Hamiltonian systems relaxation process

It is very important, in the theory of quantum relaxation processes, to understand how an atomic or molecular excited state is prepared, and to know under what circumstances it is meaningful to consider the time development of such a compound state. It is obvious, but nevertheless important to say, that an atomic or molecular system in a stationary state cannot be induced to make transitions to other states by small terms in the molecular Hamiltonian. A stationary state will undergo transition to other stationary states only by coupling with the radiation field, so that all time-dependent transitions between stationary states are radiative in nature. However, if the system is prepared in a nonstationary state of the total Hamiltonian, nonradiative transitions will occur. Thus, for example, in the theory of molecular predissociation4 it is not justified to prepare the physical system in a pure Born-Oppenheimer bound state and to force transitions to the manifold of continuum dissociative states. If, on the other hand, the excitation process produces the system in a mixed state consisting of a superposition of eigenstates of the total Hamiltonian, a relaxation process will take place. Provided that the absorption line shape is Lorentzian, the relaxation process will follow an exponential decay. 

As in Eq. (64), the electron spin spectral densities could be evaluated by expanding the electron spin tensor operators in a Liouville space basis set of the static Hamiltonian. The outer-sphere electron spin spectral densities are more complicated to evaluate than their inner-sphere counterparts, since they involve integration over the variable u, in analogy with Eqs. (68) and (69). The main simplifying assumption employed for the electron spin system is that the electron spin relaxation processes can be described by the Redfield theory in the same manner as for the inner-sphere counterpart (95). A comparison between the predictions of the analytical approach presented above, and other models of the outer-sphere relaxation, the Hwang and Freed model (HF) (138), its modification including electron spin... 

MD simulations with a constant energy is nothing but Hamiltonian dynamics. Recent accumulation of MD simulations will certainly contribute to our further understanding of Hamiltonian systems, especially in higher dimensions. The purpose of this section is to sketch briefly how the slow relaxation process emerges in the Hamiltonian dynamics, and especially to show that transport properties of phase-space trajectories reflect various underlying invariant structures. 

As mentioned in the introduction, the power spectrum density is used to probe the long-time correlation decay. Appearance of l//v-type spectra is an indication that there are, in principle, infinitely many time scales in the relaxation process. Geisel et al. [37] gave an example of mixed Hamiltonian systems... 

The underlying process generating J (f) need not be specified, but one realization of it could be a Hamiltonian system with a set of variables R. These latter variables can be infinitely many so as to result in the relaxation of the correlation properties of the system. 

The time dependence of the relaxation process necessitates the use of time-dependent perturbation theory, in which the total Hamiltonian of the system is described by... 

As soon as large ensembles of particles with statistical populations of the eigenstates and incoherent exchange and relaxation processes between these states are investigated, quantum statistical tools are necessary to describe the system. In this situation the quantum mechanical density operator p has to be employed. For the coherent evolution of the density operator under the influence of a Hamiltonian H, the following differential equation is found [80]... 

To handle the complex reactive process, we first focus on the dynamics on the Si surface to study how the system evolves towards the conical intersections. Therefore we introduce a reduced set of reactive coordinates, develop the corresponding Hamiltonian and study the time evolution of the system by means of wavepacket propagations on the calculated ah initio potential reaction surface. In the following steps, we include the nonadiabatic coupling elements as well as the laser-molecule interaction to describe the complete relaxation process. The final aim is to drive the reaction systematically through either one or the other of the two conical intersections and thus to influence the resulting product distribution. 

The commutator bracket [JfQ.Pg] describes the time-development of the system under the effects of the atomic Hamiltonian and of any static external magnetic field, while the last three terms give the evolution under the influence of depopulation pumping, repopulation pumping, and relaxation processes respectively. These processes are considered separately in the sections which follow. 

If the existence of the stationary signal at time >Ti is not determined by the interactions between quadrupole and dipole-dipole reservoirs but by the spin-lattice relaxation, we can neglect processes in which the spins absorb the quanta of the dipole-dipole interaction modulated by the RF field and can omit corresponding terms in the total Flamiltonian of the spin system. Such reduced total Hamiltonian coincides with the effective Hamiltonian Hgff in Equation (37). 

Out of the many variants of xSR, it is the Muon Spin Relaxation that has been utilized up to now for the study of dynamic processes in organometallic systems, and therefore only this aspect will be discussed here. Interested readers should find ample descriptions of the other aspects of xSR in the references provided at the end of this chapter. For the unpaired electron occupying a Is hydrogenic orbital about the muon, the electron spin S and muon spin I are coupled by a scalar product to give the Hamiltonian for the so-called hyperfine, Fermi or contact interaction. 

For multispin systems in which mixing of nuclear spin states occurs owing to dipolar couplings among deuterons, the relaxation is influenced by crosscorrelation terms between individual quadrupolar interaction tensors in addition to the auto-correlation terms [5.17, 5.30]. It is reasonable to expect that these cross-correlation terms may provide additional information on the motional processes. The static Hamiltonian Hq for two deuterons is... 

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