Limit Of Quantum Mechanics Research Articles

Unification theory of classical statistical uncertainty relation and quantum uncertainty relation and its applications

General classical statistical uncertainty relation is deduced and generalized to quantum uncertainty relation. We give a general unification theory of the classical statistical and quantum uncertainty relations, and prove that the classical limit of quantum mechanics is just classical statistical mechanics. It is shown that the classical limit of the general quantum uncertainty relation is the general classical uncertainty relation. Also, some specific applications show that the obtained theory is self-consistent and coincides with those from physical experiments.

On the classical limit of Bohmian mechanics for Hagedorn wave packets

We consider the classical limit of quantum mechanics in terms of Bohmian trajectories. For wave packets as defined by Hagedorn we show that the Bohmian trajectories converge to Newtonian trajectories in probability.

Snippets of physics

Obtaining the classical limit of quantum mechanics turns out to be conceptually and operationally non-trivial and, even today, some of the experts consider this issue to be unsettled. There is a function, originally devised by Wigner, which plays a key role in this aspect and throws some light on the w ay the classical world emerges from the quantum description.

Mapping out-of-equilibrium into equilibrium in one-dimensional transport models

Systems with conserved currents driven by reservoirs at the boundaries offer an opportunity for a general analytic study that is unparalleled in more general out-of-equilibrium systems. The evolution of coarse-grained variables is governed by stochastic hydrodynamic equations in the limit of small noise. As such it is amenable to a treatment formally equal to the semiclassical limit of quantum mechanics, which reduces the problem of finding the full distribution functions to the solution of a set of Hamiltonian equations. It is in general not possible to solve such equations explicitly, but for an interesting set of problems (the driven symmetric exclusion process and the Kipnis–Marchioro–Presutti model) it can be done by a sequence of remarkable changes of variables. We show that at the bottom of this ‘miracle’ is the surprising fact that these models can be taken through a non-local transformation into isolated systems satisfying detailed balance, with probability distribution given by the Gibbs–Boltzmann measure. This procedure can in fact also be used to obtain an elegant solution of the much simpler problem of non-interacting particles diffusing in a one-dimensional potential, again using a transformation that maps the driven problem into an undriven one.

John von Neumann's mathematical “Utopia” in quantum theory

This paper surveys John von Neumann's work on the mathematical foundations of quantum theories in the light of Hilbert's Sixth Problem concerning the geometrical axiomatization of physics. We argue that in von Neumann's view geometry was so tied to logic that he ultimately developed a logical interpretation of quantum probabilities. That motivated his abandonment of Hilbert space in favor of von Neumann algebras, specifically the type II 1 factors, as the proper limit of quantum mechanics in infinite dimensions. Finally, we present the reasons why his axiomatic program remained an “unsolved problem” in mathematical physics. A recent unpublished result by Huzimiro Araki, proving that no algebra with a tracial state defined on it, such as the type II 1 factors, can support any (regular) representation of the canonical commutation relations, is also reviewed and its consequences for von Neumann's projects are discussed.

Foldy-Wouthyusen transformation and semiclassical limit for relativistic particles in strong external fields

A general method of the Foldy-Wouthyusen (FW) transformation for relativistic particles of arbitrary spin in strong external fields has been developed. The use of the found transformation operator is not restricted by any definite commutation relations between even and odd operators. The final FW Hamiltonian can be expanded into a power series in the Planck constant which characterizes the order of magnitude of quantum corrections. Exact expressions for low-order terms in the Planck constant can be derived. Finding these expressions allows to perform a simple transition to the semiclassical approximation which defines a classical limit of the relativistic quantum mechanics. As an example, interactions of spin-1/2 and scalar particles with a strong electromagnetic field have been considered. Quantum and semiclassical equations of motion of particles and their spins have been deduced. Full agreement between quantum and classical theories has been established.

Perturbed nonlinear models from noncommutativity

Some discrete two-dimensional many particle systems are investigated inside the context of a deformed Newtonian mechanics obtained as a classical limit of a noncommutative quantum mechanics where the noncommutativity includes all degrees of freedom of the theory. Also, applying the continuous limit some alternative noncommutative generalizations of two-dimensional field theories have been constructed.

Wave Mechanics or Wave Statistical Mechanics

By comparison between equations of motion of geometrical optics andthat of classical statistical mechanics, this paper finds that there shouldbe an analogy between geometrical optics and classical statistical mechanicsinstead of geometrical mechanics and classical mechanics. Furthermore, bycomparison between the classical limit of quantum mechanics and classicalstatistical mechanics, it finds that classical limit of quantum mechanicsis classical statistical mechanics not classical mechanics, hence itdemonstrates that quantum mechanics is a natural generalization ofclassical statistical mechanics instead of classical mechanics. Thencequantum mechanics in its true appearance is a wave statistical mechanicsinstead of a wave mechanics.

