The Weak Coupling Limit

The weak coupling limit is encountered as we have stated above, when AJ 1 or 

3) In fact, even if the linear coupling is weak, it is not correct to assume that the quadratic coupling is also weak. 

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The equation of motion for the expectation < in the weak-coupling limit has a Langevin-like form... 

Our conclusions about the case of large /tls have a rather speculative character, and pursue merely an illustrative goal, since (2.41) and (2.42) are obtained in the weak-coupling limit. 

In Ref. [4], the soliton lattice configuration and energy within the SSH model were found numerically. Analytical expressions for these quantities can be obtained in the weak-coupling limit, when the gap 2A() is much smaller than the width of the re-electron band 4/0. At this point it is useful to define the lattice correlation length ... 

In the weak-coupling limit unit cell a (, 0 7a for fra/u-polyacetylene) and the Peierls gap has a strong effect only on the electron states close to the Fermi energy eF-0, i.e., stales with wave vectors close to . The interaction of these electronic states with the lattice may then be described by a continuum, model [5, 6]. In this description, the electron Hamiltonian (Eq. (3.3)) takes the form ... 

On the other hand, after the phase transition, in the weak coupling limit (A quantum decoherence and classical correlation are given approximately by... 

Near the critical point coefficients of (1) can be expanded in t = (T — Tc)/Tc. In the weak coupling limit they render ... 

Fluctuations dominate for T > For typical values fiq (350-F500) MeV and for Tc > (50 A- 70) MeV in the weak coupling limit from (26), (22) we estimate Tq< (0.6 A- 0.8)TC. If we took into account the suppression factor / of the mean field term oc e A /T, a decrease of the mass m due to the fluctuation contribution (cf. (11)), and the pseudo-Goldstone contribution (25), we would get still smaller value of T < (< 0.5TC). We see that fluctuations start to contribute at temperatures when one can still use approximate expressions (22), (20) valid in the low temperature limit. Thus the time (frequency) dependence of the fluctuating fields is important in case of CSC. 

To keep the calculation tractable, we will only consider the weak-coupling limit Ti,T2 t), in this case the advanced (and retarded) Green functions do not depend on the Ts, and for the model defined by equation (6) they are given as... 

No divergences and dependence on the contact parameters Ti 2 remain in the form for r. It shows the transmittance function (at least in the weak-coupling limit) is indeed a well-defined molecular quantity. We can rewrite equation (38), taking into account the definition of 6 (see equation (35)) and the definition of the Chebyshev polynomials of the second kind U (cos 6) — sin[(n +l)0]/sin 6 as... 

In the weak coupling limit with Eq. (32), this reduces to the well-known kinetic equation for the average number of the excited particle obtained by the X t approximation. 

In the weak coupling limit [160, 163] the transfer rate constant is given by... 

In the weak coupling limit, the effect of ZFS must be accounted for in the effective Hamiltonian ... 

From the point of view of thermodynamics we have now a microscopic model of entropy (see Eq. (52)). Therefore, we can verify that it leads to the basic expressions of thermodynamics of irreversible processes in the neighborhood of equilibrium.29 These expressions were derived until recently in the weakly coupled limit, or for dilute gases. 

The process has been treated theoretically in terms of simplified models.14 58 The quantum mechanics is one of formulating the probability of crossing from an excited to a ground state, summed over all vibrational levels. For coordination compounds, the weak coupling limit is presumably the important approximation. Here, the transition is from low lying vibrational levels of the excited state to very high vibrational levels of the ground state. 

For the case of typical ionic crystals aP 1-10, and the weak coupling limit is applicable. The most important conclusion from this treatment is that the weak coupling limit leads to a perturbed Bloch type wave function characterized by equal probability for finding the electron at any point of the medium. Thus, in the case of the ionic crystals, the current description of the polaron is that of a mobile electron followed by lattice polarization. 

In the weak coupling limit, as is the case for most molecular systems, each molecule can be treated as an independent source of nonrlinear optical effects. Then the macroscopic susceptibilities X are derived from the microscopic nonlinearities 3 and Y by simple orientationally-averaged site sums using appropriate local field correction factors which relate the applied field to the local field at the molecular site. Therefore (1,3)... 

This model, called the spin-boson Hamiltonian, is probably the only problem (except maybe for some very artificial ones) whose full solution can be obtained without any additional approximations. The equation of motion for the expectation value (crz) in the weak coupling limit has a... 

Our conclusions about the case for large pTLS are speculative in nature, and are meant to be merely illustrative, because (2.43) and (2.44) are obtained only in the weak coupling limit. 

To sum up, we have developed a general non-perturbative method that allows one to calculate the rate of relaxation processes in conditions when perturbation theory is not applicable. Theories describing non-radiative electronic transitions and multiphonon relaxation of a local mode, caused by a high-order anharmonic interaction have been developed on the basis of this method. In the weak coupling limit the obtained results agree with the predictions of the standard perturbation theory. 

In the previous section we have dealt with a simple, but nevertheless physically rich, model describing the interaction of an electronic level with some specific vibrational mode confined to the quantum dot. We have seen how to apply in this case the Keldysh non-equilibrium techniques described in Section III within the self-consistent Born and Migdal approximations. The latter are however appropriate for the weak coupling limit to the vibrational degrees of freedom. In the opposite case of strong coupling, different techniques must be applied. For equilibrium problems, unitary transformations combined with variational approaches can be used, in non-equilibrium only recently some attempts were made to deal with the problem. [139]... 

Fig. 38 Electronic transmission and corresponding current in the weak-coupling limit with ohmic dissipation (s = 1) in the bath. Parameters N = 20, Jo/wc =...

As a first step beyond the mean-field approach let s first consider the weak-coupling limit in P(E). For Jo/coc < 1 and not too high temperatures (feT/4 < 1) the main contribution to the integral in Eq. (456) comes from long times t wj1. With the change of variables z = cot, < (t) can be written as ... 

The condition EM implies a small displacement for each normal mode and therefore for the potential surfaces of the electronic states. This condition is recognized as the weak coupling limit of electronic states and Equation 6.75 gives the rate of radiationless conversion in this limit. 

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