Principle adiabatic

A formal derivation of the location of the zeros of Cg t) for a general adiabatic Hamiltonian can be given, following proofs of the adiabatic principle (e.g., [250-252]). The last source, [252] derives an evolution operator U, which is written there, with some slight notational change, in the form... 

At the basis of the building up procedure used by Bohr there lies the adiabatic principle introduced by Ehrenfest [1917]. 

This adiabatic principle was one of the corner-stones of the old quantum theory. It allowed one to find the quantum conditions when an adiabatic change was imposed on a system. It was used successfully to account for the Stark and Zeeman effects in the spectrum of atomic hydrogen, resulting from the application of an electric and magnetic field respectively (Schwartzchild [1916] Epstein [1916]). 

There are however some stringent restrictions on the applicability of the adiabatic principle (Mehra and Rechenberg [1982]). Ehrenfest himself [1917] showed that it was applicable to simply periodic systems. These are systems having two or more frequencies which are rational fractions of each other. In such systems the motion will necessarily repeat itself after a fixed interval of time. Burgers [1917], a student of Ehrenfest, showed that it was also applicable... 

Statistical rate theories often are also formulated using variational principles. Like the adiabatic principle, variational principles are intuitive and have to be proven (or disproven) by comparison with true dynamical treatments. As SACM in the previous chapters has been shown to give identical results with trajectory calculations at high temperature for the considered simple reaction system, differences between SACM and VTST would speak against the latter. The charge-dipole system, because of its simplicity, can be used particularly well for a quantitative comparison between SACM and VTST and, hence, for a quantitative test of VTST. 

An important variation of the adiabatic principle is isoperibol calorimetry. Well-defined heat leaks, minimized by efficient calorimeter construction and experiment design, are compensated for by calculation and/or extrapolation. The isoperibol design holds the temperature of the immediate environment surrounding the calorimeter constant. The word isoperibol literally means "constant temperature environment. ... 

In most of the more recent classical approaches [18], no allusion to Ehrenfest s (adiabatic) principle is employed, but rather the differential equations of motion from classical mechanics are solved, either exactly or approximately, subject to a set of initial conditions (masses, force constants, interaction potential, phase, and initial energies). The amount of energy, AE, transferred to the oscillator is obtained for these conditions. This quantity may then be averaged over all phases of the oscillating molecule. In approximate classical and semiclassical treatments, the interaction potential is expanded in a Taylor s series and only the first two terms are retained. 

E. E. Nikitin, Pathways of vibrational relaxation of diatoms in collisions with atoms Manifestation of the Ehrenfest adiabatic principle, in Gas Phase Chemical Reaction Systems, Eds. J.Wolfrum et al, Berlin-Heidelberg, Springer, 1996, p 231... 

The exponential dependence of P on velocity is the manifestation of the Ehrenfest adiabatic principle [4], once tJilti is identified with the frequency ra of a perturbed system (the BC diatom) and In/h av with the characteristic time of the perturbing force t(E) for the exponential interaction. A representation... 

We first derive the time-dependent dressed Schrodinger equation generated by the Floquet Hamiltonian, relevant for processes induced by chirped laser pulses (see Section IV.A). The adiabatic principles to solve this equation are next described in Section IV.B. 

The preceding analysis is well adapted when one considers slowly varying laser parameters. One can study the dressed Schrodinger equation invoking adiabatic principles by analyzing the Floquet Hamiltonian as a function of the slow parameters. 

The analysis of the dynamics consists of (i) the calculation of the dressed eigenenergy surfaces of the effective quasienergy operator as a function of the two Rabi frequencies flj and if, (ii) the analysis of their topology, and (iii) the application of adiabatic principles to determine the dynamics of processes in view of the topology of the surfaces. 

F. H. Chung, Quantitative interpretation of X-ray diffraction patterns of mixtures. II. Adiabatic principle of X-Ray diffraction analysis of mixtures, J. Appl. Crystallogr., 1974, 7, 526-531. 

In many cases, one can introduce a local moment in spite of the itinerancy of the electrons. This is because of the two time scales in the problem. The first one is the time scale given by the electron hopping. This determines the instantaneous moment on a site, which fluctuates rapidly. The other time scale is that of the expectation value of the moment at the site. This is a much slower time scale, and makes it possible to introduce a local moment variable [89]. This is usually called the adiabatic principle. 

