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.

The Physical World

2017 ◽  
Author(s):  
Nicholas Manton ◽  
Nicholas Mee
Keyword(s):  
Quantum Mechanics ◽  
Particle Physics ◽  
Variational Principles ◽  
Black Body ◽  
Black Body Radiation ◽  
Physical World ◽  
Atomic Scale ◽  
Solid State Physics ◽  
Least Action ◽  
Dimensional Spacetime

The book is an inspirational survey of fundamental physics, emphasizing the use of variational principles. Chapter 1 presents introductory ideas, including the principle of least action, vectors and partial differentiation. Chapter 2 covers Newtonian dynamics and the motion of mutually gravitating bodies. Chapter 3 is about electromagnetic fields as described by Maxwell’s equations. Chapter 4 is about special relativity, which unifies space and time into 4-dimensional spacetime. Chapter 5 introduces the mathematics of curved space, leading to Chapter 6 covering general relativity and its remarkable consequences, such as the existence of black holes. Chapters 7 and 8 present quantum mechanics, essential for understanding atomic-scale phenomena. Chapter 9 uses quantum mechanics to explain the fundamental principles of chemistry and solid state physics. Chapter 10 is about thermodynamics, which is built around the concepts of temperature and entropy. Various applications are discussed, including the analysis of black body radiation that led to the quantum revolution. Chapter 11 surveys the atomic nucleus, its properties and applications. Chapter 12 explores particle physics, the Standard Model and the Higgs mechanism, with a short introduction to quantum field theory. Chapter 13 is about the structure and evolution of stars and brings together material from many of the earlier chapters. Chapter 14 on cosmology describes the structure and evolution of the universe as a whole. Finally, Chapter 15 discusses remaining problems at the frontiers of physics, such as the interpretation of quantum mechanics, and the ultimate nature of particles. Some speculative ideas are explored, such as supersymmetry, solitons and string theory.

Constructing Quantum Mechanics

2019 ◽  
Author(s):  
Anthony Duncan ◽  
Michel Janssen
Keyword(s):  
Quantum Mechanics ◽  
Quantum Theory ◽  
Stark Effect ◽  
Zeeman Effect ◽  
Black Body ◽  
Black Body Radiation ◽  
Wave Mechanics ◽  
Specific Heats ◽  
Old Quantum Theory ◽  
Upper Level

This is the first of two volumes on the genesis of quantum mechanics. It covers the key developments in the period 1900–1923 that provided the scaffold on which the arch of modern quantum mechanics was built in the period 1923–1927 (covered in the second volume). After tracing the early contributions by Planck, Einstein, and Bohr to the theories of black‐body radiation, specific heats, and spectroscopy, all showing the need for drastic changes to the physics of their day, the book tackles the efforts by Sommerfeld and others to provide a new theory, now known as the old quantum theory. After some striking initial successes (explaining the fine structure of hydrogen, X‐ray spectra, and the Stark effect), the old quantum theory ran into serious difficulties (failing to provide consistent models for helium and the Zeeman effect) and eventually gave way to matrix and wave mechanics. Constructing Quantum Mechanics is based on the best and latest scholarship in the field, to which the authors have made significant contributions themselves. It breaks new ground, especially in its treatment of the work of Sommerfeld and his associates, but also offers new perspectives on classic papers by Planck, Einstein, and Bohr. Throughout the book, the authors provide detailed reconstructions (at the level of an upper‐level undergraduate physics course) of the cental arguments and derivations of the physicists involved. All in all, Constructing Quantum Mechanics promises to take the place of older books as the standard source on the genesis of quantum mechanics.

Notizen: On Phase Space and Statistics of Continua

1986 ◽  
Vol 41 (10) ◽  
pp. 1258-1260
Author(s):  
H. Tasso
Keyword(s):  
Phase Space ◽  
Radiation Spectrum ◽  
Vries Equation ◽  
Discrete Systems ◽  
Black Body ◽  
Black Body Radiation ◽  
De Vries

Problems in introducing suitable phase space and statistics occur for continua and degenerate discrete systems. The solution of these problems for the Korteweg-de Vries equation is discussed. The classical removal of the ultraviolet catastrophe in this case is contrasted with Planck’s black-body radiation spectrum.

scholarly journals The Puzzling of Zero-Point Energy Contribution to Black-Body Radiation Spectrum: The Role of Casimir Force

2017 ◽  
Vol 16 (04) ◽  
pp. 1771002 ◽  
Author(s):  
L. Reggiani ◽  
E. Alfinito
Keyword(s):  
Casimir Force ◽  
Radiation Spectrum ◽  
Mechanical Stability ◽  
Zero Point ◽  
Direct Consequence ◽  
Black Body ◽  
Thermal Fluctuations ◽  
Black Body Radiation ◽  
Point Energy ◽  
Nyquist Noise

