Relaxation quantum mechanic treatment

Wangsness and Bloch16>17 were the first to give a quantum mechanical treatment of spin relaxation using the density matrix formalism. The system considered is a spin interacting with an external magnetic field (which we suppose here to be constant) and with a heat bath. The corresponding Hamiltonian is... 

By cosidering the local field produced by one spin on the other, we can get an idea of what the relaxation parameters should be and provide their actual expressions as given by a quantum mechanical treatment which, in principle, would be required for such a biparticle interaction. Two... 

To discuss phonon relaxation processes, it is better to develop a quantum-mechanical treatment still with e>e. 

M. War ken and V. Bonadc-Koutecky, Quantum Mechanical Treatment of Stationary and Dynamical Properties of Bound Vibrational Systems. Application to the Relaxation Dynamics of Ags after an Electron Photodetachment , Chem. Phys. Lett. 272, 284 (1997). 

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). 

To summarize, the results presented for five representative examples of nonadiabatic dynamics demonstrate the ability of the MFT method to account for a qualitative description of the dynamics in case of processes involving two electronic states. The origin of the problems to describe the correct long-time relaxation dynamics as well as multi-state processes will be discussed in more detail in Section VI. Despite these problems, it is surprising how this simplest MQC method can describe complex nonadiabatic dynamics. Other related approximate methods such as the quantum-mechanical TDSCF approximation have been found to completely fail to account for the long-time behavior of the electronic dynamics (see Fig. 10). This is because the standard Hartree ansatz in the TDSCF approach neglects all correlations between the dynamical DoF, whereas the ensemble average performed in the MFT treatment accounts for the static correlation of the problem. 

Of course the proper treatment of energy transfer is via quantum mechanics. The analysis is very straightforward, and an excellent outline is presented by Cantor and Schimmel. The excitation light induces transitions in the donor to an excited (singlet) state. This decays rapidly to the lowest excited state. The donor can then relax either via fluorescence, nonradiative procesess, or interaction with the acceptor via a dipole-dipole interaction (see Fig. 3). The Hamiltonian or energy of interaction between the donor and acceptor is... 

Here k n and k m are the microscopic classical rate constants for conversion of state n to state m and vice versa. The subscript stochastic indicates that we are considering relaxations that depend on random fluctuations of the surroundings, not the oscillatory, quantum-mechanical phenomena described by Eq. (10.23). The ensemble will relax to a Boltzmann distribution of populations if the ratio k Jk m is given by tJ. —EnJk ). According to Eq. (10.27), relaxations of the diagonal elements toward thermal equilibrium do not depend on the off-diagonal elements of p, which is in accord with classical treatments of kinetic processes simply in terms of populations. 

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