scholarly journals PT -symmetric classical mechanics

2021 ◽  
Vol 2038 (1) ◽  
pp. 012003
Author(s):  
Carl M Bender ◽  
Daniel W Hook
Keyword(s):  
Classical Mechanics ◽  
Topological Transition ◽  
Work In Progress ◽  
Initial Values ◽  
Depth Analysis ◽  
Classical Trajectories ◽  
Complex Trajectories ◽  
Classical Orbits

Abstract This paper reports the results of an ongoing in-depth analysis of the classical trajectories of the class of non-Hermitian PT -symmetric Hamiltonians H = p2 + x2 (ix) ε (ε ⩾ 0). A variety of phenomena, heretofore overlooked, have been discovered such as the existence of infinitely many separatrix trajectories, sequences of critical initial values associated with limiting classical orbits, regions of broken PT -symmetric classical trajectories, and a remarkable topological transition at ε = 2. This investigation is a work in progress and it is not complete; many features of complex trajectories are still under study.

Do attractive scattering potentials concentrate particles at the origin in one, two and three dimensions? II. Potentials diverging as — C/r q

1983 ◽  
Vol 388 (1795) ◽  
pp. 419-444 ◽  
Keyword(s):  
Turning Points ◽  
Classical Mechanics ◽  
Preceding Paper ◽  
Incident Beam ◽  
Three Dimensions ◽  
One Dimension ◽  
Zero Energy ◽  
Range Parameter ◽  
Classical Trajectories ◽  
2D And 3D

In classical mechanics (c.m.), and near the semi-classical limit h →0 of quantum mechanics (s.c.l.), the enhancement factors α ≡ ρ 0 /ρ ∞ are found for scattering by attractive central potentials U(r) ; here ρ 0,∞ (and v 0,∞ ) are the particle densities (and speeds) at the origin and far upstream in the incident beam. For finite potentials ( U (0) > — ∞), and when there are no turning points, the preceding paper found both in c.m., and near the s.c.l. (which then covers high v ∞ ), α 1 = v ∞ / v 0 , α 2 = 1, α 3 = v 0 / v ∞ respectively in one dimension (1D), 2D and 3D. The argument is now extended to potentials (still without turning points), where U ( r →0) ~ ─ C/r q , with 0 < q < 1 in ID (where r ≡ | x | ), and 0 < q < 2 in 2D and 3D, since only for such q can classical trajectories and quantum wavefunctions be defined unambiguously. In c.m., α 1 (c.m.) = 0, α 3 (c.m.) = ∞, and α 2 (c.m.) = (1 —½ q ) N , where N = [integer part of (1 ─½ q ) -1 ]is the number of trajectories through any point ( r , θ) in the limit r → 0. All features of U(r) other than q are irrelevant. Near the s.c.l. (which now covers low v ∞ ) a somewhat delicate analysis is needed, matching exact zero-energy solutions at small r to the ordinary W.K.B. approximation at large r ; for small v ∞ / u it yields the leading terms α 1 (s.c.l.) = Λ 1 (q) v ∞ / u , α 2 (s.c.I) = (1 ─½ q ) -1 , α 3 (s.c.l.)= Λ 3 ( q ) u/v ∞ , where u ≡ (C/h q m 1-q ) 1/(2-q) is a generalized Bohr velocity. Here Λ 1,3 are functions of q alone, given in the text; as q →0 the α (s.c.l.) agree with the α quoted above for finite potentials. Even in the limit h = 0, α 2 (s.c.l.) and α 2 (c.m.) differ. This paradox in 2D is interpreted loosely in terms of quantal interference between the amplitudes corresponding to the N classical trajectories. The Coulomb potential ─ C/r is used as an analytically soluble example in 2D as well as in 3D. Finally, if U(r) away from the origin depends on some intrinsic range parameter α(e.g. U = ─ C exp (─r/a)/r q ) , and if, near the s.c.l., v ∞ / u is regarded as a function not of h but more realistically of v ∞ , then the expressions α (s.c.l.) above apply only in an intermediate range 1/ a ≪ mv ∞ / h ≪ ( mC/h 2 ) 1/(2- q ) which exists only if a ≫ ( h 2 / mC ) 1/(2- q ) ).

