Coupling of oscillators

The theoretical derivation of the existence of a coherently excited polar mode following from (11) can also be phrased in a field theoretical model, or in a more exact way be supported by numerical work as quoted in Ref. 7, as well as by more recent work—the present chapter is not meant to provide a complete review. It is of importance, however, that other models involving nonlinear coupling of oscillations can lead to coherent excitation of a single mode. " ... 

A simple demonstration that the quantum mechanical coupling of oscillating charges produces a non-zero stabilizing term is the Drude model see ref. [2], pp. 32-34. That some physical facts could only be explained by quantum mechanics (quantal fiat , see ref. [4], p. 200) was an epistemological scandal to many physicists of the early twentieth century. 

Two different reactions have presently been studied in the Couette flow reactor, namely the variants of the Belousov-Zhabotinsky [27-30, 32] and chlorite-iodide [29-33] reactions. The BZ reaction has revealed a rich variety of steady, periodic, quasi-periodic, frequency-locked, period-doubled and chaotic spatio-temporal patterns [27, 28], well described in terms of the diffusive coupling of oscillating reactor cells, the frequency of which changes continuously along the Couette reactor as the result of the imposed spatial gradient of constraints. This experimental observation has been successfully simulated with a schematic model of the BZ kinetics [68] and the recorded bifurcation sequences of patterns resemble those obtained when coupling two nonlinear oscillators. 

Finally, semi-classical approaches to non-adiabatic dynamics have also been fomuilated and siiccessfLilly applied [167. 181]. In an especially transparent version of these approaches [167], one employs a mathematical trick which converts the non-adiabatic surfaces to a set of coupled oscillators the number of oscillators is the same as the number of electronic states. This mediod is also quite accurate, except drat the number of required trajectories grows with time, as in any semi-classical approach. 

In equation (Cl.4.14) the saturation parameter essentially defines a criterion to compare the time required for stimulated and spontaneous processes. If I then spontaneous coupling of the atom to the vacuum modes of the field is fast compared to the stimulated Rabi coupling and the field is considered weak. If s" 1 then the Rabi oscillation is fast compared to spontaneous emission and the field is said to be strong. Setting s equal to unity defines the saturation condition... 

Calculations within tire framework of a reaction coordinate degrees of freedom coupled to a batli of oscillators (solvent) suggest tliat coherent oscillations in the electronic-state populations of an electron-transfer reaction in a polar solvent can be induced by subjecting tire system to a sequence of monocliromatic laser pulses on tire picosecond time scale. The ability to tailor electron transfer by such light fields is an ongoing area of interest [511 (figure C3.2.14). 

We shall treat coupling of modes of motion in some detail because there are fundamental mechanical and mathematical topics involved that will be useful to us in both MM and quantum mechanical calculations. In the tieatment of coupled haiinonic oscillators, matrix diagonalization and normal coordinates are encountered in a simple form. 

The variable frequencies of suites V and VIII on one side, and VI and VIII on the other, correspond to oscillations resulting from the coupling of the v(C-X) vibration with the cjf mode in the case of 2- or 5-substituted derivatives and with the mode in the case of 4-substituted derivatives. For 2,5-disubstituted thiazoles the ojf, vibration is only slightly different from that of thiazole itself and the 5 oscillation is coupled with both v(C(2iX) and vfC(5,X or Y) modes, giving rise to three frequencies, two of which are higher and classified in suites V and V, the third, being lower, is assigned to suite VIII. 

Moreover, in this linear-response (weak-coupling) limit any reservoir may be thought of as an infinite number of oscillators qj with an appropriately chosen spectral density, each coupled linearly in qj to the particle coordinates. The coordinates qj may not have a direct physical sense they may be just unobservable variables whose role is to provide the correct response properties of the reservoir. In a chemical reaction the role of a particle is played by the reaction complex, which itself includes many degrees of freedom. Therefore the separation of reservoir and particle does not suffice to make the problem manageable, and a subsequent reduction of the internal degrees of freedom in the reaction complex is required. The possible ways to arrive at such a reduction are summarized in table 1. 

The components connected between the emitter-follower and the currentsensing filter capacitor can be thought of as a resistor divider. An additional 0.17 V needs to appear at pin 7 (through a 1K resistor) so the amount of current that must be contributed to that node is 0.17 V/1K which is 170 pA. The capacitive coupling of the PNP to pin 7 essentially centers the oscillator waveform upon the current ramp. So,... 

The induced absorption band at 3 eV does not have any corresponding spectral feature in a(co), indicating that it is most probably due to an even parity state. Such a state would not show up in a(co) since the optical transition IAK - mAg is dipole forbidden. We relate the induced absorption bands to transfer of oscillator strength from the allowed 1AS-+1 (absorption band 1) to the forbidden 1 Ak - mAg transition, caused by the symmetry-breaking external electric field. A similar, smaller band is seen in EA at 3.5 eV, which is attributed to the kAg state. The kAg state has a weaker polarizability than the mAg, related to a weaker coupling to the lower 1 Bu state. 

An interesting improvement from the classical treatment of the bond under stress was proposed by Crist et al, [101], Considering the chain as a set of N-coupled Morse oscillators, these authors determined the elongation and time to failure as a function of the axial stress. The results, reported in Fig. 20, show a decreasing correlation between the total elastic strain before failure and the level of applied force with the chain length. To break a chain within some reasonable time interval (for example <10-3s) requires, however, the same level of stress (a0.7 fb) as found from the simpler de Boer s model. 

Nakanishi, S., Sakai, S., Nagai, T. and Nakato, Y. (2005) Macroscopically uniform nanoperiod alloy multilayers formed by coupling of electrodeposition with current oscillations. /. Phys. Chem. B, 109, 1750-1755. 

We will finish the example with a couple of points which will prove useful later in the chapter. If we insert the harmonic oscillator potential into the temperature correction formula (11.2), we get... 

X-ray diffraction. The mechanism by which atoms diffract or scatter electromagnetic radiation via the coupling of the electron cloud of the atom to the incident oscillating electric field was discussed in the section on SERS. The X-rays scattered by an atom are the resultant of the waves... 

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