Classical equilibrium point

In the limit n oo, the classical equilibrium point ( o, rjo) and the energy E axe determined from the system of equations ... 

Figure 1.13 Plot of potential energy, V(r), against bond length, r, for the harmonic oscillator model for vibration is the equilibrium bond length. A few energy levels (for v = 0, 1, 2, 3 and 28) and the corresponding wave functions are shown A and B are the classical turning points on the wave function for w = 28...

In the framework of DECP, the first pump pulse establishes a new potential surface, on which the nuclei start to move toward the new equilibrium. The nuclei gain momentum and reach the classical turning points of their motion at t = nT and t = (n + l/2)T. The second pump pulse then shifts the equilibrium position, either away from (Fig. 3.10b) or to the current position of the nuclei (Fig. 3.10c). The latter leads to a halt of the nuclear motion. Because photo-excitation of additional electrons can only shift the equilibrium position further in the same direction, the vibrations can only be stopped at their maximum displacement [32]. 

It is well known that self-oscillation theory concerns the branching of periodic solutions of a system of differential equations at an equilibrium point. From Poincare, Andronov [4] up to the classical paper by Hopf [12], [18], non-linear oscillators have been considered in many contexts. An example of the classical electrical non-oscillator of van der Pol can be found in the paper of Cartwright [7]. Poore and later Uppal [32] were the first researchers who applied the theory of nonlinear oscillators to an irreversible exothermic reaction A B in a CSTR. Afterwards, several examples of self-oscillation (Andronov-PoincarA Hopf bifurcation) have been studied in CSTR and tubular reactors. Another... 

The anharmonicities of the potential contribute by the terms involving the constants x, g, y,. .. as well as the energy shifts AEx = 0(h2),. .. and the frequency shifts Aw, = 0(h2),. These anharmonic constants can be calculated by the Van Vleck contact transformations [20] as well as by a semi-classical method based on an h expansion around the equilibrium point [14], which confirms that the Dunham expansion (2.8) is a series in powers of h. Systematic methods have been developed to carry out the Van Vleck contact transformations, as in the algebraic quantization technique by Ezra and Fried [21]. It should be noted that the constants x and g can also be obtained from the classical-mechanical Birkhoff normal forms [22], The energy shifts AEx,... 

The prototype potential surface invoked in chemical kinetics is a two-dimensional surface with a saddle equilibrium point and two exit channels at lower energies. The classical and quantal dynamics of such surfaces has been the object of many studies since the pioneering works by Wigner and Polanyi. Recent advances in nonlinear dynamical systems theory have provided powerful tools, such as the concepts of bifurcations and chaos, to investigate the classical dynamics from a new point of view and to perform the semiclassical... 

At energies slightly above the saddle energy, there exists a single unstable classical periodic orbit. This periodic orbit corresponds in general to symmetric stretching motion (or an equivalent mode in XYZ-type molecules). The Lyapunov exponent of this periodic orbit tends to the one of the equilibrium point as the threshold energy is reached from above. 

Experimental observations of the time evolution of externally unforced macroscopic systems on the level meSo l show that the level eth of classical equilibrium thermodynamics is not the only level offering a simplified description of appropriately prepared macroscopic systems. For example, if Cmeso is the level of kinetic theory (Sections 2.2.1, starting point. In order to see the approach 2.2.2, and 3.1.3) then, besides the level, also the level of fluid mechanics (we shall denote it here Ath) emerges in experimental observations as a possible simplified description of the experimentally observed time evolution. The preparation process is the same as the preparation process for Ath (i.e., the system is left sufficiently long time isolated) except that we do not have to wait till the approach to equilibrium is completed. If the level of fluid mechanics indeed emerges as a possible reduced description, we have then the following four types of the time evolution leading from a mesoscopic to a more macroscopic level of description (i) Mslow/ (ii) Aneso 2 -> Ath, (ui) Aneso l -> Aneso 2, and (iv) Aneso i —> Aneso 2 —> Ath- The first two are the same as (111). We now turn our attention to the third one, that is,... 