The electric dipole moment of the electron: An intuitive explanation for the evasion of Schiff’s theorem

In most experimental searches for the electron electric dipole moment, one searches for a linear Stark effect in a paramagnetic atom or molecule and interprets the result in terms of the electric dipole moment of the unpaired valence electron(s). Schiff’s theorem states that in the limit of nonrelativistic quantum mechanics, there can be no linear Stark effect to first order in the electric dipole moment. Sandars has shown that Schiff’s theorem is not applicable when special relativity is taken into account. We give a heuristic explanation for this relativistic effect, which corrects a widespread misconception in the literature.

Large N behavior of two dimensional supersymmetric Yang-Mills quantum mechanics

We analyze the N→∞ limit of supersymmetric Yang-Mills quantum mechanics (SYMQM) in two space time dimensions. To do so we introduce a particular class of SU(N) invariant polynomials and give the solutions of two-dimensional SYMQM in terms of them. We conclude that in this limit the system is not fully described by the single trace operators Tr(a†n) since there are other, bilinear operators Trn(a†a†) that play a crucial role when the Hamiltonian is free.

Non-integrability and mixing in quantum systems: On the way to quantum chaos

In spite of the increasing attention that quantum chaos has received from physicists in recent times, when the subject is considered from a conceptual viewpoint the usual opinion is that there is some kind of conflict between quantum mechanics and chaos. In this paper we follow the program of Belot and Earman, who propose to analyze the problem of quantum chaos as a particular case of the classical limit of quantum mechanics. In particular, we address the problem on the basis of our account of the classical limit, which in turn is grounded on the self-induced approach to decoherence. This strategy allows us to identify the conditions that a quantum system must satisfy to lead to non-integrability and to mixing in the classical limit.

The Problem of the Classical Limit of Quantum Mechanics and the Role of Self-Induced Decoherence

Our account of the problem of the classical limit of quantum mechanics involves two elements. The first one is self-induced decoherence, conceived as a process that depends on the own dynamics of a closed quantum system governed by a Hamiltonian with continuous spectrum; the study of decoherence is addressed by means of a formalism used to give meaning to the van Hove states with diagonal singularities. The second element is macroscopicity represented by the limit $$\hbar \rightarrow 0$$ : when the macroscopic limit is applied to the Wigner transformation of the diagonal state resulting from decoherence, the description of the quantum system becomes equivalent to the description of an ensemble of classical trajectories on phase space weighted by their corresponding probabilities.

Self-Induced Decoherence and the Classical Limit of Quantum Mechanics

In this paper we argue that the emergence of the classical world from the underlying quantum reality involves two elements: self-induced decoherence and macroscopicity. Self-induced decoherence does not require the openness of the system and its interaction with the environment: a single closed system can decohere when its Hamiltonian has continuous spectrum. We show that, if the system is macroscopic enough, after self-induced decoherence it can be described as an ensemble of classical distributions weighted by their corresponding probabilities. We also argue that classicality is an emergent property that arises when the behavior of the system is described from an observational perspective.

Semi-classical limit of relativistic quantum mechanics

It is shown that the semi-classical limit of solutions to the Klein-Gordon equation gives the particle probability density that is in direct proportion to the inverse of the particle velocity. It is also shown that in the case of the Dirac equation a different result is obtained.

Quantum Physics and Classical Physics—In the Light of Quantum Logic

In contrast to the Copenhagen interpretation we consider quantum mechanics as universally valid and query whether classical physics is really intuitive and plausible. We discuss these problems within the quantum logic approach to quantum mechanics where the classical ontology is relaxed by reducing metaphysical hypotheses. On the basis of this weak ontology a formal logic of quantum physics can be established which is given by an orthomodular lattice. By means of the Soler condition and Piron's result one obtains the classical Hilbert spaces. However, this approach is not fully convincing. There is no plausible justification of Soler's law and the quantum ontology is partly too weak and partly too strong. We propose to replace this ontology by an ontology of unsharp properties and conclude that quantum mechanics is more intuitive than classical mechanics and that classical mechanics is not the macroscopic limit of quantum mechanics.

