Limit Of Quantum Mechanics Research Articles

Covariant operators and the classical limit of the Dirac equation with a color field.

The covariant operators of the position, spin, and color in Dirac quantum mechanics with an external color field are constructed. Their expectation values have good transformation properties with respect to the Poincar\'e group in the classical limit. This set of operators solves the problem of a relativistic generalization of the classical equations of motion for a colored, spinning particle obtained as the classical limit of Dirac quantum mechanics.

The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics

Starting with a “relativistic” Schrodinger Hamiltonian for neutral gravitating particles, we prove that as the particle number N → ∞ and the gravitation constant G → 0 we obtain the well known semiclassical theory for the ground state of stars. For fermions; the correct limit is to fix GN 2/3 and the Chandrasekhar formula is obtained. For bosons the correct limit is to fix GN and a Hartree type equation is obtained. In the fermion case we also prove that the semiclassical equation has a unique solution — a fact which had not been established previously.

Classical limit of the path-integral formulation of quantum mechanics in terms of the density matrix

Instead of regarding the classical limit as the limit ℏ→0, an alternative view based on the physical interpretation of the elements of the density matrix is proposed. The derivation of the classical limit of quantum mechanics is carried out in two stages. First, the statistical classical limit is derived, and then, with an appropriate initial condition, the deterministic classical limit is obtained. The derivation hinges on the use of the Feynman path-integral formulation of quantum mechanics in terms of the density matrix. It is shown that deterministic (Hamilton’s or Newton’s) classical mehanics is not the classical limit of Schrödinger’s wave mechanics. The classical limit of Schrödinger’s wave mechanics is only statistical and corresponds to the classical Liouville equation. In order to obtain the deterministic classical limit, it is necessary to start out initially with a quantum mechanical mixture.

Path dependence of the geometric rotation of polarization in optical fibers

The geometric rotation of the polarization angle in a ideal cylindrical optical fiber without birefringence is obtained for arbitrary fiber paths in terms of the image of the path in the tangent vector space. The result extends a recent result of Chiao and Wu [Phys. Rev. Lett. 57, 933 (1986)] restricted to paths with parallel ends, which was derived from Berry's phase in the adiabatic limit of quantum mechanics [Proc. R. Soc. London Ser. A 392,45 (1984)]. The treatment given here is purely classical and uses differential geometry to extend earlier work by Ross [Opt. Quantum Electron. 16, 455 (1984)] on the uniform helix.

Analysis of the Moyal product in a flat space

This paper studies the mathematical properties of Moyal product defined in the space R2n. They correspond to the properties of quantum mechanics and permit us to consider classical mechanics as a limit of quantum mechanics when Planck’s constant vanishes. A table of formulas and two algebras give in particular the necessary results for the approximation of the Moyal product and for the spectral resolution.

Asymptotic [formula omitted] in classic particle limit of non-relativistic quantum mechanics

The unitary group describing the evolution of the quantum fluctuation around any classic phase orbit has a perturbative strongly asymptotic expansion in the small parameter h z.shtsls; 1 2 . Likewise, the mean values of the observables position and momentum in suitable time-dependent states have an asymptotic expansion in h z.shtsls; 1 2 , whose zeroth-order term satisfies the classic canonical equations. Similar expansions are found for the squared quantum dispersions.

Classical limit of quantum mechanics

The classical limit problem of quantum mechanics is a particular example of a broader issue concerning the relationship between two physical theories, one empirically valid in a subdomain of the other. It is a basic requirement that a theory superseding the previous one must be compatible with the former at a suitable limit. As a prelude to discussing the classical limit of quantum mechanics, we recall some relevant aspects of this general issue.

Classical limit of quantum mechanics in the large

AbstractHilbert space may be regarded as a convenient standard approximation by interpolation and extrapolation to a unitary space, in general decomposable into a sum of semisimple spaces. In the limit we expect a particle theory expanded in powers of h according to the number of interacting particles. In the classical limit h tending to zero, phase space, with equations of motion reducible to Hamiltonian form, replaces Hilbert space. The modification of states by observing them is taken care of by considering probability distributions of petty ensembles. If the equations for any single observer can be made autonomous by replacing empirical time by universal time with an arrow we have a causal system. We then obtain the relation between probability and negentropy required for the second law of thermodynamics. An approximate Newtonian theory provides proximate particles with internal and external variables and admits the Poincaré group. For the internal variables we have approximately the Breit interaction. For the external variables we have equivalence for observers of the homogeneous Lorentz group of relativity. We introduce grand ensembles of ultimate particles, and nebulae as proximate particles in the large. We assume the Einstein principle of equivalence for the ten parameter set of observers suggested by relativity and suppose the second law of thermodynamics holds for each observer. The Einstein law of gravitation follows in classical theory to the order of the reciprocal of the large constant, in general with positive natural curvature as well as that corresponding to mass. Replacing particle interactions by fields we include them in classical theory.

Classical limit of quantum mechanics and the quantum billiards problem

Two limiting cases follow from an algebraic formulation of quantum mechanics: Hamiltonian mechanics and quantum mechanics. The results can be used to formulate a quantum billiards problem and to study it at a qualitative level.

Feynman maps without improper integrals

The Feynman maps introduced first by Truman are examined. The domain considered here consists of the Fresnel-integrable functions in the sense of Albeverio and Hoegh-Krohn; extensions to wider classes of functions will be studied in the following paper. The original definition of the F-maps is slightly modified: we start from the underlying measures on the Hilbert space of paths in order to avoid use of improper integrals. Some new properties of the F-maps are derived. In particular, the dominated convergence theorem is shown to be not valid for the F1-map (or Feynman integral); this fact is of a certain importance for the classical limit of quantum mechanics.

