Dynamic quantum response

Now let the external force F(Z) be time-dependent. We will repeat the procedure followed in the classical case, assuming that 7 (Z) is given by the step function (11.20), that is, following the onset of a perturbation H = — FA at t = — oo, F is switched off at / = 0. We want to describe the subsequent relaxation process of the system as expressed by the evolution of the expectation of an observable B from its value (5) (equilibrium average under H = Hq + ) at t = 0 to the final 

Using the identity (11.25) we expand the thermal operators in (11.36) to first order 

As in the classical case, the important physical aspect of this result is that the time dependence in as well as the equilibrium thermal average are evaluated with respect to the Hamiltonian Hq of the unperturbed system. 

Let us now follow a different route, starting from the quantum Liouville equation for the time evolution of the density operator p 

Assume as in Eq. (11.17) that the system is at its unperturbed equilibrium state in the infinite past 

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Generally speaking, the quantum phase variables can be divided into two classes. First, we have the pure operational phases that are completely determined by the scheme of measurement. This has no contradiction with the existence of an intrinsic quantum-dynamical variable responsible for the phase properties of light [50]. In addition, there might be some inherent quantum... 

The linear response of a system is detemiined by the lowest order effect of a perturbation on a dynamical system. Fomially, this effect can be computed either classically or quantum mechanically in essentially the same way. The connection is made by converting quantum mechanical conmuitators into classical Poisson brackets, or vice versa. Suppose tliat the system is described by Hamiltonian where denotes an... 

Krause J L, Messina M, Wilson K R and Van Y J 1995 Quantum control of molecular dynamics—the strong response regime J. Phys. Chem. 99 13 736... 

In this chapter we will focus on one particular, recently developed DFT-based approach, namely on first-principles (Car-Parri-nello) molecular dynamics (CP-MD) [9] and its latest advancements into a mixed quantum mechanical/molecular mechanical (QM/MM) scheme [10-12] in combination with the calculation of various response properties [13-18] within DFT perturbation theory (DFTPT) and time-dependent DFT theory (TDDFT) [19]. 

One of the consequences of the suppression of the phase transition is the presence of a special critical point, Tc = 0 K. This point, called the quantum displacive limit, is characterized by special critical exponents. Its presence gives rise to classical quantum crossover phenomena. Quantum suppression and the response at and near this limit, Tc = 0 K, have been extensively studied on the basis of lattice dynamic models solved within the framework of both classical and quantum statistical mechanics. Figure 8 is a log-log plot of the 6 T) results for ST018 [15]. The expectation from theory is that in the quantum regime, y = 2 at 0.7 kbar, after which y should decrease. The results in Fig. 8 quantitatively show the expected behavior however, y is < 2 at 0.70 kbar. Despite the difference in the methods to suppress Tc in ST018, the results in Fig. 4a and Fig. 8 are quite similar. As shown in the results in Fig. 3b, uniaxial pressure also can be a critical parameter S for the evolution of ferroelectricity in STO. 

We presented fully self-consistent separable random-phase-approximation (SRPA) method for description of linear dynamics of different finite Fermi-systems. The method is very general, physically transparent, convenient for the analysis and treatment of the results. SRPA drastically simplifies the calculations. It allows to get a high numerical accuracy with a minimal computational effort. The method is especially effective for systems with a number of particles 10 — 10, where quantum-shell effects in the spectra and responses are significant. In such systems, the familiar macroscopic methods are too rough while the full-scale microscopic methods are too expensive. SRPA seems to be here the best compromise between quality of the results and the computational effort. As the most involved methods, SRPA describes the Landau damping, one of the most important characteristics of the collective motion. SRPA results can be obtained in terms of both separate RPA states and the strength function (linear response to external fields). 

Following selection and approval of the committee members by the Examinations Institute director, the committee begins work on the lengthy process of writing the examination. Typically, the physical chemistry committee meets at the site of each ACS national meeting until work on the examination is completed. Some of the other examination committees may meet at the Biennial Conference on Chemical Education or the ChemEd conference. At the first committee meeting, the committee will typically discuss and make decisions on several items. The first of these items is which examinations will result at the end of the process. Previous committees have opted to write only a comprehensive examination, or a suite of examinations in thermodynamics, dynamics, and quantum mechanics. A second question is how many questions are needed for each examination, and if the multiple choice format is used, how many responses will be used for each question. For example, the committee writing the 2000 set of examinations chose 40 questions with four responses, while the committee for the 2006 set chose 50 questions with four responses. This decision becomes important because it defines the number of questions that need to be written. 

The general relations among the coefficients - and Dy are presented elsewhere [179]. The quantities yj and y2 are the damping constants for the fundamental and second- harmonic modes, respectively. In Eq.(82) we shall restrict ourselves to the case of zero-frequency mismatch between the cavity and the external forces (ff>i — ff> = 0). In this way we exclude the rapidly oscillating terms. Moreover, the time x and the external amplitude have been redefined as follows x = Kf and 8F =. The s ordering in Eq.(80) which is responsible for the operator structure of the Hamiltonian allows us to contrast the classical and quantum dynamics of our system. If the Hamiltonian (77)-(79) is classical (i.e., if it is a c number), then the equation for the probability density has the form of Eq.(80) without the s terms ... 

The previous discussion shows that the relaxation processes emerge from the quantum dynamics under appropriate circumstances leading to the formation of time-dependent quasiclassical parts in the observable quantities. Let us add that quasiclassical and semiclassical methods have been recently applied to the optical response of quantum systems in several works [65, 66] where the relation to the Liouville formulation of quantum mechanics has been discussed, without however pointing out the existence of Liouvillian resonances as we discussed here above. The connection between the property of chaos and n-time correlation functions or the nth-order response of a system in multiple-pulse experiments has also been discussed [67, 68]. 

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