On the connection between quantum and classical descriptions

The review paper presents generalization of d??Alembert??s variational principle: the dynamics of a quantum system for an external observer is defined by the exact equilibrium of all acting in the system forces, including the random quantum force ?j, ??. Spatial attention is dedicated to the systems with (hidden) symmetries. It is shown how the symmetry reduces the number of quantum degrees of freedom down to the independent ones. The sin-Gordon model is considered as an example of such field theory with symmetry. It is shown why the particles S-matrix is trivial in that model.     

An interpretation within philosophy of the relationship between classical mechanics and quantum mechanics

A mapping of a finite directed graph onto a curve in space-time is considered. The mapping induces the dynamics of a free particle moving along the curve. The distinction between the Lagrangian and the Hamiltonian formulation of particle mechanics is expressed in terms of the distinction between referring to a particle in space and time and referring to the points in space which the particle occupies, respectively. These elements are combined to yield an interpretation of Feynman's path integral formulation of quantum mechanics. Describing a bound state of a system as a particle is discussed.     

Potentials of classical force fields for interactions between Na~+ and carbon nanotubes

Carbon nanotubes(CNTs) have long been expected to be excellent nanochannels for use in desalination membranes and other bio-inspired human-made channels owing to their experimentally confirmed ultrafast water flow and theoretically predicted ion rejection. The correct classical force field potential for the interactions between cations and CNTs plays a crucial role in understanding the transport behaviors of ions near and inside the CNT, which is key to these expectations. Here,using density functional theory calculations, we provide classical force field potentials for the interactions of Na+/hydrated Na+with(7,7),(8,8),(9,9), and(10,10)-type CNTs. These potentials can be directly used in current popular classical software such as nanoscale molecular dynamics(NAMD) by employing the tcl BC interface. By incorporating the potential of hydrated cation-π interactions to classical all-atom force fields, we show that the ions will move inside the CNT and accumulate, which will block the water flow in wide CNTs. This blockage of water flow in wide CNTs is consistent with recent experimental observations. These results will be helpful for the understanding and design of desalination membranes, new types of nanofluidic channels, nanosensors, and nanoreactors based on CNT platforms.     

On the classical limit of relativistic quantum theories with arbitrary spin

Using the general theory of classical limit developed by the author, we show the existence of classical limit for positive energy representations of the Poincaré group $\text{B}$ of arbitrary spin. The resulting classical phase space is an orbit of $\text{B}$ in the dual of its Lie algebra corresponding to given mass and spin.     

Relation between classical and quantum mechanics

Expressions for the Lie derivatives of functions of non-commuting variables are derived and used to reformulate classical mechanics. This is possible only if the phase space variables commute, or if they satisfy Heisenberg's commutation relations.This work was supported in part by the National Science Foundation Grant Number NSF GP-14803, and by the Air Force Office of Scientific Research, contract number AFOSR 68-1524.     

Interaction between classical and quantum systems and the measurement of quantum observables

Quantum mechanics presumes classical measuring instruments with which they interact. The problem of measurement interaction between classical and quantum systems is posed and solved. The restriction to compatible measurements comes about naturally as the condition for the integrity of the classical system. A technical device is the perspective on classical mechanics as quantum mechanics with essentially hidden dynamical variables. Work supported in part by U.S. Atomic Energy Commission, ERDA.     

On the relation between classical and quantum observables

For systems with a finite number of degrees of freedom, the relation between classical and quantum observables is analysed. In particular, a precise statement of the correspondence limit is obtained.     

Evolution equation of the optical tomogram for arbitrary quantum Hamiltonian and optical tomography of relativistic classical and quantum systems

The dynamic equation for the optical tomogram of nonrelativistic quantum system with an arbitrary Hamiltonian is obtained. The kinetic equation in the classical relativistic kinetics is discussed, and its optical tomography representation is obtained. Dynamic equations for the Wigner functions of relativistic spinless quantum particles in electromagnetic and scalar fields are obtained. Optical tomographic-distribution functions of weakly relativistic spinless quantum particles are introduced, and dynamic equations for these functions in weak electric and scalar fields are obtained.     

