Time-dependent, quantum mechanical models and theories play an important role in studies of enzyme catalysis, interactions of enzymes with chemotherapeutic agents, electron and proton tunneling in condensed media and in liquids, electron and proton transfer in biomolecular systems, photosynthesis, photodissociation processes, unimolecular decay reactions, tautomerism of nucleic acid bases, phosphorylation processes in biomolecular systems, and other molecular and biomolecular processes. Analytical models of time—dependent processes in quantum systems are limited to relatively simple objects. Several examples are discussed in the next chapter. Systems with a slightly more complicated structure can be studied using purely quantum-mechanical simulations, i.e. by numerically solving the time—dependent Schroedinger equation. In the case of larger systems, like enzymes, a quantum-classical description of their structure and dynamics is a useful strategy. A large part of such systems can be described using classical theories, whereas a small portion, typically their active sites, should be described in a quantum—dynamical way.

Quantum-Classical Molecular Dynamics. Models and Applications

It is shown that Schroedinger's equation may be derived from three postulates. The first is a kind of statistical metamorphosis of classical mechanics, a set of two relations which are obtained from the canonical equations of particle mechanics by replacing all observables by statistical averages. The second is a local conservation law of probability with a probability current which takes the form of a gradient. The third is a principle of maximal disorder as realized by the requirement of minimal Fisher information. The rule for calculating expectation values is obtained from a fourth postulate, the requirement of energy conservation in the mean. The fact that all these basic relations of quantum theory may be derived from premises which are statistical in character is interpreted as a strong argument in favor of the statistical interpretation of quantum mechanics. The structures of quantum theory and classical statistical theories are compared and some fundamental differences are identified.

/pdf/the-statistical-origins-of-quantum-mechanics-1so1r5opur.pdf

The statistical origins of quantum mechanics

We analyze the Bianchi I cosmology in the presence of a massless scalar field and describe its dynamics via a semiclassical and quantum polymer approach. We study the morphology of the Big Bounce by adopting three different sets of configurational variables: the Ashtekar connections, a set of anisotropic volume-like coordinates and the Universe volume plus two anisotropy coordinates (the latter two sets of variables would coincide in the case of an isotropic Universe). In the semiclassical analysis we demonstrate that the value of the critical matter energy density depends on the Cauchy problem for the dynamics when adopting the Ashtekar connections or the anisotropic volume-like coordinates. On the contrary, when the Universe volume is considered as a configurational coordinate, we are able to derive a polymer-modified Friedmann equation for the Bianchi I model, from which the expression of the critical energy density can be derived. This analysis shows that the Big Bounce has universal features only when the Universe volume is defined on the polymer lattice. Then, a cosmological constant is included in the Ashtekar connections' formulation and some interesting results are mentioned making a comparison between the synchronous dynamics and that one when the scalar field is taken as a relational time. From a pure quantum point of view, we investigate the Bianchi I dynamics in terms of the Ashtekar connections. We apply the ADM reduction of the variational principle and then we quantize the system. We study the resulting Schr\"{o}dinger dynamics, stressing that the behavior of the wave packet peak over time singles out common features with the semiclassical trajectories, confirming the non-universal character of the emerging Big Bounce also on a quantum level. 

/pdf/semiclassical-and-quantum-features-of-the-bianchi-i-215f8tf1.pdf

Semiclassical and quantum features of the Bianchi I cosmology in the polymer representation

This article considers the scattering of particles off complex, central-field potentials. Starting from the Schr?dinger radial-equation, we find that the partial-wave -element can be decomposed as a sum of two contributions, , due to the real and imaginary parts of the potential. We obtain element as the asymptotical limit of a single, ordinary, first-order, nonlinear, differential equation. This equation is only associated to the imaginary part of the potential. We apply the procedure, combined with the semiclassical uniform WKB method, to the scattering of and systems.

Comparison method for scattering off complex central potentials

We analyze the Bianchi I cosmology in the presence of a massless scalar field and describe its dynamics via a semiclassical and quantum polymer approach. We study the morphology of the Big Bounce by adopting three different sets of configurational variables: the Ashtekar connections, a set of anisotropic volume-like coordinates and the Universe volume plus two anisotropy coordinates (the latter two sets of variables would coincide in the case of an isotropic Universe). In the semiclassical analysis we demonstrate that the value of the critical matter energy density depends on the Cauchy problem for the dynamics when adopting the Ashtekar connections or the anisotropic volume-like coordinates. On the contrary, when the Universe volume is considered as a configurational coordinate, we are able to derive a polymer-modified Friedmann equation for the Bianchi I model, from which the expression of the critical energy density can be derived. This analysis shows that the Big Bounce has universal features only when the Universe volume is defined on the polymer lattice. Then, a cosmological constant is included in the Ashtekar connections' formulation and some interesting results are mentioned making a comparison between the synchronous dynamics and that one when the scalar field is taken as a relational time. From a pure quantum point of view, we investigate the Bianchi I dynamics in terms of the Ashtekar connections. We apply the ADM reduction of the variational principle and then we quantize the system. We study the resulting Schrodinger dynamics, stressing that the behavior of the wave packet peak over time singles out common features with the semiclassical trajectories, confirming the non-universal character of the emerging Big Bounce also on a quantum level.