Figure 2.2. Scheme of nuclear potentials in the ground electronic state E°(Q) and the excited electronic state E (Q). In the excited state, the frequency changes (i20->S2r) and the equilibrium point is shifted. The classical relaxation energy to the new nuclear configuration in the excited state is the Franck-Condon energy Efc and characterizes the linear exciton-vibration coupling. 

Figure 2. The Landau "classical VP trajectory in the, Re(/ ),Im(/j) space (bold line) The dashed line indicates the potential energy contour at the dissociation threshold. The solid circle represents the initial point of the trajectory, the open circle represents the equilibrium point of the potential arrows show the direction of the motion.

First, in contrast with the classical harmonic oscillator, the quantum harmonic oscillator in its ground state is most likely to be found at its equilibrium position. A classical harmonic oscillator spends most of its time at the classical turning points, the positions where it slows down, stops, and reverses directions. (It might... 

Particle-in-a-box models and the qnantnm harmonic oscillator illustrate a number of important features of quantum mechanics. The energy level structure depends on the natnre of the potential in the particle in a box, E n, whereas for the harmonic oscillator, E n. The probability distributions in both cases are different than for the classical analogs. The most probable location for the particle-in-a-box model in its gronnd state is the center of the box, rather than uniform over the box as predicted by classical mechanics. The most probable position for the quantum harmonic oscillator in the ground state is at its equilibrium position, whereas the classical harmonic oscillator is most likely to be fonnd at the two classical turning points. Normalization ensures that the probabilities of finding the particle or the oscillator at all positions add np to one. Finally, for large values of n, the probability distribution looks mnch more classical, in accordance with the correspondence principle. 

In the classical equilibrium thermodynamics, Stirling and Ericsson cycles have an efficiency that goes to the Carnot efficiency, as it is shown in some textbooks. These three cycles have the common characteristics, including two isothermal processes. The objection to the classical point of view is that reservoirs coupled to the engine modeled by any of these cycles do not have the same temperature as the working fluid because this working fluid is not in direct thermal contact with the reservoir. Thus, an alternative study of these cycles is using finite... 

This approximation holds locally, in the vicinity of a single equilibrium point of fhe Hamiltonian function of fhe corresponding d-dimensional classical system (see Appendix A for details). It is this equilibrium point whose phase space neighborhood is traversed by reactive trajectories on their way from the valley of reacfanfs fo fhe valley of producfs. The equilibrium poinf is... 

It is important to note here that like the flux in the classical case the CRP is determined by local properties of the Hamilton operator H in the neighborhood of the equilibrium point only. 

The strength of the coupling is controlled by the parameter e in Eq. (66). The vector field generated by the corresponding classical Hamilton function has an equilibrium point at (q p2, q, pi, p2, p ) = 0. For e sufficiently small (for given values of parameters of the Eckart and Morse potentials), the equilibrium point is of saddle-center-center stability type. 

It is useful to note that the case of two DoF Hamiltonian systems is special. In this case a classical result of Ref. [75] (see also Ref. [76]) gives convergence results for the classical Hamiltonian normal form in the neighborhood of a saddle-center equilibrium point. Recently, the first results on convergence of the QNF have appeared. In Ref. [77] convergence resulfs for a one and a half DoF system (i.e., time-periodically forced one DoF Hamiltonian system) have been given. It is not unreasonable that these results can be extended to the QNF in the neighborhood of a saddle-center equilibrium point of a fwo DoF system. 

Like in the classical case we will now assume that H(q, p fteff) (be., the classical Hamilton function) has an equilibrium point, Zo = (qo, po)/ of... 

Analogous to the classical case we will now construct a sequence of unitary-transformations which will simplify the Hamilton operator order by order in the neighborhood of the equilibrium point Zq. These unitary transformations will be of the form... 

The first two steps of the transformations (A.31) are identical to the classical case. The first step serves to shift the equilibrium point Zq to the origin according to... 

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