Quantum mechanics semiclassical approximation

The mapping approach outlined above has been designed to furnish a well-defined classical limit of nonadiabatic quantum dynamics. The formalism applies in the same way at the quantum-mechanical, semiclassical (see Section VIII), and quasiclassical level, respectively. Most important, no additional assumptions but the standard semiclassical and quasi-classical approximations are needed to get from one level to another. Most of the established mixed quantum-classical methods such as the mean-field-trajectory method or the surface-hopping approach do invoke additional assumptions. The comparison of the mapping approach to these formulations may therefore (i) provide insight into the nature of these additional approximation and (ii) indicate whether the conceptual virtues of the mapping approach may be expected to result in practical advantages. 

The preferable theoretical tools for the description of dynamical processes in systems of a few atoms are certainly quantum mechanical calculations. There is a large arsenal of powerful, well established methods for quantum mechanical computations of processes such as photoexcitation, photodissociation, inelastic scattering and reactive collisions for systems having, in the present state-of-the-art, up to three or four atoms, typically. " Both time-dependent and time-independent numerically exact algorithms are available for many of the processes, so in cases where potential surfaces of good accuracy are available, excellent quantitative agreement with experiment is generally obtained. In addition to the full quantum-mechanical methods, sophisticated semiclassical approximations have been developed that for many cases are essentially of near-quantitative accuracy and certainly at a level sufficient for the interpretation of most experiments.These methods also are com-... 

Due to the complexity of a full quantum mechanical treatment of electron impact ionization, or even a partial wave approximation, for all but relatively simple systems, a large number of semiempirical and semiclassical formulae have been developed. These often make basic assumptions which can limit their range of validity to fairly small classes of atomic or molecular systems. The more successful approaches apply to broad classes of systems and can be very useful for generating cross sections in the absence of good experimental results. The success of such calculations to reproduce experimentally determined cross sections can also give insight into the validity of the approximations and assumptions on which the methods are based. 

For larger systems, various approximate schemes have been developed, called mixed methods as they treat parts of the system using different levels of theory. Of interest to us here are quantum-semiclassical methods, which use full quantum mechanics to treat the electrons, but use approximations based on trajectories in a classical phase space to describe the nuclear motion. The prefix quantum may be dropped, and we will talk of semiclassical methods. There are a number of different approaches, but here we shall concentrate on the few that are suitable for direct dynamics molecular simulations. An overview of other methods is given in the introduction of [21]. 

How well do these quantum-semiclassical methods work in describing the dynamics of non-adiabatic systems There are two sources of errors, one due to the approximations in the methods themselves, and the other due to errors in their application, for example, lack of convergence. For example, an obvious source of error in surface hopping and Ehrenfest dynamics is that coherence effects due to the phases of the nuclear wavepackets on the different surfaces are not included. This information is important for the description of short-time (few femtoseconds) quantum mechanical effects. For longer timescales, however, this loss of information should be less of a problem as dephasing washes out this information. Note that surface hopping should be run in an adiabatic representation, whereas the other methods show no preference for diabatic or adiabatic. 

As shown in Ref. (Bulgac and Wirzba., 2001) the semiclassical result is a very good approximation of the full quantum mechanical result calculated from the exact expression (18) of the two-sphere scattering matrix when plugged into the modified Krein formula (15). 

Since and M" are equivalent in the physical subspace, both Hamiltonians generate the same quantum dynamics in this subspace. However, this is not necessarily true if approximations are employed in the evaluation of the dynamics. For example, adopting a semiclassical approximation, the quantum-mechanically equivalent Hamiltonians H and El may yield different results. Experience shows that it is useful first to transform the Hamiltonian on the... 

Field emission is a tunneling phenomenon in solids and is quantitatively explained by quantum mechanics. Also, field emission is often used as an auxiliary technique in STM experiments (see Part II). Furthermore, field-emission spectroscopy, as a vacuum-tunneling spectroscopy method (Plummer et al., 1975a), provides information about the electronic states of the tunneling tip. Details will be discussed in Chapter 4. For an understanding of the field-emission phenomenon, the article of Good and Muller (1956) in Handhuch der Physik is still useful. The following is a simplified analysis of the field-emission phenomenon based on a semiclassical method, or the Wentzel-Kramers-Brillouin (WKB) approximation (see Landau and Lifshitz, 1977). 

The potential curve for the electrons near the tip surface is shown in Fig. 1.38. The relevant dimensions are much smaller than the radius of the tip end. Therefore, a one-dimensional model is adequate. In the metal, the energy level of the electrons is lower than the vacuum level by the value of the work function c ). From the point of view of classical mechanics, the electrons cannot escape from the metal even with a very high external field, that is, the potential barrier is impenetrable. From the point of view of quantum mechanics, there is always a finite probability that the electrons can penetrate the potential barrier. In the semiclassical (WKB) approximation, the transmission coefficient for a general potential barrier is (Landau and Lifshitz, 1977) ... 

We now consider the effect of exposing a system to electromagnetic radiation. Our treatment will involve approximations beyond that of replacing (3.13) with (3.16). A proper treatment of the interaction of radiation with matter must treat both the atom and the radiation field quantum-mechanically this gives what is called quantum field theory (or quantum electrodynamics). However, the quantum theory of radiation is beyond the scope of this book. We will treat the atom quantum-mechanically, but will treat the radiation field as a classical wave, ignoring its photon aspect. Thus our treatment is semiclassical. 

The absorbing potential factor is the only nonstandard feature in Eq. (3.1). Fortunately, it is much simpler to deal with the absorbing potential semiclassically than it is quantum mechanically it cannot cause any unwanted reflections in the above semiclassical expression because we have implicitly made an infinitesimal approximation for it. Thus, it does not affect the dynamics and only causes absorption see also the disscussion by Seide-man et al. [3b]. 

W. H. Miller Tunneling is surely involved in the present case because the isomerization involves a large amount of hydrogen atom motion. Also, with no tunneling the resonances would be infinitely narrow (and thus unobservable). Our calculations, though, are a fully quantum mechanical treatment and thus do not explicitly identify what is tunneling and what is not. This would only be meaningful in an approximate (e.g., semiclassical) treatment. 

In Section 4.1 we will use the time-independent continuum basis 4//(Q E,0), defined in Section 2.5, to construct the wavepacket in the excited state and to derive (4.2). Numerical methods are discussed in Section 4.2 and quantum mechanical and semiclassical approximations based on the time-dependent theory are the topic of Section 4.3. Finally, a critical comparison of the time-dependent and the time-independent approaches concludes this chapter. 

In the first part of this introductory section, we summarize the main collective phenomena acquired by the dipolar exciton from the lattice-symmetry collectivization of molecular properties. The crystal is considered as an assembly of electrically neutral systems, the molecules, physically separated from each other and in electromagnetic interaction. This /V-body problem will be treated quantum-mechanically in the limit of low exciton densities. We redemonstrate the complete equivalence of this treatment with the theories of Lorentz and Ewald, as well as with the semiclassical approximation. In Section I.A, in a more compact but still gradual way, we establish the model of the rigid lattice of dipoles and the general theory of low-exciton-density systems in interaction with the radiation field. Coulombic excitons, photons,... 

---