Oscillator coupled 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. 

When both vibrations have high frequencies, Wa, coq, the transition proceeds along the MEP (curve 1). In the opposite case of low frequencies, rUa.s the tunneling occurs in the barrier, lowered and reduced by the symmetrically coupled vibration q, so that the position of the antisymmetrically coupled oscillator shifts through a shorter distance, than that in the absence of coupling to qs (curve 2). The cases (0 (Oq, < (Oo, and Ws Wo, (Oq, characterized by combined trajectories (sudden limit for one vibration and adiabatic for the other) are also presented in this picture. 

Kaplunovsky and Weinstein [kaplu85j develop a field-theoretic formalism that treats the topology and dimension of the spacetime continuum as dynamically generated variables. Dimensionality is introduced out of the characteristic behavior of the energy spectrum of a system of a large number of coupled oscillators. 

The eigenvalue problem was introduced in Section 7.3, where its importance in quantum mechanics was stressed. It arises also in many classical applications involving coupled oscillators. The matrix treatment of the vibrations of polyatomic molecules provides the quantitative basis for the interpretation of their infrared and Raman spectra. This problem will be addressed tridre specifically in Chapter 9. 

A simple eigenvalue problem can be demonstrated by the example of two coupled oscillators. The system is illustrated in fug. 2. It should be compared with the classical harmonic oscillator that was treated in Section 5.2.2. Here also, the system will be assumed to be harmonic, namely, that both springs obey Hooke s law. The potential energy can then be written in the form... 

The relative displacements of the masses in the two normal modes of this coupled oscillator are shown to the right in Fig. 2. This method the form of the normal modes is particularly useful in the analysis of molec vibrations (see Chapter 9). 

Not only original signals (one or several) but additional combined signals (overtones, coupled oscillations, e.g. NIR) and latent signals in form of relations between original signals (differences, e.g. MS)... 

Linear oscillator example. The general equations can now be specialized to the case on one linear oscillator coupled to a thermal bath. We will closely follow the analysis given by Lindenberg and West so that the details and derivations can be consulted in that paper [133],... 

The formulation of the preceding section is very general. We are interested, however, in rotations and vibrations of polyatomic molecules. We therefore discuss now specific applications of the algebraic method beginning with the simple case of one-dimensional coupled oscillators, presented in Section 3.3 in the Schrodinger picture. In the algebraic theory, as mentioned, one associates to each coordinate, x, and related momentum, px = — iti d/dx, an algebra. For... 

For two coupled oscillators, the second possibility is chain (II) of Eq. (4.13). A dynamical symmetry corresponding to this route implies that the Hamiltonian operator contains only invariant operators of this chain,... 

The spectrum corresponding to Eq. (4.27) is shown in Figure 4.2. One can see that this represents the usual spectrum of two normal anharmonic coupled oscillators. 

In the preceding sections we have discussed the algebraic treatment of onedimensional coupled oscillators. We now present the general theory of two three-dimensional coupled rovibrators (van Roosmalen, Dieperink, and... 

We have already discussed in Section 4.5 the local-to-normal transition for two coupled oscillators. The situation is quite analogous for two coupled rovibrators. The local-to-normal transition can be described by combining the operators of the local chain with those of the normal chain. It is convenient to introduce the Majorana operator... 

Kellman, M. E. (1982), Group Theory of Coupled Oscillators Normal Modes as Symmetry Breaking, J. Chem. Phys. 76,4528. 

Voth, G. A. (1986), On the Relationship of Classical Resonances to the Quantum Mechanics of Coupled Oscillator Systems, J. Phys. Chem. 90, 3624. 

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