scholarly journals Liouville's theorem and quantum mechanics - time quantization and reality

2020 ◽  
Vol 9 ◽  
pp. 395
Author(s):  
C. Syros ◽  
G. S. Ioannidis ◽  
G. Raptis
Keyword(s):  
Distribution Function ◽  
Quantum Mechanics ◽  
Quantum Theory ◽  
Phase Space ◽  
Statistical Mechanics ◽  
Liouville Equation ◽  
Time Variable ◽  
Planck Constant ◽  
Quantum Numbers ◽  
Liouville’S Theorem

The chrono-topology, as introduced axiomatically in a different context, is also supported by Liouville's theorem of statistical mechanics. It is shown that, if time is quantized, the distribution function (d.f.) becomes real. An elementary solution, g, of the classical Liouville equation has been found in phase-space and time, which can be used to construct any differentiable d.f, F(g), satisfying the same Liouville equation. The conditions imposed on F(g) are reality and additivity. The reality requirement, {Im F(g)=0) quantizes: (i) F(g) and makes it time-independent, (ii). The time variable, (iii) The energy. As a verification of chronotopology, the Planck constant h has been calculated on the basis of the time quantization. The d.f. F(g) becomes, after the time quantization, a real generalized Maxwell-Boltzmann d.f, F(g) = exp[g(p, g; l1,l2,..,lN)], depending on Ν quantum numbers. These facts are significant for quantum theory, because they uncover an intrinsic relationship between Liouville's theorem and quantum mechanics.

Quantum Theory

2017 ◽  
Author(s):  
Frank S. Levin
Keyword(s):  
Quantum Mechanics ◽  
Quantum Theory ◽  
Uncertainty Principle ◽  
Quantum Spin ◽  
Quantum States ◽  
Wave Functions ◽  
Symbolic Representations ◽  
Quantum Numbers ◽  
The Subject ◽  
Heisenberg's Uncertainty Principle

The subject of Chapter 8 is the fundamental principles of quantum theory, the abstract extension of quantum mechanics. Two of the entities explored are kets and operators, with kets being representations of quantum states as well as a source of wave functions. The quantum box and quantum spin kets are specified, as are the quantum numbers that identify them. Operators are introduced and defined in part as the symbolic representations of observable quantities such as position, momentum and quantum spin. Eigenvalues and eigenkets are defined and discussed, with the former identified as the possible outcomes of a measurement. Bras, the counterpart to kets, are introduced as the means of forming probability amplitudes from kets. Products of operators are examined, as is their role underpinning Heisenberg’s Uncertainty Principle. A variety of symbol manipulations are presented. How measurements are believed to collapse linear superpositions to one term of the sum is explored.

DOES LIOUVILLE'S THEOREM IMPLY QUANTUM MECHANICS?

1999 ◽  
Vol 13 (02) ◽  
pp. 161-189
Author(s):  
C. SYROS
Keyword(s):  
Quantum Mechanics ◽  
Radiation Spectrum ◽  
Black Body ◽  
Boltzmann Distribution ◽  
Black Body Radiation ◽  
Planck Constant ◽  
Klein Gordon ◽  
Energy Variable ◽  
Liouville’S Theorem ◽  
Liouville’S Equation

The essentials of quantum mechanics are derived from Liouville's theorem in statistical mechanics. An elementary solution, g, of Liouville's equation helps to construct a differentiable N-particle distribution function (DF), F(g), satisfying the same equation. Reality and additivity of F(g): (i) quantize the time variable; (ii) quantize the energy variable; (iii) quantize the Maxwell–Boltzmann distribution; (iv) make F(g) observable through time-elimination; (v) produce the Planck constant; (vi) yield the black-body radiation spectrum; (vii) support chronotopology introduced axiomatically; (viii) the Schrödinger and the Klein–Gordon equations follow. Hence, quantum theory appears as a corollary of Liouville's theorem. An unknown connection is found allowing the better understanding of space-times and of these theories.

scholarly journals General Quantum Theory: II ----Measuring & Identical Theorems, Origins & Classifications of Entanglements and Solution to Crisis of Wave Collapse

