Quantum and Classical

So far aU we have written about are classical problems, but there are definite quantum aspects to the proton transfer reaction and, in addition, to the interaction of systems with light and dissipation mechanisms. Vibrations cannot always be treated classically either. At the outset, it should be stated that, unlike the classical description, which allows a relatively simple mechanism for dissipation by means of the Fokker-Planck equation, quantum mechanics does not allow such a possibihty. In addition, aU treatments based on a mixed description of a classical and a quantum system are fundamentally flawed [29]. In the remainder of this chapter, we nevertheless introduce a possible description that looks formaUy the same in classical and quantum mechanics and has some features that make it possible to introduce an elementary mechanism of decay to the equilibrium state. And, equaUy important, it gives a mechanism for averaging over the strongly coupled vibrational modes, as weU as a unified description of the interaction with light. 

This is aU accomplished by introducing a quantity caUed the density matrix or density operator. There is a rather trivial way to introduce it by means of elementary quantum mechanics. Suppose we have a system with Hamiltonian (the caret is used to distinguish the quantum mechanical Hamiltonian operator from the 

The general formalism of quantum mechanics teUs us that, in order to get an expectation value - the average of a series of measurements on a system prepared in the state V of a Hermitian (measurable) operator A - we need to calculate (A) = (V A V . Using the expansion (9.35), this can also be written as 

Starting from the time-dependent Schrodinger equation, it is straightforward to derive the equation of motion for the density operator. This is easiest to see if we write the density operator as p = I A)( A1 2nd apply the Schrodinger equation to both sides to get 

There is a formal similarity between this equation and Eq. (9.32) the Poisson bracket in the latter is replaced by the a commutator in the former. Poisson brackets and commutators have a number of properties in common. They are Lie brackets, which means they are linear, are antisymmetric, and satisfy the Jacobi identity. Both 7f, p) and —j[H, p] can be viewed as an operations on the probability density. But here the similarity ends. The probability densities themselves hve in completely different spaces. The first is a function in classical phase space, and the second is an operator on a Hilbert state space, ft has up to now not been possible to unify those two spaces, so that, for instance, a correct classical limit can be taken. 

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In this chapter, the foundations of equilibrium statistical mechanics are introduced and applied to ideal and weakly interacting systems. The coimection between statistical mechanics and thennodynamics is made by introducing ensemble methods. The role of mechanics, both quantum and classical, is described. In particular, the concept and use of the density of states is utilized. Applications are made to ideal quantum and classical gases, ideal gas of diatomic molecules, photons and the black body radiation, phonons in a hannonic solid, conduction electrons in metals and the Bose—Einstein condensation. Introductory aspects of the density... 

Solving the Eqs. (C.6-C.8,C.12,C.13) comprise what is known as the Ehrenfest dynamics method. This method has appealed under a number of names and derivations in the literatnre such as the classical path method, eilconal approximation, and hemiquantal dynamics. It has also been put to a number of different applications, often using an analytic PES for the electronic degrees of freedom, but splitting the nuclear degrees of freedom into quantum and classical parts. 

Other considerations (such as mixed quantum and classical forces) arise when considering the complete merger of two or more methodologies in the same molecular system, but since they are not yet available in this release of HyperChem they need not be considered here. 

Depending on the desired level of accuracy, the equation of motion to be numerically solved may be the classical equation of motion (Newton s), a stochastic equation of motion (Langevin s), a Brownian equation of motion, or even a combination of quantum and classical mechanics (QM/MM, see Chapter 11). 

Figure 1 Schematic diagram depicting the partitioning of an enzymatic system into quantum and classical regions. The side chains of a tyrosine and valine are treated quantum mechanically, whereas the remainder of the enzyme and added solvent are treated with a classical force field.

For QM-MM methods it is assumed that the effective Hamiltonian can be partitioned into quantum and classical components by writing [9]... 

This term is essential to obtain the correct geometry, because there is no Pauli repulsion between quantum and classical atoms. The molecular mechanics energy tenn, E , is calculated with the standard potential energy term from CHARMM [48], AMBER [49], or GROMOS [50], for example. 

Figure 2 A glutamate side chain partitioned into quantum and classical regions. The terminal CH2C02 group IS treated quantum mechanically, and the backbone atoms are treated with the molecular mechanics force field.

BW Beck, IB Koerner, RB Yelle, T Ichiye. Unusual hydrogen bonding ability of sulfurs m Fe-S redox sites Ah initio quantum and classical mechanical studies. I Phys Chem B, submitted. 

The first term on the right hand side of Eq. (1-3) stands for the van der Waals interaction between quantum and classical atoms... 

In centroid path integral, the canonical QM partition function of a hybrid quantum and classical system, consisting of one quantized atom for convenience, can be written as follows ... 

