Weak coupling approximation

It is instructive to compare the approximate weak-coupling theory to essential exact, numerical (density matrix renormalization group) calculations on the same model (namely the Pariser-Parr-Pople model). The numerical calculations are performed on polymer chains with the polyacetylene geometry. Since these chains posses inversion symmetry the many-body eigenstates are either even (Ag) or odd By). As discussed previously, the singlet exciton wave function has either even or odd parity when the particle-hole eigenvalue is odd or even. Conversely, the triplet exciton wavefunction has either even or odd parity when the particle-hole eigenvalue is even or odd. As a consequence, we can express a B state as... 

The mean field teclmique is one of the most robust and simple methods used to handle larger molecules in gas and liquid enviromnents [M, 134. 135 and 136]. The basic premise of all mean-field methods is that the fiill wavefiinction represents N very weakly coupled modes (2 ) and can be approximated as... 

In tills weakly coupled regime, ET in an encounter complex can be described approximately using a two-level system model [23]. As such, tlie time-dependent wave function is... 

A second approximation neglects coupling between the spin of an electron and its orbital momentum but assumes that coupling between orbital momenta is strong and that between spin momenta relatively weak but appreciable. This represents the opposite extreme to the 77-coupling approximation. It is known as the Russell-Saunders coupling approximation and serves as a useful basis for describing most states of most atoms and is the only one we shall consider in detail. 

Solving now the Heisenberg equations of motion for the a operators perturbatively in the same way as in the weak-coupling case, one arrives (at = 0) at the celebrated non-interacting blip approximation [Dekker 1987b Aslangul et al. 1985]... 

Working in the same weak-coupling approximation, it takes little effort to produce the expression for the rate constant in the asymmetric case, by simply replacing J in (2.42)-(2.44) by the energy bias . 

Uncoupled solutions for current and electric field give simple and explicit descriptions of the response of piezoelectric solids to shock compression, but the neglect of the influence of the electric field on mechanical behavior (i.e., the electromechanical coupling effects) is a troublesome inconsistency. A first step toward an improved solution is a weak-coupling approximation in which it is recognized that the effects of coupling may be relatively small in certain materials and it is assumed that electromechanical effects can be treated as a perturbation on the uncoupled solution. 

Fig. 4.5. The degree of approximation for the increase of current in time for uncoupled and weakly coupled solutions for impact-loaded, x-cut quartz and z-cut lithium niobate is shown by comparison to the numerically predicted, fully coupled case. In the figure, the initial current is set to the value of 1.0 at the measured value (after Davison and Graham [79D01]).

The determination of piezoelectric constants from current pulses is based on interpretation of wave shapes in the weak-coupling approximation. It is of interest to use the wave shapes to evaluate the degree of approximation involved in the various models of piezoelectric response. Such an evaluation is shown in Fig. 4.5, in which normalized current-time wave forms calculated from various models are shown for x-cut quartz and z-cut lithium niobate. In both cases the differences between the fully coupled and weakly coupled solutions are observed to be about 1%, which is within the accuracy limits of the calculations. Hence, for both quartz and lithium niobate, weakly coupled solutions appear adequate for interpretation of observed current-time waveforms. On the other hand, the adequacy of the uncoupled solution is significantly different for the two materials. For x-cut quartz the maximum error of about 1%-1.5% for the nonlinear-uncoupled solution is suitable for all but the most precise interpretation. For z-cut lithium niobate the maximum error of about 8% for the nonlinear-uncoupled solution is greater than that considered acceptable for most cases. The linear-uncoupled solution is seriously in error in each case as it neglects both strain and coupling. 

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

Using the long time-weak coupling approximation and the hypothesis of random phases for the thermostat, Bloch and Wangsness find, after taking the trace over the heat bath, the following equations for the reduced density matrix a ... 

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. 

For the weak coupling case with Eq. (32), our master equation reduces to the well-known quantum master equation, obtained through the approximation, widely used in quantum optics. This equation describes, among other things, quantum decoherence due to Brownian motion. Hence, we have derived an exact quantum master equation for the transformed density operator p that describes exact decoherence. Furthermore, our master equation cannot keep the purity of the transformed density matrix. Indeed, one can show that if p(t) is factorized into a product of transformed wave functions at t = 0, it will not be factorized into their product for t > 0. This is consistent the nondistributivity of the nonunitary transformation (18). 

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. 

The resulting expression is especially simple in the weak coupling case. In this case, the two propagators in Eq. (12) can be approximated by their first order (i.e, single hop) terms. (The zeroth order term makes no contribution of Kjf as long as i 5 f) In this weak coupling limit, the expression for Pif (t) can be expressed as ... 

As discussed above, the evaluation of the condensed phase probability for transitions between the vibrational states of a solute molecule is a problem in the weak coupling regime for which the short x approximation should be valid. We have performed calculations of the probability for the transition from the first excited vibrational state to the ground vibrational state of Br2 in a dense Ar fluid employing a forward-backward surface hopping method similar to the one described in the previous section. The simulation system contains one Br2 molecule and 107 Ar... 

We now resort to the crucial approximation that a it) varies slower than either e(t) or 0(t). This approximation is justifiable in the weak-coupling regime (to second order in Hj) as discussed below. Under this approximation, Eq. (4.41) is transformed into a differential equation describing relaxation at a time-dependent rate ... 

Radical formation in a mixed crystal system of cytosine monohydrate doped with small amounts of thiocytosine (ca. 0.5%) was investigated on order to gain insight into hole transfer in a well-defined crystalline system.31 Also of interest was whether the protonation state of the thiocytosine radical(s) was the same as that of the cystosine radical(s). Crystals were X-irradiated (ca. 30 kGy) and ESR and ENDOR spectra recorded at ca. 15 K. After irradiation, many types of free radicals were formed. Among these, the low field resonance from a sulfur centered radical (42), with g-tensor (2.132, 2.004, 2.002), was clearly visible. Radical 42 constituted approximately 10% of the total cohort of radicals formed in the crystal and is apparently the only sulfur-centred radical observed in this experiment. Six weakly coupled protons were observed, two of which are shown... 

This equation is, of course, well known and often called the Pauli equation. We recognize on the right-hand side the familiar gain and loss terms. The transition probabilities which appear in the Pauli equation correspond to the Born approximation for one-photon processes. For further reference let us summarize the main properties of this weakly coupled approximation. 

Viewed in the time domain, the replacement of M(a>) by M washes out the details of the time variation within Q space. For this approximation to be useful, all strongly coupled states should be included in the P space and the Q space should not include any states that couple strongly to the P space (weak coupling assumption). We now find that the population dynamics of the m levels within the P space is governed by the equations of motion... 

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