Mechanisms of electron emission from solids, related to surface bombardment by different particles, are called secondary electron emission. The most important mechanism from this group is secondary ion-electron emission. According to the adiabatic principle (see Section 2.10), heavy ions are not efficient for transferring energy to light electrons to provide ionization. This general statement can be referred to the direct electron emission from solid surfaces induced by ion impact. As one can see from Fig. 2-16 (Dobretsov ... 

The classical significance of the E(R) vs. R curves for diatomic molecules (and the E< (Q ) vs. Q hypersurfaces for polyatomic molecules) can be qualitatively seen as follows, if it is assumed that any infinitesimally small motion of the nuclei is instantaneously followed by the electrons (adiabatic principle) If R is changed from Ro, the equilibrium position, to Ro- -(iR, it is seen that the energy of the s tem increases to E(Ro)-f-(iE. From this it follows that an external force F must be present to change the nuclear positions rfR away from their positions of equilibrium. The work is equal to F. iR according to classical mechanics, where... 

A solution of either (1.1) or (2.1) is impossible without in-, troducing some approximations even for the simplest chemical reactions. A great simplification of the problem is achieved on the basis of the EHRENFEST adiabatic principle, which states that a system remains in the same quantum state if a change in its surroundings occurs sufficiently slowly. Consequently, the electronic state will be not affected if the motions of nuclei are very slow compared with the motions of the electrons. This is usually the real situation, as first recognized in the molecular spectroscopy by BORN and OPPENHEIMER /22/ and, subsequently, in chemical dynamics by LONDON /23/ Therefore, in (la.I) T. ... 

Introduced by Paul Ehrenfest in 1917, the adiabatic principle allows one to find the quantum conditions when an adiabatic or gradual change is imposed on a system. However, it depends on the possibihty of deriving the new motion from the known one by means of an adiabatic transformation. For example, the quantum states of a particular system are known, the new quantum states that result from a gradual change, such as the apphcation of an electric or a magnetic field, can be calculated. The quantities that preserve their values after such a transformation are known as adiabatic invariants. Ehrenfest showed that for any arbitrary periodic motion, the following quantity is an adiabatic invariant ... 

There are stringent restrictions on the apphcabihty of the adiabatic principle. Ehrenfest himself showed that it was apphcable to simply periodic systems. These... 

Bohr was well aware of this limitation of the adiabatic principle but continued to use it even for many-electron atoms in the hope that it might stiU remain vahd for such aperiodic systems. He repeatedly acknowledged this point in his writings ... 

This does not alter the issue, however, since the system remains aperiodic, and Pauli was using the adiabatic principle where it did not strictly apply. But as often happens in science, taking a step that is not rigorous can often pay dividends, at least temporarily, as it did in this case. 

In a loss-Iess cavity both the frequency / and the energy W of the (Ji-wave field will go up according to the adiabatic principle i. e.,... 

Pathways of Vibrational Relaxation of Diatoms in Collisions with Atoms Manifestation of the Ehrenfest Adiabatic Principle... 

Different pathways of vibrational relaxation of diatoms in thermal collisions with atoms are discussed in the framework of the Ehrenfest adiabatic principle and generalized Landau-Teller model. Since the efficiency of different energy-transfer channels depend very strongly on the value of the Ehrenfest exponent, it is possible to assign, for given collision partners and the heat-bath temperature, the vibrational energy transfer events to VT, VRT or VR processes. 

The process of vibrational excitation and deexcitation of a diatom in a collision with an atom represents a simplest example from the host of processes which are relevant to gas-phase chemical kinetics. Experimental techniques available now allow one to measure directly state-to-state energy transfer rate coefficients. Theoretically, it is possible to accomplish completely ab initio calculation of these coefficients. One can therefore, regard the existing models of the vibrational relaxation from a new standpoint as a means for helping to understand more clearly the dynamics of the energy transfer provided that all the models are related to a single fundamental principle. This is the Ehrenfest adiabatic principle as formulated by Landau and Teller in the application to the collisional vibrational transitions of diatomic molecules. 

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