2021 ◽  
Author(s):  
C. Huang ◽  
Yong-Chang Huang ◽  
Yi-You Nie
Keyword(s):  
Wave Function ◽  
Quantum Mechanics ◽  
Quantum Theory ◽  
Statistical Mechanics ◽  
Particle System ◽  
Square Root ◽  
Classical Statistical Mechanics ◽  
Wave Particle Duality ◽  
Wave Collapse ◽  
Modern Physics

This paper derives measurement and identical principles, then makes the two principles into measurement and identical theorems of quantum mechanics, plus the three theorems derived earlier, we deduce the axiom system of current quantum mechanics, the general quantum theory no axiom presumptions not only solves the crisis to understand in current quantum mechanics, but also obtains new discoveries, e.g., discovers the velocities of quantum collapse and entanglement are instantaneously infinitely large. We deduce the general Schrȍdinger equation of any n particles from two aspects, and the wave function not only has particle properties of the complex square root state vector of the classical probability density of any n particles, but also has the plane wave properties of any n particles. Thus, the current crisis of the dispute about the origin of wave- particle duality of any n microscopic particles is solved. We display the classical locality and quantum non-locality for any n particle system, show entanglement origins, and discover not only any n-particle wave function system has the original, superposition and across entanglements, but also the entanglements are of interactions preserving conservation or correlation, three kinds of entanglements directly give lots of entanglement sources. This paper discovers, one of two pillars of modern physics, quantum mechanics of any n particle system is a generalization ( mechanics ) theory of the complex square root ( of real density function ) of classical statistical mechanics, any n particle system’s quantum mechanics of being just a generalization theory of the complex square root of classical statistical mechanics is both a revolutionary discovery and key new physics, which are influencing people’s philosophical thinking for modern physics, solve all the crisises in current quantum theories, quantum information and so on, and make quantum theory have scientific solid foundations checked, no basic axiom presumption and no all quantum strange incomprehensible properties, because classical statistical mechanics and its complex square root have scientific solid foundations checked. Thus, all current studies on various entanglements and their uses to quantum computer, quantum information and so on must be further updated and classified by the new entanglements. This and our early papers derive quantum physics, solve all crisises of basses of quantum mechanics, e.g., wave-particle duality & the first quantization origins, quantum nonlocality, entanglement origins & classifications, wave collapse and so on.Key words: quantum mechanics, operator, basic presumptions, wave-particle duality, principle of measurement, identical principle, superposition principle of states, entanglement origin, quantum communication, wave collapse, classical statistical mechanics, classical mechanics

Phase-space dynamics and ergodicity

2021 ◽  
pp. 79-98
Author(s):  
James P. Sethna
Keyword(s):  
Phase Space ◽  
Statistical Mechanics ◽  
Dissipative Systems ◽  
Planetary Motion ◽  
Equilibrium Statistical Mechanics ◽  
Average Behavior ◽  
Time Average ◽  
Space Dynamics ◽  
Mathematical Justification ◽  
Liouville’S Theorem

This chapter provides the mathematical justification for the theory of equilibrium statistical mechanics. A Hamiltonian system which is ergodic is shown to have time-average behavior equal to the average behavior in the energy shell. Liouville’s theorem is used to justify the use of phase-space volume in taking this average. Exercises explore the breakdown of ergodicity in planetary motion and in dissipative systems, the application of Liouville’s theorem by Crooks and Jarzynski to non-equilibrium statistical mechanics, and generalizations of statistical mechanics to chaotic systems and to two-dimensional turbulence and Jupiter’s great red spot.

scholarly journals Irreversibility in quantum mechanics

2004 ◽  
Vol 2004 (1) ◽  
pp. 75-83 ◽  
Author(s):  
R. C. Bishop ◽  
A. Bohm ◽  
M. Gadella
Keyword(s):  
Quantum Mechanics ◽  
Quantum Theory ◽  
Boundary Conditions ◽  
Liouville Equation ◽  
Quantum Systems ◽  
Arrow Of Time ◽  
Time Asymmetry ◽  
Classical Physics ◽  
Von Neumann ◽  
Signal Features

