Classical mechanics’ fundamental view

Q.7.9 What two words succinctly describe classical mechanics fundamental view of state space ... 

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... 

Chaos provides an excellent illustration of this dichotomy of worldviews (A. Peres, 1993). Without question, chaos exists, can be experimentally probed, and is well-described by classical mechanics. But the classical picture does not simply translate to the quantum view attempts to find chaos in the Schrodinger equation for the wave function, or, more generally, the quantum Liouville equation for the density matrix, have all failed. This failure is due not only to the linearity of the equations, but also the Hilbert space structure of quantum mechanics which, via the uncertainty principle, forbids the formation of fine-scale structure in phase space, and thus precludes chaos in the sense of classical trajectories. Consequently, some people have even wondered if quantum mechanics fundamentally cannot describe the (macroscopic) real world. 

If quantum mechanics is really the fundamental theory of our world, then an effectively classical description of macroscopic systems must emerge from it - the so-called quantum-classical transition (QCT). It turns out that this issue is inextricably connected with the question of the physical meaning of dynamical nonlinearity discussed in the Introduction. The central thesis is that real experimental systems are by definition not isolated, hence the QCT must be viewed in the relevant physical context. 

The Hamilton-Jacobi form of the classical equations of motion has been shown to have provided the basis for the quantum-mechanical formulations according to Sommerfeld, Heisenberg, Schrodinger and Bohm. Each of these formulations inspired its own peculiar interpretation of quantum effects, despite their common basis. Each of the different points of view still has its adherents and the debates about their relative merits continue. Closer scrutiny shows that the Sommerfeld and Heisenberg systems assume quanta to be particles in the classical sense, although Heisenberg considered electronic positions to be fundamentally unobservable. 

The approach to be used here is, to be sure, well known in parts of theoretical physics, but is novel as far as chemistry is concerned. It is based on the view that macroscopic matter is to be described by a suitably generalized formulation of quantum mechanics, namely Quantum Field Theory the traditional postulate that matter is made up or composed of microscopic elementary constituents (in the classical building-block sense) is given up, and instead the fundamental postulate of the quantum theory of matter is, to paraphrase Gertrude Stein, Matter is Matter is Matter. Then if our interest is chemistry we have of course to confront the obvious question as to how we may construct the particles we call atoms and molecules i.e. we must establish how the notions of atom and molecule emerge from quantum theory construed in a general and modem way as the theory of matter. This is the subject matter of the next section of the review... 

The interpretation of QM formalism in terms of the complementarity principle leads to puzzling situations the particle-wave view seems to be fundamentally flawed to the extent classical concepts do not belong to an interpretive framework to quantum mechanics. 

Does the particle nature of light cause its wave aspects Or vice versa All these questions may only be asked from the point of view of classical physics, they only have meaning from the classical view. Once quantum mechanical physics enters the scene, no one even attempts to answer the questions on the classical level, if my guess that brain and mind are parallel aspects of a more fundamental reality is nebulous, perhaps it will take on some relevance when a "quantum mechanics of philosophy" will be available, whether a process of mind studying mind will accomplish such a feat is still an open question. 

Here we conclude our account of Bohr s theory. Although it has led to an enormous advance in our knowledge of the atom, and in particular of the laws of line spectra, it involves many difficulties of principle. At the very outset, the fundamental assumption of the validity of Bohr s frequency condition amounts to a. direct and unexplained contradiction of the laws of the classical theory. Again, the purely formal quantisation rule, which stands at the head of the theory, is a foreign element which in the first instance is absolutely unintelligible from the physical point of view. We shall see later how both of these difficulties are removed in a perfectly natural way in wave mechanics. 

Quantum mechanics employs several mathematical tools and physical concepts that may be unfamiliar. Some of these concepts deal with radiation rather than matter because the interaction between matter and radiation is critical to many applications of physical chemistry. To smooth the way a bit before we discuss quantum mechanics, let s first briefly summarize the fundamental properties of matter and radiation as viewed from a classical perspective. 

The tunnel correction is not now a fundamentally defined number rather it is defined by the equation Q = kobJk, where kobs is the observed rate constant for a chemical reaction and k is that calculated on the basis of some model which is as good as possible except that it does not allow tunnelling. In this chapter the definition used for k is that calculated by absolute reaction rate theory [3], i.e., k = KRT/Nh)K where X is the equilibrium constant for the formation of the transition state. The factor k, the transmission coefficient, is also a quantum correction on the barrier passage process, but it is in the other direction, that is k < 1. We shall here follow the customary view (though it is not solidly based) that k is temperature-independent and not markedly less than unity. The term k is used following Bell [1] the s stands for semi-classical, that is quantum mechanics is applied to vibrations and rotations, but translation along the reaction coordinate is treated classically. 

Initially, to explain these processes, some purely conventional, hypothetical and apparently unusual ideas were put forward, and in fact they were unusual from the point of view of the classical concepts of chemistry and physics. Nevertheless, as their mechanism was gradually revealed by theoretical and experimental analysis of these ideas, the real processes proved to be still more unusual, and their discovery marked a completely new era in biology, the discovery of the nature of fundamental biological processes. Three principal stages can be distinguished in this development. 

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