Phase Space, Density Matrices, Energy Densities, and Exchange Holes

2001 ◽  
Vol 215 (10) ◽  
Author(s):  
M. Springborg ◽  
J. P. Perdew ◽  
K. Schmidt
Keyword(s):  
Quantum Mechanics ◽  
Phase Space ◽  
Momentum Space ◽  
Classical Mechanics ◽  
Quantum Mechanical ◽  
Phase Space Density ◽  
Density Matrices ◽  
Space Density

In the general case, quantum-mechanical quantities are represented by operators in position- or momentum-space representations, but in phase space they are represented by functions. The correspondence between classical mechanics and quantum mechanics is non-unique as a consequence of [

scholarly journals QUANTUM-MECHANICAL DUALITIES ON THE TORUS

2004 ◽  
Vol 19 (23) ◽  
pp. 1733-1744 ◽  
Author(s):  
JOSÉ M. ISIDRO
Keyword(s):  
Quantum Mechanics ◽  
Phase Space ◽  
Classical Mechanics ◽  
Quantum Mechanical ◽  
Differentiable Structure ◽  
Recent Developments ◽  
The Creation ◽  
Classical Phase Space ◽  
Creation Operators

On classical phase spaces admitting just one complex-differentiable structure, there is no indeterminacy in the choice of the creation operators that create quanta out of a given vacuum. In these cases the notion of a quantum is universal, i.e. independent of the observer on classical phase space. Such is the case in all standard applications of quantum mechanics. However, recent developments suggest that the notion of a quantum may not be universal. Transformations between observers that do not agree on the notion of an elementary quantum are called dualities. Classical phase spaces admitting more than one complex-differentiable structure thus provide a natural framework to study dualities in quantum mechanics. As an example we quantise a classical mechanics whose phase space is a torus and prove explicitly that it exhibits dualities.

Quantization and Group Representations

1977 ◽  
Vol 29 (6) ◽  
pp. 1264-1276 ◽  
Author(s):  
R. Cressman
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Phase Space ◽  
Mechanical System ◽  
Correspondence Principle ◽  
Classical Mechanics ◽  
Quantum Mechanical ◽  
Group Representations ◽  
Additional Requirement ◽  
Adjoint Operators

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.

On the error of approximations in quantum mechanics. I. General theory

1965 ◽  
Vol 257 (1081) ◽  
pp. 309-326 ◽  
Keyword(s):  
Quantum Mechanics ◽  
General Theory ◽  
Computer Applications ◽  
Quantum Mechanical ◽  
Quantum Mechanical Calculations ◽  
Density Matrices ◽  
Single Error

General formulas for estimating the errors in quantum-mechanical calculations are given in the formalism of density matrices. Some properties of the traces of matrices are used to simplify the estimating and to indicate a way of obtaining a better approximation. It is shown that the simultaneous correction of all the equations to be fulfilled leads in most cases to a faster convergence than the exact fulfilment of some of the equations and approximating stepwise to some of the others. The corrective formulas contain only direct operations of the matrices occurring and so they are advantageous in computer applications. In the last section a ‘subjective error’ definition is given and by taking into account the weight of the errors of the several equations a faster convergence and a single error quantity is obtained. Some special applications of the method will be published later.

scholarly journals Cooling of Sr to high phase-space density by laser and sympathetic cooling in isotopic mixtures

2006 ◽  
Vol 73 (2) ◽  
Author(s):  
G. Ferrari ◽  
R. E. Drullinger ◽  
N. Poli ◽  
F. Sorrentino ◽  
G. M. Tino
Keyword(s):  
Phase Space ◽  
Phase Space Density ◽  
Sympathetic Cooling ◽  
Space Density ◽  
High Phase

scholarly journals Comparison of energetic electron flux and phase space density in the magnetosheath and in the magnetosphere

2012 ◽  
Vol 117 (A5) ◽  
pp. n/a-n/a ◽  
Author(s):  
Bingxian Luo ◽  
Xinlin Li ◽  
Weichao Tu ◽  
Jiancun Gong ◽  
Siqing Liu
Keyword(s):  
Phase Space ◽  
Electron Flux ◽  
Energetic Electron ◽  
Phase Space Density ◽  
Space Density

scholarly journals Phase-space density in heavy-ion collisions revisited

2003 ◽  
Vol 30 (4) ◽  
pp. 517-523 ◽  
Author(s):  
Q. H. Zhang ◽  
J. Barrette ◽  
C. Gale
Keyword(s):  
Phase Space ◽  
Heavy Ion Collisions ◽  
Heavy Ion ◽  
Phase Space Density ◽  
Space Density ◽  
Ion Collisions