Time asymmetry and irreversibility are signal features of our world. They are the reason of our aging and the basis for our belief that effects are preceded by causes. These features have many manifestations called arrows of time. In classical physics, some of these arrows are described by the increase of entropy or probability, and others by time-asymmetric boundary conditions of time-symmetric equations (e.g., Maxwell or Einstein). However, there is some controversy over whether probability or boundary conditions are more fundamental. For quantum systems, entropy increase is usually associated with the effects of an environment or measurement apparatus on a quantum system and is described by the von Neumann-Liouville equation. But since the traditional (von Neumann) axioms of quantum mechanics do not allow time-asymmetric boundary conditions for the dynamical differential equations (Schrödinger or Heisenberg), there is no quantum analogue of the radiation arrow of time. In this paper, we review consequences of a modification of a fundamental axiom of quantum mechanics. The new quantum theory is time asymmetric and accommodates an irreversible time evolution of isolated quantum systems.

Complementary variational principles for ∇2φ = f(φ) with applications to the Thomas–Fermi and Liouville equations

1969 ◽  
Vol 65 (2) ◽  
pp. 535-541 ◽  
Author(s):  
A. M. Arthurs ◽  
P. D. Robinson
Keyword(s):  
Quantum Mechanics ◽  
Statistical Mechanics ◽  
Boundary Value Problem ◽  
Liouville Equation ◽  
Variational Principles ◽  
Boundary Value ◽  
Equilibrium Statistical Mechanics ◽  
Thomas Fermi

AbstractComplementary variational principles associated with the boundary-value problem ∇2φ = f(φ) in ∇,φ = φ0 on the boundary of ∇, are presented. The theory is applied to the Thomas–Fermi equation in quantum mechanics and the Liouville equation in equilibrium statistical mechanics.

scholarly journals Elliptical orbits in the phase-space quantization

2016 ◽  
Vol 38 (3) ◽  
Author(s):  
Leonardo Andreta de Castro ◽  
Carlos Alexandre Brasil ◽  
Reginaldo de Jesus Napolitano
Keyword(s):  
Quantum Mechanics ◽  
Quantum Theory ◽  
Phase Space ◽  
Energy Levels ◽  
Radial Component ◽  
Original Method ◽  
Direct Integration ◽  
Old Quantum Theory ◽  
Complex Integration ◽  
Elliptical Orbits

The energy levels of hydrogen-like atoms are obtained from the phase-space quantization, one of the pillars of the old quantum theory, by three different methods - (i) direct integration, (ii) Sommerfeld's original method, and (iii) complex integration. The difficulties come from the imposition of elliptical orbits to the electron, resulting in a variable radial component of the linear momentum. Details of the calculation, which constitute a recurrent gap in textbooks that deal with phase-space quantization, are shown in depth in an accessible fashion for students of introductory quantum mechanics courses.

The formulation of quantum mechanics in terms of phase space functions

1964 ◽  
Vol 60 (3) ◽  
pp. 581-586 ◽  
Author(s):  
D. B. Fairlie
Keyword(s):  
Wave Function ◽  
Distribution Function ◽  
Quantum Mechanics ◽  
Phase Space ◽  
Schrödinger Equation ◽  
Time Dependence ◽  
Schrodinger Equation ◽  
Average Energy ◽  
Weighted Mean

AbstractA relationship between the Hamiltonian of a system and its distribution function in phase space is sought which will guarantee that the average energy is the weighted mean of the Hamiltonian over phase space. This relationship is shown to imply the existence of a wave function satisfying the Schrödinger equation, and dictates the possible forms of time-dependence of the distribution function.

COULD FRESNEL OPTICS BE QUANTUM MECHANICS IN PHASE SPACE?

2004 ◽  
Vol 04 (01) ◽  
pp. L43-L51 ◽  
Author(s):  
O. CRASSER ◽  
H. MACK ◽  
W. P. SCHLEICH
Keyword(s):  
Distribution Function ◽  
Quantum Mechanics ◽  
Phase Space ◽  
Space Distribution ◽  
Quantum Mechanical ◽  
Phase Space Distribution ◽  
Intimate Connection

We formulate and argue in favor of the following conjecture: There exists an intimate connection between Wigner's quantum mechanical phase space distribution function and classical Fresnel optics.

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