The classical and commutative limits of noncommutative quantum mechanics: a superstar * Wigner-Moyal equation

We are interested in the similarities and differences between the quantum-classical (Q-C) and the noncommutative-commutative (NC-Com) correspondences. As one useful platform to address this issue we derive the superstar Wigner-Moyal equation for noncommutative quantum mechanics (NCQM). A superstar *-product combines the usual phase space * star and the noncommutative * star-product. Having dealt with subtleties of ordering present in this problem we show that the Weyl correspondence of the NC Hamiltonian has the same form as the original Hamiltonian, but with a non-commutativity parameter theta-dependent, momentum-dependent shift in the coordinates. Using it to examine the classical and the commutative limits, we find that there exist qualitative differences between these two limits. Specifically, if <FONT FACE=Symbol>q ¹</FONT> 0 there is no classical limit. Classical limit exists only if <FONT FACE=Symbol>q ®</FONT> 0 at least as fast as h ->0, but this limit does not yield Newtonian mechanics, unless the limit of theta/h vanishes as <FONT FACE=Symbol>q ®</FONT> 0. For another angle towards this issue we formulate the NC version of the continuity equation both from an explicit expansion in orders of theta and from a Noether's theorem conserved current argument. We also examine the Ehrenfest theorem in the NCQM context.

THE QUANTUM-CLASSICAL TRANSITION: THE FATE OF THE COMPLEX STRUCTURE

According to Dirac, fundamental laws of Classical Mechanics should be recovered by means of an "appropriate limit" of Quantum Mechanics. In the same spirit it is reasonable to enquire about the fundamental geometric structures of Classical Mechanics which will survive the appropriate limit of Quantum Mechanics. This is the case for the symplectic structure. On the contrary, such geometric structures as the metric tensor and the complex structure, which are necessary for the formulation of the Quantum theory, may not survive the Classical limit, being not relevant in the Classical theory. Here we discuss the Classical limit of those geometric structures mainly in the Ehrenfest and Heisenberg pictures, i.e. at the level of observables rather than at the level of states. A brief discussion of the fate of the complex structure in the Quantum-Classical transition in the Schrödinger picture is also mentioned.

Smoothed Wigner functions: a tool to resolve semiclassical structures

The Wigner and Husimi distributions are the usual phase space representations of a quantum state. The Wigner distribution has structures of order ħ2. On the other hand, the Husimi distribution is a Gaussian smearing of the Wigner function on an area of size ħ and then, it only displays structures of size ħ. We have developed a phase space representation which results a Gaussian smearing of the Wigner function on an area of size ħσ, with σ≥1. Within this representation, the Husimi and Wigner functions are recovered when σ=1 and $ \sigma \gtrsim 2 $ respectively. We treat the application of this intermediate representation to explore the semiclassical limit of quantum mechanics. In particular we show how this representation uncover semiclassical hyperbolic structures of chaotic eigenstates.

GENERALIZATION OF CLASSICAL STATISTICAL MECHANICS TO QUANTUM MECHANICS AND STABLE PROPERTY OF CONDENSED MATTER

Classical statistical average values are generally generalized to average values of quantum mechanics. It is discovered that quantum mechanics is a direct generalization of classical statistical mechanics, and we generally deduce both a new general continuous eigenvalue equation and a general discrete eigenvalue equation in quantum mechanics, and discover that a eigenvalue of quantum mechanics is just an extreme value of an operator in possibility distribution, the eigenvalue f is just classical observable quantity. A general classical statistical uncertain relation is further given, and the general classical statistical uncertain relation is generally generalized to the quantum uncertainty principle; the two lost conditions in classical uncertain relation and quantum uncertainty principle, respectively, are found. We generally expound the relations among the uncertainty principle, singularity and condensed matter stability, discover that the quantum uncertainty principle prevents the appearance of singularity of the electromagnetic potential between nucleus and electrons, and give the failure conditions of the quantum uncertainty principle. Finally, we discover that the classical limit of quantum mechanics is classical statistical mechanics, the classical statistical mechanics may further be degenerated to classical mechanics and we discover that merely stating that the classical limit of quantum mechanics is classical mechanics is a mistake. As application examples, we deduce both the Schrödinger equation and the state superposition principle, and deduce that there exists a decoherent factor from a general mathematical representation of the state superposition principle; the consistent difficulty between statistical interpretation of quantum mechanics and determinant property of classical mechanics is overcome.

A new approach on studying electron behavior under very intense laser beam

The recent emergence of the field studying interactions between very intense laser fields and charged particles, brings about the problem of determining if, and under what conditions, the quantum equations which describe these interactions are at the classical limit. We present an affirmative answer to this question in the case of an electron in electromagnetic field, corresponding to intensities of the electrical field of the order one atomic unit, or greater. We neglect the spin interactions, and show that with very good approximation (the relative error being less than 10/sup -7/), the solution of the Klein-Gordon equation is a function only of the classical action S, written for the same system. Our approach does not use the semiclassical approximation, and it implies that the solution of the Klein-Gordon equation is at the classical limit of quantum mechanics for the systems discussed here. The results lead to a simplified computational model for analysing these systems.