We can calculate the thermal rate constants at low temperatures with the cross-sections for the HD and OH rotationally excited states, using Eqs. (34) and (35), and with the assumption that simultaneous OH and HD rotational excitation does not have a strong correlated effect on the dynamics as found in the previous quantum and classical trajectory calculations for the OH + H2 reaction on the WDSE PES.69,78 In Fig. 13, we compare the theoretical thermal rate coefficient with the experimental values from 248 to 418 K of Ravishankara et al.7A On average, the theoretical result... 

Grossman, J. C. Schwegler, E. Galli, G., Quantum and classical molecular dynamics simulations of hydrophobic hydration structure around small solutes, J. Phys. Chem. B 2004,108, 15865-15872... 

The observation of the SIM behaviour of TbPc2 and of TbPc 2 derivatives obtained by substituting the periphery of the phthalocyanine macrocycles combined with a considerable blocking temperature attracted much attention from both chemists and physicists. It was soon realized that these systems proved to be ideal testing grounds for theories of the coexistence of quantum and classical... 

These interference patterns are wonderful manifestations of wave function behavior, and are not found in classical electronics or electrodynamics. Since the correspondence principle tells us that quantum and classical systems should behave similarly in the limit of Planck s constant vanishing, we suspect that adequate decoherence effects will change the quantum equation into classical kinetics equations, and so issues of crosstalk and interference would vanish. This has been... 

An electron is a particle of such small dimensions that its mechanical behaviour can be correctly described only by medium of quantum mechanics. The aim of this discussion is therefore to introduce quantum mechanics and to investigate the mechanical behaviour of electrons in situations with implications for chemistry. Since quantum and classical mechanics yield the same results in the so-called classical limit, the more complicated quantum system can be developed by analogy with the simpler classical system. 

The next step forward has yet to be taken The clash between relativity and quantum mechanics - the choice between causality and unitarity - awaits resolution. However, on a less grand scale, the tension between fundamentally different points of view is already apparent in the discord between quantum and classical mechanics. Unlike special relativity, where v/c —> 0 smoothly transitions between Einstein and... 

Newton, the limit h —> 0 is singular. The symmetries underlying quantum and classical dynamics - unitarity and symplecticity, respectively - are fundamentally incompatible with the opposing theory s notion of a physical state quantum-mechanically, a positive semi-definite density matrix classically, a positive phase-space distribution function. 

Keeping the lesson of the above example in mind, we will explore three different dynamical possibilities below isolated evolution, where the system evolves without any coupling to the external world, unconditioned open ev olution, where the system evolves coupled to an external environment but where no information regarding the system is extracted from the environment, and conditioned open evolution where such information is extracted. In the third case, the evolution of the physical state is driven by the system evolution, the coupling to the external world, and by the fact that observational information regarding the state has been obtained. This last aspect - system evolution conditioned on the measurement results via Bayesian inference - leads to an intrinsically nonlinear evolution for the system state. The conditioned evolution provides, in principle, the most realistic possible description of an experiment. To the extent that quantum and classical mechanics are eventually just methodological tools to explain and predict the results of experiments, this is the proper context in which to compare them. 

As mentioned already, quantum and classical mechanics are fundamentally incompatible in many ways, yet the macroscopic world is well-described by classical dynamics. Physicists have struggled with this quandary ever since the laying of the foundations of quantum theory. It is fair to say that, even today, not everyone is satisfied with the state of affairs - including many seasoned practitioners of quantum mechanics. 

To illustrate an application of nonlinear quantum dynamics, we now consider real-time control of quantum dynamical systems. Feedback control is essential for the operation of complex engineered systems, such as aircraft and industrial plants. As active manipulation and engineering of quantum systems becomes routine, quantum feedback control is expected to play a key role in applications such as precision measurement and quantum information processing. The primary difference between the quantum and classical situations, aside from dynamical differences, is the active nature of quantum measurements. As an example, in classical theory the more information one extracts from a system, the better one is potentially able to control it, but, due to backaction, this no longer holds true quantum mechanically. 

Tapia, O. Solvent effect theories quantum and classical formalisms and their applications in chemistry and biochemistry, J.Math. Chem., 10 (1992), 139-181... 

The similar appearance of the quantum and classical Liouville equations has motivated several workers to construct a mixed quantum-classical Liouville (QCL) description [27 4]. Hereby a partial classical limit is performed for the heavy-particle dynamics, while a quantum-mechanical formulation is retained for the light particles. The quantities p(f) and H in the mixed QC formulation are then operators with respect to the electronic degrees of freedom, described by some basis states 4> ), and classical functions with respect to the nuclear degrees of freedom with coordinates x = x, and momenta p = pj — for example. 

Quantum and Classical Dynamics in Condensed Phase Simulations, B. J. Berne, G. Ciccotti, and D. F. Coker, eds.. World Scientific, Singapore, 1998. 

QUANTUM AND CLASSICAL DYNAMICS OF NONINTEGRABLE SYSTEMS 141 and the relations for... 

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