Harmonic oscillator damped

In an early study of lysozyme ([McCammon et al. 1976]), the two domains of this protein were assumed to be rigid, and the hinge-bending motion in the presence of solvent was described by the Langevin equation for a damped harmonic oscillator. The angular displacement 0 from the equilibrium position is thus governed by... 

Equation (30) is the well-known differential equation of the damped harmonic oscillator, the solution of which is... 

Under moderate (pJ/cm2) photoexcitation, where the photoexcited carrier density is comparable or less than the intrinsic density, time evolution of coherent A g and Eg phonons is respectively described by a damped harmonic oscillation... 

This phase shift is akin to the phase shift experienced by a damped harmonic oscillator driven in the vicinity of the oscillator s eigenfreqnency Hence, the presence of a vibrational resonance not only changes the amplitnde of the signal field, bnt also its phase. To incorporate this effect, the resonant nonlinear susceptibility is no longer real as it contains imaginary contributions ... 

The return to equilibrium of a polarized region is quite different in the Debye and Lorentz models. Suppose that a material composed of Lorentz oscillators is electrically polarized and the static electric field is suddenly removed. The charges equilibrate by executing damped harmonic motion about their equilibrium positions. This can be seen by setting the right side of (9.3) equal to zero and solving the homogeneous differential equation with the initial conditions x = x0 and x = 0 at t = 0 the result is the damped harmonic oscillator equation ... 

Exercise. A damped harmonic oscillator with delta-correlated fluctuations in the frequency is given by the Langevin equation ... 

Basically the perturbative techniques can be grouped into two classes time-local (TL) and time-nonlocal (TNL) techniques, based on the Nakajima-Zwanzig or the Hashitsume-Shibata-Takahashi identity, respectively. Within the TL methods the QME of the relevant system depends only on the actual state of the system, whereas within the TNL methods the QME also depends on the past evolution of the system. This chapter concentrates on the TL formalism but also shows comparisons between TL and TNL QMEs. An important way how to go beyond second-order in perturbation theory is the so-called hierarchical approach by Tanimura, Kubo, Shao, Yan and others [18-26], The hierarchical method originally developed by Tanimura and Kubo [18] (see also the review in Ref. [26]) is based on the path integral technique for treating a reduced system coupled to a thermal bath of harmonic oscillators. Most interestingly, Ishizaki and Tanimura [27] recently showed that for a quadratic potential the second-order TL approximation coincides with the exact result. Numerically a hint in this direction was already visible in simulations for individual and coupled damped harmonic oscillators [28]. 

Memory effects play an important role for the description of dynamical effects in open quantum systems. As mentioned above, Meier and Tannor [32] developed a time-nonlocal scheme employing the numerical decomposition of the spectral density. The TL approach as discussed above as well as the approaches by Yan and coworkers [33-35] use similar techniques. Few systems exist for which exact solutions are available and can serve as test beds for the various theories. Among them is the damped harmonic oscillator for which a path-integral solution exists [1], In the simple model of an initially excited... 

Fig. 1 Population dynamics of a damped harmonic oscillator. The populations of the ground state (n = 0) up to the third excited state (n = 3) are shown while initially all population is in the third excited state. The parameters are o-v = tao/2, / = 0.544ivo, and (3 = l/ivo. The results for the TNL theory are shown by the solid curve, those for the TL approach by the dotted curve and those of the Markovian limit by the dashed curve. (Reproduced from Ref. [29]. Copyright 2004, American Institute of Physics.)...

In the spectral region below the electronic band-to-band transitions, the Cauchy approximation (transparency region, (3.24)) or the damped harmonic oscillator function (both transparency as well as absorption region) are often utilized as MDF approaches. 

The DF spectra of wurtzite-structure ZnO within the VIS-to-VUV spectral region contain CP structures, which can be assigned to band-gap-related electronic band-to-band transitions Eq with a = A, B,C and to above-band-gap band-to-band transitions E13 with (3 = 1,..., 7. The F -related structures can be described by lineshape functions of the 3DMo-type (3.9 and 3.10), the CP structures with (3 = 3,4 by lineshape functions of the 2DMo-type (3.11), and the CP structures with (3=1,2,5,6,7 can be described by Lorentzian-damped harmonic oscillator functions (3.13). The CP structures Eq are supplemented by discrete (3.14) and continuum (3.16) excitonic contributions. Tables 3.9 and 3.10 summarize typical parameters of the CPs Eq and E, respectively, of ZnO [15]. 

Any linear dielectric response can be described as the sum (or integral) of damped harmonic oscillators in the form of Eq. (L2.318),... 

The interaction of a light wave and electrons in atoms in a solid was first analysed by H. A. Lorentz using a classical model of a damped harmonic oscillator subject to a force determined by the local electric field in the medium, see Equation (2.28). Since an atom is small compared with the wavelength of the radiation, the electric field can be regarded as constant across the atom, when the equation of motion becomes ... 

In order to investigate solids or polymer systems with free carriers by IR spectroscopy, it is very convenient to measure the reflectivity instead of absorbance or transmittance. Thus, the problems to be discussed in this context are usually described by a linear response formalism. In its simplest form, this means that the response function (dielectric function) s(u ) of a damped harmonic oscillator is used to describe the interaction between light and matter. The complex form of this function is... 

Since the reduced spectrum x"( ) clearly shows the low-ftequency Raman modes, we introduced a simple model to analyze the spectral profile of x"(.v) for obtaining the quantitative information. The model is composed of two damped harmonic oscillator modes and one Debye type relaxation mode (liquid water) or one Cole-Cole type relaxation mode (aqueous solution). Cole-Cole type relaxation is usually adopted in analyzing the dielectric relaxation. The formula of Cole-Cole type relaxation is represented as ... 

In Eq. (4-31), the first three terms describe a simple damped harmonic oscillator the first term is due to molecular accelerations, the second is due to viscous drag, and the third is due to the restoring force. Qq is the oscillator frequency, which is of order 10 sec", and p is a viscous damping coefficient. The crucial term producing the dynamic glass transition is, of course, the fourth term, which has the form of a memory integral, in which molecular motions produce a delayed response. The kernel m(t — t ) is determined self-consistently by the time-dependent structure. One simple choice relating m(s) to the structure is ... 

A quantum-mechanical treatment has been given for the coherent excitation and detection of excited-state molecular vibrations by optical absorption of ultrashort excitation and probe pulses [66]. Here we present a simplified classical-mechanical treatment that is sufficient to explain the central experimental observations. The excited-state vibrations are described as damped harmonic oscillations [i.e., by Eq. (11) with no driving term but with initial condition Q(0) < 0.] We consider the effects of coherent vibrational oscillations in Si on the optical density OD i at a single wavelength k within the Sq -> Si absorption spectrum. Due to absorption from Sq to Si and stimulated emission from Si and Sq,... 

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