The harmonic oscillator is a model system in both quantum and classical mechanics. Classically, it is a particle moving back and forth at a frequency, - k/m, where k is the force constant and m is the particle mass. Its quantum mechanical description shows the system existing only with certain allowed energies, each separated by Planck s constant times the oscillator s frequency. A system of many particles connected by harmonic springs can exhibit normal mode vibration, whereby all the particles move with the same frequency. For a system of N particles, there are 3N - 6 normal modes, or 3N - 5 if the system is linear. [Pg.188]

MSN. 149. I. Prigogine, Classical and quantum mechanics of unstable dynamical systems, in Proceedings, International Conference on Dynamical Systems and Chaos, Y. Aizawa, S. Saito, and K. Shiraiwa, eds.. World Scientific, Singapore, Vol. 2, 1995. [Pg.60]

MSN. 153.1. Prigogine and T. Petrosky, Poincare Resonances and the Extension of Classical and Quantum Mechanics, in Nonlinear, deformed and irreversible quantum systems, H. D. Doebner, V. K. Dobrev, and P. Nattermann, eds.. World Scientific, Singapore, pp. 3-21, 1995. [Pg.60]

The electronic polarization may be treated by using classical mechanics, where the system is regarded as a simple harmonic oscillator. There are three forces acting on the electron (1) elastic restoring force - Kx, where K is the elastic constant and x is the displacement of the electron from its equilibrium position, (2) viscosity force - ydxfdt, and (3) the electric force - eE e , where Eo and are the amplitude and frequency of the apphed electric field, respectively. The dynamic equation is [Pg.34]

Equation 13.12 with K = 1 is a typical expression for a rate constant if it is assumed that the nuclei are moving by classical mechanics with the ground state PES as potential. This is the simplest approximation for the rate. However, in ET reactions, we have to remember that the barrier is associated with an avoided crossing between two different PES, and that there is a probability that the system jumps to the upper surface. In that case, k < 1 in Equation 13.12. Thus, the system is in a formally different electronic state, but the character of the wave function does not [Pg.351]

The above scheme suggests that all Hamiltonian systems are inte-grable, since all Hamiltonian systems seem to possess a maximal set k = 1,...,2/ of constants of the motion. Indeed, this is the impression conveyed by traditional textbooks on classical mechanics (see, e.g., Landau and Lifschitz (1970), Symon (1971), Goldstein (1976)). But if all Hamiltonian systems have a maximal set of constants of the motion, how can we reconcile this fact with the occurrence of chaos in most Hamiltonian systems [Pg.80]

As discussed above, to identify states of the system as those for the reactant A, a dividing surface is placed at the potential energy barrier region of the potential energy surface. This is a classical mechanical construct and classical statistical mechanics is used to derive the RRKM k(E) [4]. [Pg.1011]

No system is exactly unifomi even a crystal lattice will have fluctuations in density, and even the Ising model must pemiit fluctuations in the configuration of spins around a given spin. Moreover, even the classical treatment allows for fluctuations the statistical mechanics of the grand canonical ensemble yields an exact relation between the isothemial compressibility K j,and the number of molecules Ain volume V [Pg.647]

In reality, atoms and molecules in solid materials are far from static unless the temperature is low but even at 0 K, vibrational motion remains. For ionically conductive materials, atomic movement is the subject of major interest. allows us to simulate the dynamics of the particles in a well-defined system to gain greater insights into local structure and local dynamics - such as ion transport in solid materials. In an MD simulation, atomic motion in a chemical system is described in classical mechanics terms by solving Newton s equations of motion [Pg.316]

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