The role played by zero-point contribution in black-body radiation spectrum is investigated in connection with the presence of Casimir force. We assert that once mechanical stability for the physical system is established, there is no further role for zero-point contribution to the spectrum in full agreement with experimental evidence. As a direct consequence, Johnson–Nyquist noise in dissipative conductors, should be interpreted just in terms of thermal fluctuations only, thus neglecting quantum fluctuations predicted by [H. Callen and T. Welton, Irreversibility and generalized noise, Phys. Rev. 83 (1951) 34]. Casimir force between opposite metallic plates can be independently measured by its equilibration through application of a mechanical force and measuring it at a mechanical equilibrium.

scholarly journals Simulation of Radiation Calculation of Black Body by Using the Interpolation Method

2019 ◽  
Vol 5 (1) ◽  
pp. 23
Author(s):  
Feli Cianda Adrin Burhendi ◽  
Rizky Dwi Siswanto ◽  
Wahyu Dian Laksanawati
Keyword(s):  
Programming Languages ◽  
Radiation Spectrum ◽  
Interpolation Method ◽  
Line Graph ◽  
Black Body ◽  
Black Body Radiation ◽  
Small Error ◽  
Light Radiation ◽  
Computational Processes

Simulation of radiation calculation of black body by using the interpolation method is designed to facilitate the determination of radiation in black matter efficiency. Fortran programming languages are chosen for computational processes. The calculation program that has been designed is able to calculate the efficiency of black body radiation easily and quickly with a fairly small error rate of 0.5\%. The light radiation spectrum of objects is around 1000, 1100, 1200, and 1300 $^{\circ}$C. The $x$ axis shows the wavelength, while the $y$ axis shows the intensity or strength of light. If we pay attention to the curvature of 1000 $^{\circ}$C, along with the increasing frequency of light, the intensity of light is also getting stronger aka more bright. But at certain light frequencies, the line reaches the peak, and after that the light intensity drops dramatically. At temperatures of 1200 $^{\circ}$C and 1300 $^{\circ}$C, even though the temperature rises, the outline of the line graph is similar to the line 1000 $^{\circ}$C. This is in accordance with the existing theoretical and experimental results.

Are smaller microplastics missing?

2020 ◽  
Author(s):  
Kunihiro Aoki ◽  
Ryo Furue
Keyword(s):  
Size Distribution ◽  
Gradual Increase ◽  
Black Body ◽  
Boltzmann Distribution ◽  
Black Body Radiation ◽  
Rapid Decrease ◽  
Size Distributions ◽  
Biological Uptake ◽  
Data Source ◽  
Planck’S Law

Abstract The size distribution of marine microplastics (< 5 mm) provides a fundamental data source for understanding the dispersal, break down, and biotic impacts of the microplastics in the ocean. The observed size distribution generally shows, from large to small sizes, a gradual increase followed by a rapid decrease. This decrease has led to the hypothesis that the smallest fragments are selectively removed by sinking or biological uptake. Here we propose a new model of size distribution without any removal of material from the system. The model uses an analogy with black-body radiation and the resultant size distribution is analogous to Planck's law. In this model, the original large plastic piece is broken into smaller pieces once by the application of “energy” or work by waves or other processes, under two assumptions, one that fragmentation into smaller pieces requires larger energy and the other that the probability distribution of the “energy” follows the Boltzmann distribution. Our formula well reproduces observed size distributions over wide size ranges from micro- (< 5 mm) to mesoplastics ( > 5 mm). According to this model, the smallest fragments are fewer because large “energy” required to produce such small fragments occurs more rarely.

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.

Black-Body Radiation

2019 ◽  
pp. 322-330
Author(s):  
Robert H. Swendsen
Keyword(s):  
Quantum Mechanics ◽  
Energy Spectrum ◽  
Electromagnetic Radiation ◽  
Black Body ◽  
Black Body Radiation ◽  
Perfect Absorber ◽  
Max Planck ◽  
Planck’S Constant ◽  
Light Waves ◽  
Planck's Constant

A black body is a perfect absorber of electromagnetic radiation. The energy spectrum was correctly calculated by Max Planck under the assumption that the energy of light waves only came in discrete multiples of a constant (called Planck’s constant) times the frequency. This was perhaps the first achievement of quantum mechanics. The derivation is presented here. The purpose of the current chapter is to calculate the spectrum of radiation emanating from a black body. The calculation was originally carried out by Max Planck in 1900 and published the following year. This was before quantum mechanics had been invented, or perhaps it could be regarded the first step in its invention.

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