Quantum/semiclassical approaches Hamiltonian approach

Rigorous Quantum Rate Theory Versus the Quantized ARRKM Theory A Semiclassical Approximation to the Rigorous Quantum Rate Theory Effective Hamiltonian Approach to Unimolecular Dissociation Wave Packet Dynamics Approach VII. Quantum Transport in Classically Chaotic Systems... 

The mapping procedure introduced in Sec. 6 results in a quantum-mechanical Hamiltonian with a well-defined classical limit, and therefore extends the applicability of the established semiclassical approaches to nonadiabatic dynamics. The thus obtained semiclassical version of the mapping approach, as well as the equivalent formulation that is obtained by requantizing the classical electron analog model of Meyer and Miller, have been applied to a variety of systems with nonadiabatic dynamics in the recent years. It appears that this approach is so far the only fully semiclassical method that allows a numerical treatment of truly multidimensional nonadiabatic dynamics at conical intersections. 

In general, in the study of systems at equihbrium the main advantage of the semiclassical approaches is that they can be utilized in simulation work by following essentially the same techniques as in the classical case (i.e., one works with Ng particles). Moreover, whenever they can be applied, semiclassical approaches can work excellently and represent a substantial saving in computational effort as compared to the exact PI xP-calculations. On the negative side, even the best semiclassical approaches cannot cope with very large quantum diffraction effects. Actually, they do not capture all the information needed to characterize the quantum system (e.g., some g (r) structures cannot be computed). In the literature of this field there are different names for these pair potentials (e.g., classical effective potentials, semiclassical effective potentials, quantum effective pair potentials, etc.). The absence of external fields will be assumed in this section, and the system will be ruled by the Hamiltonian... 

Nonlinear interaction can be studied on a full quantum basis by expressing the Hamiltonian in terms of quantum operators for the material and the radiation degrees of freedom, but the most fruitful treatment has been the semiclassical approach in which the electric field operator is replaced by its expectation value and considered as a function, though quantum operators are used for the material. Only electric dipole polarization will be addressed. 

At the beginning of this section it is important to point out that the theoretical treatment we will briefly discuss now is sometimes called semiclassical just because the correlation functions are classical. A quantum mechanical treatment has been proposed (2, p. 284) which presents many formal similarities to the semiclassical treatment but which fundamentally differs from it by the way the correlation functions are defined (in terms of time-dependent operators and not of time functions). This distinction is rarely done in review articles about relaxation where the semiclassical approach is generally presented as the only way to handle the problem. To introduce the correlation functions and spectral densities, we will consider a spin system characterized by eigenstates a, b, c. and corresponding energies E, E. A perturbation, time-dependent Hamiltonian H(t) acts on this system. H(t) corresponds to the coupling of the spin system with the lattice. We shall make the assumption that H(t) can be written as a product A f(t). 

In a recent analysis carried out for a bounded open system with a classically chaotic Hamiltonian, it has been argued that the weak form of the QCT is achieved by two parallel processes (B. Greenbaum et.al., ), explaining earlier numerical results (S. Habib et.al., 1998). First, the semiclassical approximation for quantum dynamics, which breaks down for classically chaotic systems due to overwhelming nonlocal interference, is recovered as the environmental interaction filters these effects. Second, the environmental noise restricts the foliation of the unstable manifold (the set of points which approach a hyperbolic point in reverse time) allowing the semiclassical wavefunction to track this modified classical geometry. 

Our approach is based on a systematic semiclassical study of the Dirac equation. After separating particles and anti-particles to arbitrary powers in h, a semiclassical expansion of the quantum dynamics in the Heisenberg picture is developed. To leading order this method produces classical spin-orbit dynamics for particles and anti-particles, respectively, that coincide with the findings of Rubinow and Keller Hamiltonian relativistic (anti-) particles drive a spin precession along their trajectories. A modification of that method leads to a semiclassical equivalent of the Foldy-Wouthuysen transformation resulting in relativistic quantum Hamiltonians with spin-orbit coupling. 

In the last few years we have witnessed the successful development of several methods for the numerical solution of multi-dimensional quantum Hamiltonians Monte Carlo methods centroid methods,mixed quantum-classical methods, and recently a revival of semiclassical methods. We have developed another approach to this problem, the exponential resummation of the evolution operator. - The rest of this Section will explain briefly this method. 

Another theory which is used to describe scattering problems and which blends together classical and quantum mechanics is the semiclassical wavepacket approach [53]. The basic procedure comes from the fact that wavepackets which are initially Gaussian remain Gaussian as a function of time for potentials that are constant, linear or quadratic functions of the coordinates. In addition, the centres of such wavepackets evolve in time in accord with classical mechanics. We have already seen one example of this with the free particle wavepacket of equation (A3.11.7). Consider the general quadratic Hamiltonian (still in one dimension but the generalization to many dimensions is straightforward)... 

The microscopic (hyper)polarizabilities are studied by means of the so-called theory of the response functions which is of importance for aU molecular and cluster entities (Roman et al. 2006). The most commonly used approach in studying the linear and nonlinear optical properties of clusters is the so-caUed semiclassical one. According to this approach a classical treatment is used to describe the response of the cluster to an external field (radiation) while the system itself is treated using the lows and techniques of quantum mechanics. This is done by using a Hamiltonian which combines both of the above treatments ... 

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