Quantum classical limit

To take the quantum-classical limit of this general expression for the transport coefficient we partition the system into a subsystem and bath and use the notation 1Z = (r, R), V = (p, P) and X = (r, R,p, P) where the lower case symbols refer to the subsystem and the upper case symbols refer to the bath. To make connection with surface-hopping representations of the quantum-classical Liouville equation [4], we first observe that AW(X 1) can be written as... 

The quantum-classical limit of the transport coefficient is obtained by evaluating the evolution equation for the matrix elements of W in the quantum-classical limit. This limit was taken in Ref. [68] and the result is... 

The property of the new (Lie) brackets (44) of being correct in the known full quantum and full classical limits may reasonably convince ourselves that the intermediate situation, in which hi —> h and h2 —> 0, generates quantum-classical dynamics. If the assumedly quantum-classical limit is performed on Ph1,h2(9i, 92), we obtain... 

We are now ready to deduce the consequences of the quantum-classical limit derived in section 5. 

The quantum-classical limit h —> h, I12 — 0 can be replaced by the equivalent limit heg — 0 in which the new effective Planck constant appears. The representation of the group becomes... 

In the theory based on the group B", as presented in section 5, the third point is missing. In the present section, we removed the inconsistency of the whole construction and showed the classical nature of the presumed quantum-classical limit, which was hidden when the inconsistent, although formally equivalent, representation of the group D" where adopted. 

The most common traditional definition of the quantum/classical limit is the point at which Planck s constant h - 0. However, this is an unreasonable stipulation [33] because h is not dimensionless and its value can therefore not be varied. A possible operational condition could be formulated in terms of a dimensionless parameter of the form h/S 1, where S is the action quantity in a given situation. It could be argued that for S sufficiently large compared to h, measurement at the macroscopic level cannot detect quantum effects because of limited instrument resolution. This argument implies that the coarse-grained appearance of a classical world is simply a question of experimental accuracy and that every physical system ultimately displays quantum features and that there is no classical limit. 

The Wigner form of the quantum evolution operator iLw X ) in (62) for the equation of motion for W X, X2,t) can be rewritten in a form that is convenient for the passage to the quantum-classical limit. Recalling that the system may be partitioned into S and S subspaces, the Poisson bracket operator A can be written as the sum of Poisson bracket operators acting in each of these subspaces as A Xi) = A xi) + A Xi). Thus, we may write... 

A. Sergi and R. Kapral (2004) Quantum-Classical Limit of Quantum Correlation Functions. J. Chem. Phys. 121, pp. 7565-7576... 

The present author is no exception, having spent many hours in fruitless pursuit of the elusive quantum/classical limit, commonly defined by the unphysical condition h Stepping back from the traditional perspective, in this case, suggests some unexpected aspects of vacuum structure. 

Concepts in theoretical science, which may appear difficult to comprehend on first encounter, often become accepted through familiarity, rather than insight. Such concepts end up as dogmatic belief, which casts a shadow on the understanding of related theories. The notion of a quantum/classical limit is discussed as an example. Analysed as a measure of quantum potential it is shown to clear up the related concepts of Compton wavelength, fine-structure constant, wave structures and the nature of the vacuum. In a Riemannian... 

In this quantum-classical formulation, the rate constant can be determined by taking the quantum-classical limit of the dynamics while retaining the full quantum equilibrium structure in the correlation function expression in eqn (10.6). In this limit the rate coefficient takes the form ... 

In the classical limit, the triplet of quantum numbers can be replaced by a continuous variable tiirough the transformation... 

If z = exp(pp) l, one can also consider the leading order quantum correction to the classical limit. For this consider tlie thennodynamic potential cOq given in equation (A2.2.144). Using equation (A2.2.149). one can convert the sum to an integral, integrate by parts the resulting integral and obtain the result ... 

Figure A3.8.3 Quantum activation free energy curves calculated for the model A-H-A proton transfer reaction described 45. The frill line is for the classical limit of the proton transfer solute in isolation, while the other curves are for different fully quantized cases. The rigid curves were calculated by keeping the A-A distance fixed. An important feature here is the direct effect of the solvent activation process on both the solvated rigid and flexible solute curves. Another feature is the effect of a fluctuating A-A distance which both lowers the activation free energy and reduces the influence of the solvent. The latter feature enliances the rate by a factor of 20 over the rigid case.

Miller W H 1971 Semiclassical nature of atomic and molecular collisions Accounts Chem. Res. 4 161-7 Miller W H 1974 Classical-limit quantum mechanics and the theory of molecular collisions Adv. Chem. Phys. 25 69-177... 

Miller W H 1974 Classical-limit quantum mechanics and the theory of molecular collisions Adv. Chem. 

As mentioned above, the correct description of the nuclei in a molecular system is a delocalized quantum wavepacket that evolves according to the Schrbdinger equation. In the classical limit of the single surface (adiabatic) case, when effectively 0, the evolution of the wavepacket density... 

This proves that the pseudoparticles in the quantum fluid obey classical mechanics in the classical limit. 

For the case of intramolecular energy transfer from excited vibrational states, a mixed quantum-classical treatment was given by Gerber et al. already in 1982 [101]. These authors used a time-dependent self-consistent field (TDSCF) approximation. In the classical limit of TDSCF averages over wave functions are replaced by averages over bundles of trajectories, each obtained by SCF methods. 

Approximation Properties and Limits of the Quantum-Classical Molecular Dynamics Model... 

Bornemann, F. A., Schiitte, Ch. On the Singular Limit of the Quantum-Classical Molecular Dynamics Model. Preprint SC 97-07 (1997) Konrad-Zuse-Zentrum Berlin. SIAM J. Appl. Math, (submitted)... 

Abstract. The overall Hamiltonian structure of the Quantum-Classical Molecular Dynamics model makes - analogously to classical molecular dynamics - symplectic integration schemes the methods of choice for long-term simulations. This has already been demonstrated by the symplectic PICKABACK method [19]. However, this method requires a relatively small step-size due to the high-frequency quantum modes. Therefore, following related ideas from classical molecular dynamics, we investigate symplectic multiple-time-stepping methods and indicate various possibilities to overcome the step-size limitation of PICKABACK. 

F.A. Bornemann and Ch. Schiitte. On the singular limit of the quantum-classical molecular dynamics model. Preprint SC 97-07, ZIB Berlin, 1997. Submitted to SIAM J. Appl. Math. 

The classical bath sees the quantum particle potential as averaged over the characteristic time, which - if we recall that in conventional units it equals hjk T- vanishes in the classical limit h- Q. The quasienergy partition function for the classical bath now simply turns into an ordinary integral in configuration space. 

As emphasized by Sadovskii and Zhilinskii [2], this latter point is important for quantum systems for which the lattice is too small to allow the constmction in Fig. 16a, because there is still a systematic reorganization of the spectra, involving transfer of individual levels or groups of levels from lower to upper bands, as y increases from 0 to 1. Figure 17 shows examples for n = A and i = j, 1, and, which illustrate the influence of quantum monodromy far from the classical limit. 

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