ψ-function, the Generalized Ehrenfest Theorem re- veals some interesting exact relationships between quantum and classical expectation values. These general results seem to imply a deep-rooted unity (in spite of apparent radical differences in conceptual structures) between classical and quantum mechan- ics. Some significant consequences and important physical insights which follow from the general formulation are discussed with examples. Most im- portant is that it offers, under reasonable approx i- mations, a pure quantum mechanical description of the Stern-Gerlach experiment with realistic inhomo- geneous magnetic field (∇ ∇.B = 0). EVEN almost seventy-five years after the first formul a- tion of quantum mechanics (QM), its relation with clas - sical mechanics (CM) remains obscure. The usual belief that QM has superseded CM is also not well founded. In all cases where a new theory supersedes an old one, there is a universal prescription of recovering the mathematical structure of the old theory in the doma in of its validity, i.e. where it is empirically established. This is true for both special and general theory of rel a- tivity 1 . But for QM, no such universal approximation procedure can be offered. About the success and failure of the two mechanics we can make two positive asser- tions: (i) There are experimental results in the micr o- scopic domain (such as hydr ogen spectrum) or in the macroscopic domain (such as specific heat of so lids at low temperature) which cannot be interpreted within the framework of CM. (ii) In all such cases, the mathemat i- cal formalism of QM provides the correct empirical results. Not a single failure has yet been reported. These empirical evidences show that CM cannot be the universal mechanics and that QM has such a pote n- tiality. However, since it is not yet possible to recover CM in all the domains of its validity, we cannot assert that QM has superseded CM. A second alternative that

Significance of Ehrenfest theorem in quantum-classical relationship

The classical and quantum physics seem to divide nature into two domains macroscopic and microscopic. It is also certain that they accurately predict experimental results in their respective regions. However, the reduction theory, namely, the general derivation of classical results from the quantum mechanics is still a far cry. The outcome of some recent investigations suggests that there possibly does not exist any universal method for obtaining classical results from quantum mechanics. In the present work we intend to investigate the problem phenomenonwise and address specifically the phenomenon of scattering. We suggest a general approach to obtain the classical limit formula from the phase shiftδl, in the limiting value of a suitable parameter on whichδl depends. The classical result has been derived for three different potential fields in which the phase shifts are exactly known. Unlike the current wisdom that the classical limit can be reached only in the high energy regime it is found that the classical limit parameter in addition to other factors depends on the details of the potential fields. In the last section we have discussed the implications of the results obtained.

Classical limit of scattering in quantum mechanics—A general approach

According to Bohr’s complementarity principle, two distinct types of complementarity exist-one of complementary variables and other in the so-called wave-particle complementarity experiments. Some authors have claimed that mutual exclusiveness (ME) in both the cases arise due to uncertainty principle and have analysed the second type in terms of Fourier space analysis and consequent putative “momentum kick” distribution. Some others, on the other hand, have identified the collapse hypothesis as the actual quantum mechanical principle responsible for ME in the interferometry experiments. In this paper the momentum space analysis is thoroughly examined vis-a-vis the general quantum mechanical description in terms of the changes in the wave function. It is argued that such alternative explanations are not in full conformity with the strict quantum mechanical description.

Origin of mutual exclusiveness in Bohr’s complementarity principle

Abstract It is argued that two distinct types of complementarity are implied in Bohr's complementarity principle. While in the case of complementary variables it is the quantum mechanical uncertainty relation which is at work, the collapse hypothesis ensures this exclusiveness in the so-called wave-particle complementarity experiments. In particular it is shown that the conventional analysis of the double slit experiment which invokes the uncertainty principle to explain the absence of the simultaneous knowledge of the which-slit information and the interference pattern is incorrect and implies consequences that are quantum mechanically inconsistent.

Is the Uncertainty Relation at the Root of all Mutual Exclusiveness

Wave-particle duality and bohr's complementarity principle in quantum mechanics

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