Effects of complex time on periodic and nonperiodic classical trajectories of one-dimensional Hamiltonian systems

2013 ◽  
Vol 91 (4) ◽  
pp. 293-299 ◽  
Author(s):  
Asiri Nanayakkara
Keyword(s):  
Hamiltonian Systems ◽  
Complex Function ◽  
Classical Mechanics ◽  
Complex Time ◽  
One Dimensional ◽  
Specific Complex ◽  
Classical Trajectories ◽  
Fixed Phase ◽  
Countably Infinite ◽  
Infinite Set

In recent years, much research has been carried out on extending both quantum mechanics and classical mechanics into the complex domain by making parameters of real hermitian Hamiltonians or total energy of the system complex. In this paper we investigate the effects of complex time on periodic and nonperiodic trajectories of both hermitian and nonhermitian one-dimensional classical Hamiltonian systems. Most of the periodic classical trajectories of real hermitian systems turn into nonperiodic and open when the energy or the parameters of the potential become complex. We show that when time is taken as a complex quantity with a specific fixed phase angle or as a specific complex function, nonperiodic trajectories become periodic and closed. Furthermore, we show that real hermitian systems, such as H = p2/2m + x4 + bx3 + cx2 + dx (b, c, and d are real quantities) possess classical periodic trajectories for real energies even when time is complex (i.e., t = treiτ). It was found that there is a discrete set of τ values for which the trajectories of the preceding system are closed and periodic and periods associated with them form a countably infinite set.

A QUANTUM-CLASSICAL APPROACH TO MOLECULAR DYNAMICS

2003 ◽  
Vol 02 (01) ◽  
pp. 73-90 ◽  
Author(s):  
G. D. BILLING
Keyword(s):  
Wave Function ◽  
Classical Mechanics ◽  
Basis Set ◽  
Discrete Variable ◽  
Discrete Variable Representation ◽  
Molecular Systems ◽  
Order Expansion ◽  
Classical Trajectories ◽  
Grid Points ◽  
Variable Representation

We present a new method for treating the dynamics of molecular systems. The method has been named "quantum dressed" classical mechanics and is based on an expansion of the wave function in a time-dependent basis-set, the Gauss–Hermite basis-set. From here it is possible to proceed in two ways, one is in principle exact and the other approximate. In the exact approach one constructs a discrete variable representation (DVR) in which the grid points are defined by the Hermite part of the Gauss–Hermite basis set. In the approximate method a second order expansion of the potential around the classical trajectories is introduced and the quantum dymamics solved in a second quantization rather than a wave-function representation.

scholarly journals Toward a classification of dynamical symmetries in classical mechanics

1983 ◽  
Vol 27 (1) ◽  
pp. 53-71 ◽  
Author(s):  
Geoff Prince
Keyword(s):  
Conservation Law ◽  
Kepler Problem ◽  
Classical Mechanics ◽  
Dynamical Symmetry ◽  
Lagrangian System ◽  
Modern Approach ◽  
Parameter Group ◽  
Classical Trajectories ◽  
Invariance Groups

A one-parameter group on evolution space which permutes the classical trajectories of a Lagrangian system is called a dynamical symmetry. Following a review of the modern approach to the “symmetry-conservation law” duality an attempt is made to classify such invariance groups according to the induced transformation of the Cartan form. This attempt is fairly successful inasmuch as the important cases of Lie, Noether and Cartan symmetries can be distinguished. The theory is illustrated with a presentation of results for the classical Kepler problem.

Bimolecular Reactions, Dynamics of Collisions

2018 ◽  
Author(s):  
Niels Engholm Henriksen ◽  
Flemming Yssing Hansen
Keyword(s):  
Cross Sections ◽  
Classical Mechanics ◽  
Reaction Probability ◽  
Many Body ◽  
Zener Model ◽  
Quantum Mechanical Description ◽  
Oppenheimer Approximation ◽  
Classical Trajectories ◽  
Bimolecular Collisions ◽  
Three Body

This chapter discusses the dynamics of bimolecular collisions within the framework of (quasi-)classical mechanics as well as quantum mechanics. The relation between the cross-section and the reaction probability, which can be calculated theoretically from a (quasi-)classical or quantum mechanical description of the collision, is described in terms of classical trajectories and wave packets, respectively. As an introduction to reactive scattering, classical two-body scattering is described and used to formulate simple models for chemical reactions, based on reasonable assumptions for the reaction probability. Three-body (and many-body) quasi-classical scattering is formulated and the numerical evaluation of the reaction probability is described. The relation between scattering angles and differential cross-sections in various frames is emphasized. The chapter concludes with a brief description of non-adiabatic dynamics, that is, situations beyond the Born–Oppenheimer approximation where more than one electronic state is in play. A discussion of the so-called Landau–Zener model is included.

