Dynamics of the Harmonic Oscillation

Quantum dynamics of the harmonic oscillator 2.9.1 Elementary considerations... 

DYNAMICS OF THE HARMONIC OSCILLATION 2.4.1 Differential equations of harmonic oscillations... 

The perpendicular slice through the phase portrait provides the stroboscopic phase portrait or Poincare section (e.g. [3]). This is in the case of the harmonic oscillator one point in the phase portrait. A further powerful method for the analysis of nonlinear dynamical systems is the determination of the Fourier spectrum of the response function >2. 

In Sections 2.2 and 2.9 we have discussed the dynamics of the two-level system and of the harmonic oscillator, respectively. These exactly soluble models are often used as prototypes of important classes of physical system. The harmonic oscillator is an exact model for a mode of the radiation field (Chapter 3) and provides good starting points for describing nuclear motions in molecules and in solid environments (Chapter 4). It can also describe the short-time dynamics of liquid environments via the instantaneous normal mode approach (see Section 6.5.4). In fact, many linear response treatments in both classical and quantum dynamics lead to harmonic oscillator models Linear response implies that forces responsible for the return of a system to equilibrium depend linearly on the deviation from equilibrium—a harmonic oscillator property We will see a specific example of this phenomenology in our discussion of dielectric response in Section 16.9. 

We will now develop the equations used to compute the local mode energies. After defining the local mode Hamiltonian, we will convert it into the normal coordinates that were used to define the basis sets used for the dynamical calculations. The local mode Hamiltonian, for mode /, is defined in terms of the harmonic oscillator potential (except for the initially excited stretch, as described later)... 

Fig. 7 Molecular dynamic trajectories of a particle coupled to a heat bath on the potential-energy surface of Fig. 1. The saddle point is indicated by the dashed line, (a) and (b) are for relatively high friction and (c) low friction. A = 0.05 eV. The unit of time has been chosen such that the period of the harmonic oscillator is 2jr, and the friction coefficient y is given in terms of this unit.

The book thus embraces an extended study on a variety of issues within the theory of orientational ordering and phase transitions in two-dimensional systems as well as the theory of anharmonic vibrations in low-dimensional crystals and dynamic subsystems interacting with a phonon thermostat. For the sake of readability, the main theoretical approaches involved are either presented in separate sections of the corresponding chapters or thoroughly scrutinized in appendices. The latter contain the basic formulae of the theory of local and resonance states for a system of bound harmonic oscillators (Appendix 1), the theory of thermally activated reorientations and tunnel relaxation of orientational... 

Semiclassical techniques like the instanton approach [211] can be applied to tunneling splittings. Finally, one can exploit the close correspondence between the classical and the quantum treatment of a harmonic oscillator and treat the nuclear dynamics classically. From the classical trajectories, correlation functions can be extracted and transformed into spectra. The particular charm of this method rests in the option to carry out the dynamics on the fly, using Born Oppenheimer or fictitious Car Parrinello dynamics [212]. Furthermore, multiple minima on the hypersurface can be treated together as they are accessed by thermal excitation. This makes these methods particularly useful for liquid state or other thermally excited system simulations. Nevertheless, molecular dynamics and Monte Carlo simulations can also provide insights into cold gas-phase cluster formation [213], if a reliable force field is available [189]. 

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