Development of the analogy between classical and quantum mechanics

A quantum-mechanical generalisation of Carathéodory’s theorem in classical dynamics is established. Several related properties of classical canonical transformations are also generalised to the quantum case.     

Zwitters: Particles between quantum and classical

We describe both quantum particles and classical particles in terms of a classical statistical ensemble, with a probability distribution in phase space. By use of a wave function in phase space both can be treated in the same quantum formalism. Quantum particles are characterized by a specific choice of observables and time evolution of the probability density. Then interference and tunneling are found within classical statistics. Zwitters are (effective) one-particle states for which the time evolution interpolates between quantum and classical particles. Experimental bounds on a small parameter can test quantum mechanics.     

On the relation between classical and quantum statistical mechanics

Classical and quantum statistical mechanics are compared in the high temperature limit =1/kT0. While this limit is rather trivial for spin systems, we obtain some rigorous results which suggest (and sometimes prove) different asymptotics for continuous systems, depending on the behaviour of the two-body potential for small distances: the difference between suitable classical and quantum variables vanishes as ² for smooth potentials and as for potentials with hard cores.Supported in part by FAPESP. Permanent address: Instituto de Fisica, Universidade de Sao Paulo, Sao Paulo, Brazil     

Comparison between the classical and quantum theories of incoherent scattering

Acta physica Academiae Scientiarum Hungaricae - The incoherent scattering of light by a volume of gas can be treated by classical methods in a straightforward manner. Certain difficulties are...     

Simulation of Pake doublet with classical spins and correspondence between the quantum and classical approaches

Problems of interacting quantum magnetic moments become exponentially complex withincreasing number of particles. As a result, classical equations are often used to modelspin systems. In this paper we show that a classical spins based approach can be used todescribe the phenomena essentially quantum in nature such as of the Pake doublet.     

A link between quantum and classical Potts models

We study ground states of quantum Potts models. We construct ground states of certaind-dimensional quantum models as Gibbs measures of ad-dimensional classical spin system. Our results imply that various phenomena of classical spin systems can also be found in quantum ground states.     

The correspondence principle between classical and quantum quantities

Questions of the reestablishment of a classical quantity from the known quantum operator are investigated in this article, in that case when the operators are constructed according to the rule proposed in article [15]. The existence of a twofold limiting transition is proved, in which the investigated rule satisfies the requirements of the correspondence principle.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 18–23, June, 1974.The author expresses his deep gratitude to V. V. Kuryshkin for suggesting the topic and for systematic aid in the work.     

Ultrafast electroabsorption at the transition between classical and quantum response

We report the first observation of unusually large induced absorption in semiconductors strongly driven by intense ultrashort midinfrared laser fields. This ultrafast electroabsorption has the largest extent below the band edge ( approximately 1 eV) ever observed, to our knowledge, which we interpret as being a manifestation of the dynamical Franz-Keldysh effect. The electroabsorption is observable when the ponderomotive potential is comparable to the photon energy of the applied field, i.e., when the applied field is at the transition between the classical and quantum regimes.     

Two examples illustrating the differences between classical and quantum mechanics

Two examples are presented: The first shows that a potentialV(x) can be in the limit circle case at even if the classical travel time to is infinite. The second shows thatV(x) can be in the limit point case at even though the classical travel time to infinity is finite. The first example illustrates the reflection of quantum waves at sharp steps. The second example illustrates the tunnel effect.This research was partially supported by the National Science Foundation under Grant GP-34260.For help with the diagrams, the authors thank Bob Johnson.     

Performance comparison between classical and quantum control for a simple quantum system

Brańczyk et al. pointed out that the quantum control scheme is superior to the classical control scheme for a simple quantum system using simulation [A.M. Brańczyk, P.E.M.F. Mendonca, A. Gilchrist, A.C. Doherty, S.D. Barlett, Quantum control theory of a single qubit, Physical Review A 75 (2007) 012329 or arXiv e-print quant-ph/0608037]. Here we rigorously prove the result. Furthermore we will show that any quantum operation does not universally “correct” the dephasing noise.