Semi-classical mechanics in phase space: A study of Wigner’s function

1977 ◽  
Vol 287 (1343) ◽  
pp. 237-271 ◽  
Keyword(s):  
Phase Space ◽  
Classical Limit ◽  
Degrees Of Freedom ◽  
Classical Mechanics ◽  
Delta Function ◽  
Full Detail ◽  
Integrable Case ◽  
Classical Motion ◽  
Classical Path ◽  
Path Structure

We explore the semi-classical structure of the Wigner functions Ψ( q,p ) representing bound energy eigenstates | Ψ 〉 for systems with f degrees of freedom. If the classical motion is integrable, the classical limit of Ψ is a delta function on the f -dimensional torus to which classical trajectories corresponding to |Ψ〉 are confined in the 2 f -dimensional phase space. In the semi-classical limit of Ψ ( ℏ small but not zero) the delta function softens to a peak of order Ψ−  f and the torus develops fringes of a characteristic ‘Airy’ form. Away from the torus,Ψ can have semi-classical singularities that are not delta functions; these are discussed (in full detail when f = 1) using Thom's theory of catastrophes. Brief consideration is given to problems raised when is calculated in a representation based on operators derived from angle coordinates and their conjugate momenta. When Ψ the classical motion is non-integrable, the phase space is not filled with tori and existing semi-classical methods fail. We conjecture that (a) For a given value of non-integrability parameter ⋲ ,the system passes through three semi-classical régimes as ℏ diminishes. (b) For states |Ψ〉 associated with regions in phase space filled with irregular trajectories, Ψ will be a random function confined near that region of the ‘energy shell’ explored by these trajectories (this region has more thanks dimensions). (c) For ⋲ ≠ 0, ℏ blurs the infinitely fine classical path structure, in contrast to the integrable case ⋲ = 0, where ℏ imposes oscillatory quantum detail on a smooth classical path structure.

scholarly journals A QUANTUM IS A COMPLEX STRUCTURE ON CLASSICAL PHASE SPACE

2005 ◽  
Vol 02 (04) ◽  
pp. 633-655
Author(s):  
JOSÉ M. ISIDRO
Keyword(s):  
Quantum Mechanics ◽  
Phase Space ◽  
Degrees Of Freedom ◽  
Complex Structure ◽  
Classical Mechanics ◽  
Elementary Excitation ◽  
Differentiable Structure ◽  
Duality Transformations ◽  
Kähler Potential ◽  
Classical Phase Space

Duality transformations within the quantum mechanics of a finite number of degrees of freedom can be regarded as the dependence of the notion of a quantum, i.e., an elementary excitation of the vacuum, on the observer on classical phase space. Under an observer we understand, as in general relativity, a local coordinate chart. While classical mechanics can be formulated using a symplectic structure on classical phase space, quantum mechanics requires a complex-differentiable structure on that same space. Complex-differentiable structures on a given real manifold are often not unique. This article is devoted to analysing the dependence of the notion of a quantum on the complex-differentiable structure chosen on classical phase space. For that purpose we consider Kähler phase spaces, endowed with a dynamics whose Hamiltonian equals the local Kähler potential.

scholarly journals Complex Covariance

10.14311/1809 ◽  
2013 ◽  
Vol 53 (3) ◽  
Author(s):  
Frieder Kleefeld
Keyword(s):  
Quantum Theory ◽  
Phase Space ◽  
Special Relativity ◽  
Degrees Of Freedom ◽  
Classical Mechanics ◽  
Space Time ◽  
Complex Phase ◽  
Dirac Equations ◽  
Klein Gordon ◽  
Complex Valued

According to some generalized correspondence principle the classical limit of a non-Hermitian quantum theory describing quantum degrees of freedom is expected to be the well known classical mechanics of classical degrees of freedom in the complex phase space, i.e., some phase space spanned by complex-valued space and momentum coordinates. As special relativity was developed by Einstein merely for real-valued space-time and four-momentum, we will try to understand how special relativity and covariance can be extended to complex-valued space-time and four-momentum. Our considerations will lead us not only to some unconventional derivation of Lorentz transformations for complex-valued velocities, but also to the non-Hermitian Klein-Gordon and Dirac equations, which are to lay the foundations of a non-Hermitian quantum theory.

