Interference effects quantum theory

The obvious defect of classical trajectories is that they do not describe quantum effects. The best known of these effects is tunnelling tln-ough barriers, but there are others, such as effects due to quantization of the reagents and products and there are a variety of interference effects as well. To circumvent this deficiency, one can sometimes use semiclassical approximations such as WKB theory. WKB theory is specifically for motion of a particle in one dimension, but the generalizations of this theory to motion in tliree dimensions are known and will be mentioned at the end of this section. More complete descriptions of WKB theory can be found in many standard texts [1, 2, 3, 4 and 5, 18]. 

The initial state of the excited system has been represented as a superposition of the (time-independent) molecular eigenstates, each of which is a superposition of BO basis functions. The decay process is then described in terms of the time evolution of the amplitudes of the molecular eigenstates. The general theory of quantum mechanics implies that the decay of the state (10-4) will exhibit interference effects. 

Stark R.W. and Reifenberg R. (1977) Quantitative Theory of Quantum Interference Effect./. Low Temp. Phys. 26,763. 

Recent years have also witnessed exciting developments in the active control of unimolecular reactions [30,31]. Reactants can be prepared and their evolution interfered with on very short time scales, and coherent hght sources can be used to imprint information on molecular systems so as to produce more or less of specified products. Because a well-controlled unimolecular reaction is highly nonstatistical and presents an excellent example in which any statistical theory of the reaction dynamics would terribly fail, it is instmctive to comment on how to view the vast control possibihties, on the one hand, and various statistical theories of reaction rate, on the other hand. Note first that a controlled unimolecular reaction, most often subject to one or more external fields and manipulated within a very short time scale, undergoes nonequilibrium processes and is therefore not expected to be describable by any unimolecular reaction rate theory that assumes the existence of an equilibrium distribution of the internal energy of the molecule. Second, strong deviations Ifom statistical behavior in an uncontrolled unimolecular reaction can imply the existence of order in chaos and thus more possibilities for inexpensive active control of product formation. Third, most control scenarios rely on quantum interference effects that are neglected in classical reaction rate theory. Clearly, then, studies of controlled reaction dynamics and studies of statistical reaction rate theory complement each other. 

There are two corrections to equation (12) that one might want to make. The first has to do with dynamical factors [19,20] i.e., trajectories leave Ra, crossing the surface 5/3, but then immediately return to Ra. Such a trajectory contributes to the transition probability Wfia, but is not really a reaction. We can correct for this as in variational transition-state theory (VTST) by shifting Sajj along the surface normals. [8,9] The second correction is for some quantum effects. Equation (14) indicates one way to include them. We can simply replace the classical partition functions by their quantum mechanical counterparts. This does not correct for tunneling and interference effects, however. 

The phenomenon of optical interference is commonly describable in completely classical terms, in which optical fields are represented by classical waves. Classical and quantum theories of optical interference readily explain the presence of an interference pattern, but there are interference effects that distinguish the quantum (photon) nature of light from the wave nature. In this section, we present elementary concepts and definitions of both the classical and quantum theories of optical interference and illustrate the role of optical coherence. 

The correlation functions (28) described by the field operators are similar to the correlation functions (6) and (20) of the classical field. A closer look into Eqs. (6), (20), and (28) could suggest that the only difference between the classical and quantum correlation functions is that the classical amplitudes E (R, f) and E(R, f) are replaced by the field operators E (R, t) and eW(R,(). This is true as long as the first-order correlation functions are considered, where the interference effects do not distinguish between the quantum and classical theories of the electromagnetic field. However, there are significant differences between the classical and quantum descriptions of the field in the properties of the second-order correlation function [16]. 

In many cases, too, the semiclassical model provides a quantitative description of the quantum effects in molecular systems, although there will surely be situations for which it fails quantitatively or is at best awkward to apply. From the numerical examples which have been carried out thus far— and more are needed before a definitive conclusion can be reached—it appears that the most practically useful contribution of classical S-matrix theory is the ability to describe classically forbidden processes i.e. although completely classical (e.g. Monte Carlo) methods seem to be adequate for treating classically allowed processes, they are not meaningful for classically forbidden ones. (Purely classical treatments will not of course describe quantum interference effects which are present in classically allowed processes, but under most practical conditions these are quenched.) The semiclassical approach thus widens the class of phenomena to which classical trajectory methods can be applied. 

It is interesting to compare the Fano theory of autoionisation, which is an interference effect, with another kind of interference effect which is responsible for pronounced variations of intensity in certain molecular bands. Homogeneous perturbations arise when two excited electronic-vibrational states of a molecule are coupled by a perturbation in a manner which does not depend on the rotational quantum numbers of a molecule, but only on the rotational term and vibrational state, so that complete rotational bands can be enhanced or depressed in intensity. 

Interest in the polarization correlation of photons goes back to the early measurements of the linear polarization correlation of the two photons produced in the annihilation of para-positronium which were carried out as a result of a suggestion by Wheeler that these photons, when detected, have orthogonal polarizations. Yang subsequently pointed out that such measurements are capable of giving information on the parity state of nuclear particles that decay into two photons. In addition, the polarization correlation observed in the two-photon decay of atoms is considered to be one of the few phenomena where semiclassical theories of radiation are inadequate and it is necessary to invoke a full quantum theory of radiation. The effect has also been used to demonstrate the phenomenon of quantum interference. ... 

It is important to appreciate clearly the distinction between SC theory and the older or classical valence bond theory. In classical VB theory, the orbitals are taken to be predetermined, either as simple atomic orbitals or hybrids of atomic orbitals. These hybrids, moreover, are fixed, for example, either as sp, sp, or sp, etc., -type orbitals. In SC theory, in contrast, no such preconceptions are imposed. The orbitals are optimized as linear combinations of basis functions (usually approximate AOs) much as in MO-based approaches. However, in common with classical VB theory, the SC orbitals in general overlap with one another (except, of course, in the case of orbitals of different symmetry), or, since the SC orbitals are often localized, by virtue of the physical separation between them. Generally speaking no constraints, apart from normalization, are applied to the SC orbitals and as a result they may be as localized or as delocalized as the situation demands. Bearing in mind that the SC orbitals are always singly occupied, this last means that their shapes are determined by whatever produces the optimum balance between the greatest extent of avoidance of the electrons in different orbitals and quantum interference effects, which arise from the overlap between orbitals. In practice, we have found that this invariably means that the SC orbitals turn out to be localized and indeed often resemble atomic or hybrid atomic orbitals, or semi-localized, meaning that the SC orbitals spread over two or, at most, three centres. 

Quantum mechanical effects—tunneling and interference, resonances, and electronic nonadiabaticity— play important roles in many chemical reactions. Rigorous quantum dynamics studies, that is, numerically accurate solutions of either the time-independent or time-dependent Schrodinger equations, provide the most correct and detailed description of a chemical reaction. While hmited to relatively small numbers of atoms by the standards of ordinary chemistry, numerically accurate quantum dynamics provides not only detailed insight into the nature of specific reactions, but benchmark results on which to base more approximate approaches, such as transition state theory and quasiclassical trajectories, which can be applied to larger systems. 

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