Harmonic oscillators coupled

Figure A.4 Visualization of lattice vibrations as coupled harmonic oscillations of spheres connected by springs.

Another analysis method was based on the local wave vector estimation (LFE) approach applied on a field of coupled harmonic oscillators.39 Propagating media were assumed to be homogeneous and incompressible. MRE images of an agar gel with two different stiffnesses excited at 200 Hz were successfully simulated and compared very well to the experimental data. Shear stiffnesses of 19.5 and 1.2 kPa were found for the two parts of the gel. LFE-derived wave patterns in two dimensions were also calculated on a simulated brain phantom bearing a tumour-like zone and virtually excited at 100-400 Hz. Shear-stiffnesses ranging from 5.8 to 16 kPa were assumed. The tumour was better detected from the reconstructed elasticity images for an input excitation frequency of 0.4 kHz. 

I. Sack, J. Bernarding and J. Braun, Analysis of wave patterns in MR elastography of skeletal muscle using coupled harmonic oscillator simulations, Magn. reson. Imaging, 2002, 20, 95-104. 

The concept of resonance was introduced into quantum mechanics by Heisenberg16 in connection with the discussion of the quantum states of the helium atom. He pointed out that a quantum-mechanical treatment somewhat analogous to the classical treatment of a system of resonating coupled harmonic oscillators can be applied to many systems. The resonance phenomenon of classical mechanics is observed, for example, for a system of two tuning forks with the same characteristic frequency of oscillation and attached to a common base, hich... 

Lattice vibrations are fundamental for the understanding of several phenomena in solids, such as heat capacity, heat conduction, thermal expansion, and the Debye-Waller factor. To mathematically deal with lattice vibrations, the following procedure will be undertaken [7] the solid will be considered as a crystal lattice of atoms, behaving as a system of coupled harmonic oscillators. Thereafter, the normal oscillations of this system can be found, where the normal modes behave as uncoupled harmonic oscillators, and the number of normal vibration modes will be equal to the degrees of freedom of the crystal, that is, 3nM, where n is the number of atoms in the unit cell and M is the number of units cell in the crystal [8],... 

In order to solve this problem, it is possible to use the Hamiltonian procedure of classical mechanics [8], Hence, the classical Hamiltonian of a system of coupled harmonic oscillators can be written as follows [7] ... 

The deduction is based upon assuming that one of the oscillators is a weak bond which will break when energy E is present in it. For a molecule that consists of n weakly coupled harmonic oscillators the chance that, when the molecule has energy E, at least E of it will be localized in one oscillator is given by (1 — E /E) . The rate at which such an event happens, k E), is then presumed proportional to this ratio, the constant of proportionality being A, the mean rate of internal energy transfer in the molecule. This derivation may be justified for a classical and for a quantized molecule. 

A solvable model which we have not investigated is the one of coupled harmonic oscillators. This was introduced by Ford and his collaborators [Ford 1965] and also by Ref. [Ullersma 1966], This model provides a formally exact derivation of the Master Equation. Many features of irreversible evolution can be investigated exactly within this model for example see Ref. [Haake 1985 Strunz 2003]. The result is also equivalent with the approaches in Refs. [Cal-deira 1983 Unruh 1989],... 

In the second part of this work, we addressed the problem of how to use the above described effects of a space-dependent interaction to steer molecular transition. We proposed a model that allows us to induce a space-dependent coupling between two molecular potentials via a steady-state coupling to a third potential surface. By changing the frequency and intensity of the steady state laser, we can shape the space-dependence of the coupling. We illustrated the method with an example of three coupled harmonic oscillators and showed how displacement and width of the excited wave packet can be controlled. 

Linear oscillator Civil engineering, Coupled harmonic oscillators... 

We now we summarize some of the procedures that are used in analyzing multidimensional IR data. Constants factors are often omitted from the formulas as are the transition dipole factors which are easily incorporated [74] when the modes are a collection of coupled harmonic oscillators. More generally the variations of transition dipole with nuclear displacement should be incorporated. It is often useful to compare the 2D-IR results with the results of other nonlinear experiments because it turns out that various manipulations of these multidimensional signals provide all of the common nonlinear results such as echoes, gratings, degenerate four wave effects, and pump-probe spectroscopy. 

