Quantum versus classical mechanics

Since the birth of quantum theory, there has been considerable interest in the transition from quantum to classical mechanics. Because the two formulations are given in a different theoretical framework (nonlinear classical trajectories versus expectation values of linear operators), this transition is far more involved than the naive limit —> 0 suggests. By exploring the classical limit of quantum mechanics, new theoretical concepts have been developed, including path integrals [1], various phase-space representations of quantum mechanics [2], the semiclassical propagator and the trace formula [3], and the notion of quantum... 

It should also be frankly acknowledged here that there are a variety of theoretical challenges associated with these problems that are not highlighted at all in this chapter. These range from formulation questions involving quantum versus classical issues in calculating rates (see, for example, Chapter 16) to the quantum chemical electronic structure issues of solute intramolecular force fields. These and other difficulties certainly impede the theoretical ability to confidently predict VET rates and mechanisms, but not the desire to try. 

As for the quantum versus classical electrodynamics, QED description is of course the correct and complete theory to describe all the molecular plasmonics phenomena. Nevertheless, it requires the definition and the manipulation of quantities that are often not as intuitive as their classical counterparts. Moreover, classical electrodynamics is able to explain most of the molecular plasmonics phenomena, and, with a few expedients, even intrinsically quantum-mechanical phenomena such as spontaneous emission. Therefore, in the following we shall stick to a classical electrodynamics description of the system, and we refer the reader to other works for a QED treatment [60]. 

Einstein became a member of the Scientific Committee and took an active part in the discussions on quantum mechanics, which was the theme of the 1927 Conference. Indeed, the famous Einstein-Bohr discussions on classical determinism versus the quantum statistical causality took place in Brussels at the Solvay Conferences of 1927 and 1930, and continued thereafter. 

Fig. 6.9. Left-hand side Vibrational excitation function N(ro) and weighting function W(ro) versus the initial oscillator coordinate ro for three values of the coupling parameter e. The equilibrium separation of the free BC molecule is f = 0.403 A and the equilibrium value within the parent molecule is re = 0.481 A. Right-hand side Final vibrational state distributions P(n) for fixed energy E the quantum mechanical and the classical distributions are normalized to the same height at the maxima. The classical distributions are obtained with the help of (6.32). The lowest part of the figure contains also the pure Franck-Condon (FC) distribution (

Provided k can be estimated, a graph of the LHS of the equation versus [M] yields a value for fcj i which may be compared with the rate coeflBcient for spontaneous decomposition or isomerisation derived from the classical HRRKM theory or the quantum mechanical theory. ... 

The quest for absolute zero under classical versus quantum mechanics... 

Classical versus Quantum Mechanics Time-Independent Structure... 

Exponents derived from the analytic theories are frequently called classical as distinct from modem or nonclassical although this has nothing to do with classical versus quantum mechanics or classical versus statistical thermodynamics. The important thermodynamic exponents are defined here, and their classical values noted the values of the more general nonclassical exponents, determined from experiment and theory, will appear in later sections. The equations are expressed in reduced units in order to compare the amplitude coefficients in subsequent sections. 

It is also interesting to consider the classical/quantal correspondence in the number of energized molecules versus time N(/, E), Eq. (8.22), for a microcanonical ensemble of chaotic trajectories. Because of the above zero-point energy effect and the improper treatment of resonances by chaotic classical trajectories, the classical and quantal I l( , t) are not expected to agree. For example, if the classical motion is sufficiently chaotic so that a microcanonical ensemble is maintained during the decomposition process, the classical N(/, E) will be exponential with a rate constant equal to the classical (not quantal) RRKM value. However, the quantal decay is expected to be statistical state specific, where the random 4i s give rise to statistical fluctuations in the k and a nonexponential N(r, E). This distinction between classical and quantum mechanics for Hamiltonians, with classical f (/, E) which agree with classical RRKM theory, is expected to be evident for numerous systems. 

In order to place our theoretical framework in the most general position with respect to our contradiction, i.e. positing irreversible thermodynamics versus reversible dynamics, we need to direct our focus towards a relativistically invariant (or covariant in the general case) theory. It would perhaps be more pertinent to use the word consistent, since our more general complex symmetric framework, allows broken symmetry solutions, while relevant symmetries are properly embedded if necessary. In a previous communication [12], see also [25], we have derived a Global Superposition Principle that applies to both classical and quantum mechanical interpretations. 

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