Ehrenfest average

This method has also been used for obtaining semiclassical equations of motion for atom-surface scattering by Jackson [108]. If only the equations for R(t) and P (r) are used we obtain classical dynamical equations of motions for the translational motion of the atom moving in an Ehrenfest average potential, i.e., a potential which has been averaged over the quantum coordinates of the system (here the solid). Thus the equations of motion are solved in a mean field potential of the solid. This mean field potential will be considered below. 

According to the correspondence principle as stated by N. Bohr (1928), the average behavior of a well-defined wave packet should agree with the classical-mechanical laws of motion for the particle that it represents. Thus, the expectation values of dynamical variables such as position, velocity, momentum, kinetic energy, potential energy, and force as calculated in quantum mechanics should obey the same relationships that the dynamical variables obey in classical theory. This feature of wave mechanics is illustrated by the derivation of two relationships known as Ehrenfest s theorems. 

The classical-path approximation introduced above is common to most MQC formulations and describes the reaction of the quantum DoF to the dynamics of the classical DoF. The back-reaction of the quantum DoF onto the dynamics of the classical DoF, on the other hand, may be described in different ways. In the mean-field trajectory (MFT) method (which is sometimes also called Ehrenfest model, self-consistent classical-path method, or semiclassical time-dependent self-consistent-field method) considered in this section, the classical force F = pj acting on the nuclear DoF xj is given as an average over the quantum DoF... 

I agree with Ehrenfest and particularly enjoy questions from children. But I sometimes wonder if scientists will be able to answer humanity s deepest questions. Perhaps our brains, which evolved to make us run from lions on the African savanna, are not constructed to penetrate the fabric of the universe. Imagine an alien with an IQ a hundred times our own. What profound concepts might be available to this creature in areas of awareness to which we are now totally closed The average poodle cannot understand Fourier transforms, the Sapir-Whorf hypothesis, or gravitational wave theory. Human forebrains are a few ounces bigger than a poodle s, and we can ask many more questions than a poodle. Are there facets of the universe we can never know Are there questions we can t even ask Michael Murphy discusses a related idea in The Future of the Body. 

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. 

It follows trivially that dS/dt = 0 for all time in the equilibrium state. Thus, there is no purely microscopic mechanism that will give rise to entropy production, dS/dt > 0. The reason for this is that the phase space density / q accounts for all of the microstructure of phase space. In reality, we can never know this full microstructure, as was recognized by Ehrenfest, who suggested that one should work with a coarse-grained density, /eq, obtained by averaging over suitably small cells in phase space. Then one can define an entropy in analogy with Eq. [50], but in terms of /eq. In this way, entropy production can be realized microscopically, as discussed in detail in Ref. 23. Thus, the fine-grained entropy as defined in Eq. [50] always has a zero time derivative. 

The integration implied by dr in eqn (6,16) averages this force on the electron at r over the motions (i.e. positions) of all of the remaining particles in the system and the result is the force density F(r), the force exerted on the electron at r by the average distribution of the remaining particles in the total system. Integration of this force density over the basin of the atom 1 then yields the average electronic or Ehrenfest force exerted on the atom in the system. Even... 

The atomic statements of the Ehrenfest force law and of the virial theorem establish the mechanics of an atom in a molecule. As was stressed in the derivations of these statements, the mode of integration used to obtain an atomic average of an observable is determined by the definition of the subsystem energy functional i2]. It is important to demonstrate that the definition of this functional is not arbitrary, but is determined by the requirement that the definition of an open system, as obtained from the principle of stationary action, be stated in terms of a physical property of the total system. This requirement imposes a single-particle basis on the definition of an atom, as expressed in the boundary condition of zero flux in the gradient vector field of the charge density, and on the definition of its average properties. 

Averaging of this final operator expression for the virial of V in the manner indicated in eqn (6.69) for the potential energy density TFbWi which is the virial of the Ehrenfest force eqn (6.29), yields... 

Though it has been successfully applied, for example to describe energy transfer processes at metal surfaces, the Ehrenfest method fails when it becomes important to monitor different paths for different electronic states rather than a trajectory determined by an average over the different surfaces. This problem is particularly serious if one is interested in studying state specific nuclear pathways, such as those present in scattering events, or those determining low probability products in a chemical reaction. 

In mean-field or Ehrenfest methods, " the forces result from the contribution of two terms the first is related to the nonadiabatic coupling, the second is an average of the gradients of the potentials of the populated electronic states. Therefore, the forces acting on the nuclei depend directly on the population of the electronic states. The electronic problem and the nuclear dynamics have to be solved simultaneously. The time step must be sufficiently small to account for the time variation of the electronic wave functions. In this case, the solution of the TDSE can be propagated as ... 

If we consider a classical-mechanical particle, its wave function will be large only in a very small region corresponding to its position, and we may then drop the averages in (7.114) to obtain Newton s second law. Hius classical mechanics is a special case of quantum mechanics. Equation (7.114) is known as Ehrenfest s theorem, after the physicist who derived it in 1927. 

Turning back to practical application of the generalized Landau-Teller model, one can assert that with the help of the improved semiclassical approximation for the transition probability and the asymptotic method for calculation of the exchange interaction it is possible to get a reliable estimate of the adiabatic Ehrenfest exponent for a collinear atom-diatom collision. This implies of course, a non-empirical prediction, within the exponential accuracy, of the temperature dependence of the average transition probability. It remains to be seen how well these ideas can be extended for a three-dimensional collision. 

Another basic theory of nonadiabatic transitions is the semiclassical Ehrenfest theory (SET). Although it can cope with multidimensional nonadiabatic electronic-state mixing, it inevitably produces a nuclear path that runs on an averaged potential energy surface after having passed across the nonadiabatic region, which is totally unphysical. Unfortunately, since SET seems intuitively correct, a naive and conventional derivation of this theory obscures how this critical difficulty arises. 

We next turn to much simpler and easily calculable methods. The semiclassical Ehrenfest theory (SET) is based on rather an intuitive combination of the electronic dynamics on time-dependent nuclear configuration and the Newtonian dynamics of point-like nuclei using the wavepacket average of the Hellmann-Feynman force. 

Upon comparing this force with that in Eq. (4.43), gives rise to the Ehrenfest (mean-field) force. Therefore, the mean-field force does not arise from a physical origin but from our mathematical or artificial operation of averaging. However, this averaging is not necessarily bad in case where the nonadiabatic interaction is so strong that the electronic states may largely fluctuate from time to time. 

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