Classical orbits from the wave function in the large-quantum-number limit

2003 ◽  
Vol 81 (7) ◽  
pp. 929-939
Author(s):  
James D Bonnar ◽  
Jeffrey R Schmidt
Keyword(s):  
Wave Function ◽  
Probability Density Function ◽  
Probability Density ◽  
Quantum Number ◽  
Coulomb Potential ◽  
Correspondence Principle ◽  
Classical Mechanics ◽  
Classical Trajectories ◽  
Large Quantum ◽  
Large Quantum Number

Classical trajectories for the Coulomb potential are obtained from the large principle quantum-number limit of solutions to the nonrelativistic Schrödinger equation, by use of integral equations satisfied by the radial probability density function. These trajectories are found to be in excellent agreement with those computed directly from classical mechanics, in accordance with a statement of the Bohr Correspondence principle, except in a region very close to the center of force. PACS No.: 05.45.Mt

Meridional Circulation, Turbulence, and Diffusion Processes in Stars

1976 ◽  
Vol 32 ◽  
pp. 109-116 ◽  
Author(s):  
S. Vauclair
Keyword(s):  
Convection Zone ◽  
Diffusion Processes ◽  
Meridional Circulation ◽  
Diffusion Time ◽  
Work In Progress ◽  
First Results ◽  
Stellar Envelopes ◽  
And Diffusion ◽  
Turbulence And Diffusion ◽  
Hr Diagram

This paper gives the first results of a work in progress, in collaboration with G. Michaud and G. Vauclair. It is a first attempt to compute the effects of meridional circulation and turbulence on diffusion processes in stellar envelopes. Computations have been made for a 2 Mʘstar, which lies in the Am - δ Scuti region of the HR diagram.Let us recall that in Am stars diffusion cannot occur between the two outer convection zones, contrary to what was assumed by Watson (1970, 1971) and Smith (1971), since they are linked by overshooting (Latour, 1972; Toomre et al., 1975). But diffusion may occur at the bottom of the second convection zone. According to Vauclair et al. (1974), the second convection zone, due to He II ionization, disappears after a time equal to the helium diffusion time, and then diffusion may happen at the bottom of the first convection zone, so that the arguments by Watson and Smith are preserved.

TEM/AEM characterization of fine-grained clay minerals in very-low-grade rocks: Evaluation of contamination by empa involving celadonite family minerals

1996 ◽  
Vol 54 ◽  
pp. 694-695
Author(s):  
Gejing Li ◽  
D. R. Peacor ◽  
D. S. Coombs ◽  
Y. Kawachi
Keyword(s):  
Electron Microscopy ◽  
Volcanic Rocks ◽  
Analytical Electron Microscopy ◽  
Metamorphic Rocks ◽  
New Mineral ◽  
Chemical Compositions ◽  
Low Grade ◽  
Fine Grained ◽  
Depth Analysis ◽  
Mineral Aggregates

Recent advances in transmission electron microscopy (TEM) and analytical electron microscopy (AEM) have led to many new insights into the structural and chemical characteristics of very finegrained, optically homogeneous mineral aggregates in sedimentary and very low-grade metamorphic rocks. Chemical compositions obtained by electron microprobe analysis (EMPA) on such materials have been shown by TEM/AEM to result from beam overlap on contaminant phases on a scale below resolution of EMPA, which in turn can lead to errors in interpretation and determination of formation conditions. Here we present an in-depth analysis of the relation between AEM and EMPA data, which leads also to the definition of new mineral phases, and demonstrate the resolution power of AEM relative to EMPA in investigations of very fine-grained mineral aggregates in sedimentary and very low-grade metamorphic rocks.Celadonite, having end-member composition KMgFe3+Si4O10(OH)2, and with minor substitution of Fe2+ for Mg and Al for Fe3+ on octahedral sites, is a fine-grained mica widespread in volcanic rocks and volcaniclastic sediments which have undergone low-temperature alteration in the oceanic crust and in burial metamorphic sequences.

Doping effects on the geometric and electronic structure of tin clusters

2019 ◽  
Vol 21 (44) ◽  
pp. 24478-24488 ◽  
Author(s):  
Martin Gleditzsch ◽  
Marc Jäger ◽  
Lukáš F. Pašteka ◽  
Armin Shayeghi ◽  
Rolf Schäfer
Keyword(s):  
Density Functional Theory ◽  
Electronic Structure ◽  
Density Functional ◽  
Beam Deflection ◽  
Functional Theory ◽  
Doping Effects ◽  
Depth Analysis ◽  
Trajectory Simulations

In depth analysis of doping effects on the geometric and electronic structure of tin clusters via electric beam deflection, numerical trajectory simulations and density functional theory.

iPLEDGE Remains A Work in Progress

2007 ◽  
Vol 38 (3) ◽  
pp. 1-92
Author(s):  
CHRISTINE KILGORE
Keyword(s):  
Work In Progress
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