Nonrelativistic limit of relativistic quantum mechanics

The nonrelativistic limit of a relativistic quantum mechanics of Stueckelberg type is studied. A new degree of freedom, which is a purely quantum effect, emerges as a residue of the relativistic theory; its role is to impose Galilean simultaneity. It is shown that the usual nonrelativistic Schr\"odinger equation is obtained as the asymptotic form of the relativistic evolution equation, and some new insight is provided into its interpretation from the point of view of the relativistic theory.

New approach to the semiclassical limit of quantum mechanics

We propose a new approach for the estimate of the rate of degeneracy of the lowest eigenvalues of the Schrodinger operator in the presence of tunneling based on the theory of diffusion processes. Our method provides lower and upper bounds for the energy splittings and the rates of localization of the wave functions and enables us to discuss cases which, as far as we know, have never been treated rigorously in the literature. In particular we give an analysis of the effect on eigenvalues and eigenfunctions of localized deformations of 1) symmetric double well potentials 2) potentials periodic and symmetric over a finite interval. Theses situations are characterized by a remarkable dependence on such deformations. Our probabilistic techniques are inspired by the theory of small random perturbations of dynamical systems.

Probabilistic interpretation, group contraction and the Galilean limit of relativistic quantum mechanics

The problem of getting the Galilei group as a limit of the Poincare group has been solved in a rigorous manner. We use the generalized contraction introduced by Doebner and Melsheimer. Good physical reasons are given for keepingE3 invariant in the contraction procedure, in spite of other attempts made in this direction. Group representations are contracted by using well-defined mathematical methods studied by Mickelsson and Niederle. Ill-defined contracted states do not appear in this way. An intriguing connection between localizability and contraction has energed through this work. Some physical conclusions are associated to this feature.

Quantization and Group Representations

A quantization of a fixed classical mechanical system is firstly an association between quantum mechanical observables (preferably self-adjoint operators on Hilbert space) and classical mechanical observables (i.e. real-valued functions on phase space). Secondly, a quantization should permit an interpretation of the correspondence principle that ‘classical mechanics is the limit of quantum mechanics as Planck's constant approaches zero'. With these two underlying precepts, Section 2 states the four basic requirements, I to IV, of a quantization along with an additional requirement V that characterizes the subclass of special quantizations.

Oscillatory integrals and the method of stationary phase in infinitely many dimensions, with applications to the classical limit of quantum mechanics I

We give a theory of oscillatory integrals in infinitely many dimensions which extends, for a class of phase functions, the finite dimensional theory. In particular we extend the method of stationary phase, the theory of Lagrange immersions and the corresponding asymptotic expansions to the infinite dimensional case. A particular application of the theory to the Feyman path integrals defined in previous work by the authors yields asymptotic expansions to all orders of quantum mechanical quantities in powers of Planck's constant.

Notes on the classical and the nonrelativistic limits in quantum mechanics

A method is presented which can be used to discuss both the classical and also the nonrelativistic limit of quantum mechanics. A one-to-one correspondence may be established between the asymptotic convergence of the resolvent and that of the timedependent solution. In so far as the question of dynamics is concerned we investigate the relation between families of nonrelativistic Hamiltonians and the corresponding Dirac-Hamiltonians when c→±∞ or when c→±0. The nonrelativistic ‘free’ theory formally shows the same pattern when ħ→±0 (the classical limit) or when ħ→±∞. The investigation finally shows how the asymptotic convergence of the relativistic theory can take place under some fairly general conditions of the radiation field.

Comments on the Classical Limit of Quantum Mechanics

First Page

The classical S-matrix: a more detailed study of classically forbidden transitions in inelastic collissions

Procedures are investigated for describing classically forbidden collision processes within the framework of the classical limit of quantum mechanics. The goal is to use exact classical mechanics (numerically computed trajectories) to treat classically forbidden transitions in a complex collision (such as an atom plus diatom) in a manner analogous to the way one uses it to treat barrier transmission by a single particle in one dimension (a classically forbidden process); in this latter example the transmission coefficient is exp(−2 O), θ being the classical action integral through the barrier (a classically forbidden region). Numerical application of these procedures is made to the linear atom-diatom collision system (without reaction), and the resulting vibrational probabilities for the classically forbidden transitions are in as good agreement with the exact quantum mechanical values are the transition probabilities for classically allowed transitions. It appears, therefore, that the dynamics of even these classically forbidden transitions is accurately described in this semiclassical framework.

Classical Mechanics as a Limit of Quantum Mechanics

AbstractIn this paper a formulation of classical mechanics is given with the help of linear operators in HILBERT space, which is different from the formalism of v. NEUMANN and KOOPMAN, i.e. the observables are represented by selfadjoint operators instead of real functions. It is shown that classical mechanics in this form fulfils the same axioms as quantum mechanics except those two ones which contain ħ. But the corresponding classical axioms are got from the quantum mechanical ones substituting ħ by O.

Weyl Transformation and the Classical Limit of Quantum Mechanics

The Weyl correspondence for obtaining quantum operators from functions of classical coordinates and momenta is known to be incorrect. To calculate quantum-mechanical expectation values as phase-space averages with the Wigner density function, one cannot use classical functions but must use Weyl transforms. These transforms are defined and their properties derived from quantum mechanics. Their properties are expressed in terms of a Hermitian operator Δ(Q, K) whose Weyl transform is a δ function. The Wigner function is the transform of the density operator. Every Weyl transform is exhibited as a difference of two functions which are nonnegative on the phase spece. Weyl transforms do not obey the algebra of classical functions. In the classical limit ℏ → 0, Weyl transforms become classical functions, the Wigner function becomes nonnegative throughout the phase space, and the Hilbert space is spanned by an orthonormal set of vectors which are simultaneous eigenkets of the commuting coordinate and momentum operators.