scholarly journals Path Integrals for (Complex) Classical and Quantum Mechanics

10.14311/1414 ◽  
2011 ◽  
Vol 51 (4) ◽  
Author(s):  
R. J. Rivers
Keyword(s):  
Quantum Mechanics ◽  
Phase Space ◽  
Degrees Of Freedom ◽  
Path Integrals ◽  
Classical Mechanics ◽  
Classical Particle ◽  
Extended Phase Space ◽  
Extended Phase ◽  
Complex Extension ◽  
Classical And Quantum Mechanics

An analysis of classical mechanics in a complex extension of phase space shows that a particle in such a space can behave in a way redolent of quantum mechanics; additional dimensions permit ‘tunnelling’ without recourse to instantons and time/energy uncertainties exist. In practice, ‘classical’ particle trajectories with additional degrees of freedom arise in several different formulations of quantum mechanics. In this talk we compare the extended phase space of the closed time-path formalism with that of complex classical mechanics, to suggest that ℏ has a role in our understanding of the latter. However, differences in the way that trajectories are used make a deeper comparison problematical. We conclude with some thoughts on quantisation as dimensional reduction.

Rigorous theories

2003 ◽  
Keyword(s):  
Quantum Mechanics ◽  
Classical Limit ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Practical Applications ◽  
Classical Path ◽  
Mechanical Formulation ◽  
S Matrix ◽  
Quantal Theory ◽  
Classical Picture

In this chapter we discuss theories which are rigorous in their formulation but which in order to be useful need to be modified by introducing approximations of some kind. The approximations we are interested in are those which involve introduction of classical mechanical concepts, that is, the classical picture and/or classical mechanical equations of motion in part of the system. At this point, we wish to distinguish between “the classical picture,” which is obtained by taking the classical limit ħ → 0 and the appearance of “classical equations of motion.” The latter may be extracted from the quantum mechanical formulation without taking the classical limit—but, as we shall see later by introducing a certain parametrization of quantum mechanics. Thus there are two ways of introducing classical mechanical concepts in quantum mechanics. In the first method, the classical limit is defined by taking the limit ħ → 0 either in all degrees of freedom (complete classical limit) or in some degrees of freedom (semi-classical theories). We note in passing that the word semi-classical has been used to cover a wide variety of approaches which have also been referred to as classical S-matrix theories, quantum-classical theories, classical path theory, hemi-quantal theory, Wentzel Kramer-Brillouin (WKB) theories, and so on. It is not the purpose of this book to define precisely what is behind these various acronyms. We shall rather focus on methods which we think have been successful as far as practical applications are concerned and discuss the approximations and philosophy behind these. In the other approach, the ħ-limit is not taken—at least not explicitly— but here one introduces “classical” quantities, such as, trajectories and momenta as parameters, and derives equations of motion for these parameters. The latter method is therefore one particular way of parameterizing quantum mechanics. We discuss both of these approaches in this chapter. The Feynman path-integral formulation is one way of formulating quantum mechanics such that the classical limit is immediately visible [3]. Formally, the approach involves the introduction of a quantity S, which has a definition resembling that of an action integral [101].

Shape Dynamics and the Linking Theory

2018 ◽  
Author(s):  
Flavio Mercati
Keyword(s):  
Phase Space ◽  
Degrees Of Freedom ◽  
Line Element ◽  
Hamiltonian Formulation ◽  
Operational Definition ◽  
Extended Phase Space ◽  
Extended Phase ◽  
Problem Of Time ◽  
Definition Of ◽  
Matter Fields