Phase space density analysis of outer radiation belt electron energization and loss during geoeffective and non-geoeffective sheath regions

2021 ◽  
Author(s):  
Milla Kalliokoski ◽  
Emilia Kilpua ◽  
Adnane Osmane ◽  
Allison Jaynes ◽  
Drew Turner ◽  
...  
Keyword(s):  
Phase Space ◽  
Energetic Electron ◽  
Electron Content ◽  
Phase Space Density ◽  
Geomagnetic Disturbances ◽  
Interplanetary Coronal Mass Ejections ◽  
Outer Radiation Belt ◽  
Space Density ◽  
Sheath Region ◽  
Chorus Waves

<p>The energetic electron content in the Van Allen radiation belts surrounding the Earth can vary dramatically on timescales from minutes to days, and these electrons present a hazard for spacecraft traversing the belts. The outer belt response to solar wind driving is however yet largely unpredictable. Here we investigate the driving of the belts by sheath regions preceding interplanetary coronal mass ejections. Electron dynamics in the belts is governed by various competing acceleration, transport and loss processes. We analyzed electron phase space density to compare the energization and loss mechanisms during a geoeffective and a non-geoeffective sheath region. These two case studies indicate that ULF-driven inward and outward radial transport, together with the incursions of the magnetopause, play a key role in causing the outer belt electron flux variations. Chorus waves also likely contribute to energization during the geoeffective event. A global picture of the wave activity is achieved through a chorus proxy utilizing POES measurements. We highlight that also the non-geoeffective sheath presented distinct changes in outer belt electron fluxes, which is also evidenced by our statistical study of outer belt electron fluxes during sheath events. While not as intense as during geoeffective sheaths, significant changes in outer belt electron fluxes occur also during sheaths that do not cause major geomagnetic disturbances.</p>

scholarly journals An analysis on Quantum mechanical Stability of Regular Polygons on a Point Base Using Heisenberg Uncertainty Principle

2021 ◽  
Vol 9 (10) ◽  
pp. 74-79
Author(s):  
Anurag Chapagain
Keyword(s):  
Quantum Mechanics ◽  
Uncertainty Principle ◽  
Stability Problem ◽  
Classical Mechanics ◽  
Quantum Mechanical ◽  
Heisenberg Uncertainty Principle ◽  
Heisenberg Uncertainty ◽  
Degree Of Stability ◽  
The Stability ◽  
Heisenberg's Uncertainty Principle

Abstract: It is a well-known fact in physics that classical mechanics describes the macro-world, and quantum mechanics describes the atomic and sub-atomic world. However, principles of quantum mechanics, such as Heisenberg’s Uncertainty Principle, can create visible real-life effects. One of the most commonly known of those effects is the stability problem, whereby a one-dimensional point base object in a gravity environment cannot remain stable beyond a time frame. This paper expands the stability question from 1- dimensional rod to 2-dimensional highly symmetrical structures, such as an even-sided polygon. Using principles of classical mechanics, and Heisenberg’s uncertainty principle, a stability equation is derived. The stability problem is discussed both quantitatively as well as qualitatively. Using the graphical analysis of the result, the relation between stability time and the number of sides of polygon is determined. In an environment with gravity forces only existing, it is determined that stability increases with the number of sides of a polygon. Using the equation to find results for circles, it was found that a circle has the highest degree of stability. These results and the numerical calculation can be utilized for architectural purposes and high-precision experiments. The result is also helpful for minimizing the perception that quantum mechanical effects have no visible effects other than in the atomic, and subatomic world. Keywords: Quantum mechanics, Heisenberg Uncertainty principle, degree of stability, polygon, the highest degree of stability

scholarly journals NIHAO project II: halo shape, phase-space density and velocity distribution of dark matter in galaxy formation simulations

2016 ◽  
Vol 462 (1) ◽  
pp. 663-680 ◽  
Author(s):  
Iryna Butsky ◽  
Andrea V. Macciò ◽  
Aaron A. Dutton ◽  
Liang Wang ◽  
Aura Obreja ◽  
...  
Keyword(s):  
Dark Matter ◽  
Phase Space ◽  
Velocity Distribution ◽  
Galaxy Formation ◽  
Phase Space Density ◽  
Space Density
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