The one-exciton Hamiltonian for a particular polypeptide, n in a distribution of structures, was chosen as M coupled harmonic oscillators ... 

The simulation of the isotopically substituted linear and 2D-IR spectra of helices is based on one- and two-exciton Hamiltonians, Eq. (61), which describe the frequencies and delocalization of amide-1 modes of a helix with N = 25 coupled harmonic oscillators and two isotpomers. The zero-order isotope shifts were incorporated into the energy of the residues of the isotopomer modes and the naturally abundant modes also included by sampling... 

Figure 2. The system and the bath behave like coupled harmonic oscillators if the bath mass is rather small.

Other interesting treatments of the solid motion have been developed in which the motion of the solid s atoms is described by quantum mechanics [Billing and Cacciatore 1985, 1986]. This has been done for a harmonic solid in the context of treatment of the motion of the molecule by classical mechanics and use of a TDSCF formalism to couple the quantum and classical subsystems. The impetus for this approach is the fact that, if the entire solid is treated as a set of coupled harmonic oscillators, the quantum solution can be evaluated directly in an operator formalism. Then, the effect of solid atom motion can be incorporated as an added force on the gas molecule. Another advantage is the ability to treat the harmonic degrees of freedom of the solid and the harmonic electron -hole pair excitations on the same footing. The simplicity of such harmonic degrees of freedom can also be incorporated into the previously defined path-integral formalism in a simple manner to yield influence functionals (Feynman and Hibbs 1965). 

Here Ef is the amplitude, t the duration, and co the frequency of the ith pulse. This scheme has been applied in Ref [46] to a generic two-dimensional HT model which incorporated a H-atom reaction coordinate as well as a low-frequency H-bond mode. In a subsequent work [47] the approach has been specified to a simple model of HT in thioacetylacetone. The Hamiltonian was tailored to the form of Eq. (4.1) based on the information available for the stationary points, that is, the energetics as well as the normal modes of vibration. From these data an effective two-dimensional potential was constructed including the H-atom coordinate as well as a coupled harmonic oscillator, which describes the 0-S H-bond motion. Although perhaps oversimplified, this model allowed the study of some principle aspects of laser-driven H-bond motion in an asymmetric low-barrier system. 

In the preceding text we have presented a unified theory of regularization of the perturbed Kepler motion. Quaternion algebra allows for an elegant treatment of the spatial case in a way completely analogous to the way the planar case is traditionally handled by means of complex numbers. As a consequence of the linearity of the regularized equations of the perturbed Kepler motion, the problem of satellite encounters reduces to a linear perturbation problem, the problem of coupled harmonic oscillators. Orbital elements based on the oscillators may lead to a simpified discussion of ordered and chaotic behavior in repeated satellite encounters. This has been demonstrated by means of an instructive example. 

Fig. 41-2.—Energy levels for coupled harmonic oscillators left, with X 0 right, with X vJ/5.

Table 49-1.—Sets of Quantum Numbers for Five Coupled Harmonic Oscillators with Total Quantum Number 10...

Fiu. 49-1.—The probability values Pn for system-part a in a system of five coupled harmonic oscillators with total quantum number n 10 (closed circles), and values calculated by the Boltzmann distribution law (open circles). 

In order for the oscillators to behave identically with respect to external perturbations as well as mutual interactions they would have to occupy the same position in space that is, to oscillate about the same point. A system such as a crystal is often treated approximately as a set of coupled harmonic oscillators (the atoms oscillating about their equilibrium positions). The Boltzmann statistics would be used for this set of oscillators, inasmuch as the interactions depend on the positions of the oscillators in space in such a way as to make them non-identical. 

Eq. (A.52) is identical to the Langevin equation for a set of n—p coupled harmonic oscillators each of unit mass with coordinates y(t), dynamical matrix and friction matrix p. 

First, let us consider a system of weakly coupled harmonic oscillators of frequencies v, v, each having gi, g2, , gm degeneracies, respec-... 

Numerical simulations have shown that the presence of the perturbation V in Eq. (4.52) is not sufficient for ergodic behavior. It was discovered that a system of N weakly coupled harmonic oscillators will not freely exchange energy as long as there is no collection of integers nj for which... 

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