This chapter explains in detail the current Hamiltonian formulation of SD, and the concept of Linking Theory of which (GR) and SD are two complementary gauge-fixings. The physical degrees of freedom of SD are identified, the simple way in which it solves the problem of time and the problem of observables in quantum gravity are explained, and the solution to the problem of constructing a spacetime slab from a solution of SD (and the related definition of physical rods and clocks) is described. Furthermore, the canonical way of coupling matter to SD is introduced, together with the operational definition of four-dimensional line element as an effective background for matter fields. The chapter concludes with two ‘structural’ results obtained in the attempt of finding a construction principle for SD: the concept of ‘symmetry doubling’, related to the BRST formulation of the theory, and the idea of ‘conformogeometrodynamics regained’, that is, to derive the theory as the unique one in the extended phase space of GR that realizes the symmetry doubling idea.

scholarly journals Semi-Classical Localisation Properties of Quantum Oscillators on a Noncommutative Configuration Space

2015 ◽  
Vol 22 (04) ◽  
pp. 1550021 ◽  
Author(s):  
Fabio Benatti ◽  
Laure Gouba
Keyword(s):  
Phase Space ◽  
Configuration Space ◽  
Classical Limit ◽  
Coherent States ◽  
Point Of View ◽  
Quantum Mechanical ◽  
Wigner Functions ◽  
Quantum Mechanical Oscillators ◽  
Mechanical Oscillators ◽  
Space Noncommutativity

When dealing with the classical limit of two quantum mechanical oscillators on a noncommutative configuration space, the limits corresponding to the removal of configuration-space noncommutativity and position-momentum noncommutativity do not commute. We address this behaviour from the point of view of the phase-space localisation properties of the Wigner functions of coherent states under the two limits.

NONEXISTENCE OF QUANTUM NAMBU MECHANICS

1994 ◽  
Vol 09 (29) ◽  
pp. 2727-2732 ◽  
Author(s):  
DEBENDRANATH SAHOO ◽  
M. C. VALSAKUMAR
Keyword(s):  
Phase Space ◽  
Classical Mechanics ◽  
Nambu Mechanics ◽  
Quantum Analog

We investigate the problem of quantization of Nambu mechanics — a problem posed by Nambu [Phys. Rev.D7, 2405 (1973)] — along the line of Wigner–Weyl–Moyal (WWM) phase-space quantization of classical mechanics and show that the quantum analog of Nambu mechanics does not exist.

ROTATIONS AND VIBRATIONS OF FULLERENES IN THE MOLECULAR COMPLEX C20@C80

2021 ◽  
pp. 35-48
Author(s):  
M.A. Bubenchikov ◽  
◽  
A.M. Bubenchikov ◽  
D.V. Mamontov ◽  
◽  
...  
Keyword(s):  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Classical Mechanics ◽  
Rotation Frequency ◽  
Dynamic State ◽  
Large Molecules ◽  
Rotational Degrees Of Freedom ◽  
Rotational Motions ◽  
Centers Of Mass ◽  
The Moment

The aim of this work is to apply classical mechanics to a description of the dynamic state of C20@C80 diamond complex. Endohedral rotations of fullerenes are of great interest due to the ability of the materials created on the basis of onion complexes to accumulate energy at rotational degrees of freedom. For such systems, a concept of temperature is not specified. In this paper, a closed description of the rotation of large molecules arranged in diamond shells is obtained in the framework of the classical approach. This description is used for C20@C80 diamond complex. Two different problems of molecular dynamics, distinguished by a fixing method for an outer shell of the considered bimolecular complex, are solved. In all the cases, the fullerene rotation frequency is calculated. Since a class of possible motions for a single carbon body (molecule) consists of rotations and translational displacements, the paper presents the equations determining each of these groups of motions. Dynamic equations for rotational motions of molecules are obtained employing the moment of momentum theorem for relative motions of the system near the fullerenes’ centers of mass. These equations specify the operation of the complex as a molecular pendulum. The equations of motion of the fullerenes’ centers of mass determine vibrations in the system, i.e. the operation of the complex as a molecular oscillator.

Classical limit of fermions in phase space

2001 ◽  
Vol 42 (9) ◽  
pp. 4020-4030 ◽  
Author(s):  
A. O. Bolivar
Keyword(s):  
Phase Space ◽  
Classical Limit
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