential energy surface on which the nuclear motion unfolds, and there are standard quantum chemistry software packages for solving the electronic Schrödinger equation. For the nuclear Schrödinger equation, however, there are no widely applicable quantum-mechanical approaches, and most simulations are performed using classical Newtonian mechanics which is often adequate due to large nuclear masses. However, the nuclear quantum effects are significant for chemical processes involving light nuclei at low energies, and including these effects into simulation, even approximately, is highly desirable. In this dissertation, an approximate methodology of including quantum-mechanical effects within the quantum trajectory or the de Broglie-Bohm formulation of the Schrödinger equations is developed. Use of the trajectory framework makes the approach scalable to hundreds of degrees of freedom. The methodology is applied to study high-dimensional systems (solid He4 and others) relevant to chemistry.
The approach of defining quantum corrections on nuclear dynamics of molecular systems incorporated approximately into selected degrees of freedom, is described. The approach is based on the Madelung-de-Broglie-Bohm formulation of time-dependent quantum mechanics which represents a wavefunction in terms of an ensemble of trajectories. The trajectories follow classical laws of motion except that the quantum potential, dependent on the wavefunction amplitude and its derivatives, is added to the external, classical potential. In this framework the quantum potential, determined approximately for practical reasons, is included only into the "quantum" degrees of freedom describing light particles such as protons, while neglecting with the quantum force for the heavy, nearly classical nuclei. The entire system comprised of light and heavy particles is described by a single wavefunction of full dimensionality. The coordinate space of heavy particles is divided into spatial domains or subspaces. The quantum force acting on the light particles is determined for each domain of similar configurations of the heavy nuclei. This approach effectively introduces parametric dependence of the reduced dimensionality quantum force, on classical degrees of freedom. This strategy improves accuracy of the quantum force and does not restrict interaction between the domains. The concept is illustrated for two-dimensional scattering systems, where the quantum force is required to reproduce vibrational energy of the quantum degree of freedom.
We study the back-reaction of quantum systems onto classical ones. Taking the starting point that semi-classical physics should be described at all times by a point in classical phase space and a quantum state in Hilbert space, we consider an unravelling approach, describing the system in terms of a classical-quantum trajectory. We derive the general form of the dynamics under the assumptions that the classical trajectories are continuous and the evolution is autonomous, and the requirement that the dynamics is linear and completely positive in the combined classical-quantum state. This requirement is necessary in order to consistently describe probabilities, and forces the dynamics to be stochastic when the back-reaction is non-zero. The resulting equations of motion are natural generalisations of the standard semi-classical equations of motion, but since the resulting dynamics is linear in the combined classical-quantum state, it does not lead to the pathologies which usually follow from evolution laws based on expectation values. In particular, the evolution laws we present account for correlations between the classical and quantum system, which resolves issues associated with other semi-classical approaches. In addition, despite a breakdown of predictability in the classical degrees of freedom, the quantum state evolves deterministically conditioned on the classical trajectory, provided a trade-off between decoherence and diffusion is saturated. As a result, the quantum state remains pure when conditioned on the classical trajectory. To illustrate these points, we numerically simulate a number of semi-classical toy models, including one of vacuum fluctuations as a source driving the expansion of the universe. Finally, we discuss the application of these results to semi-classical gravity, and the black-hole information problem.