81. Concepts: Statistical Mechanics made of. Statistical mechanics averages properties of particles tofind the properties of the material they form. For example, the http://necsi.org/guide/concepts/statisticalmechanics.html

Concepts in Complex Systems Yaneer Bar-Yam Statistical Mechanics Statistical mechanics begins as an effort to explain the macroscopic laws of thermodynamics by considering the microscopic application of Newton's laws to the particles that a material is made of. Statistical mechanics averages properties of particles to find the properties of the material they form. For example, the temperature of a gas is found to be related to the random motion of the gas molecules. The faster they move on average, the higher the temperature. Heat transfer is the transfer of Newtonian energy of the particles of one object to the particles of the other object. In this way, the statistical treatment of the many particles of a material, with a key set of assumptions, reveals that thermodynamic laws are a natural consequence of many microscopic particles interacting with each other. Since the study of complex systems is about understanding the relationship of the behavior of parts to the behavior of a system as a whole, many of the tools developed in statistical mechanics are useful in the study of complex systems. Related concepts: thermodynamics Newtonian mechanics , separation of scales, phase transitions.

82. 2.3 Identical Particles Figure 2.1 Indistinguishable particles in quantum mechanics (left) initially thereare two particles at A and B, later on two particles are found at C and D http://www.tcm.phy.cam.ac.uk/~pdh1001/thesis/node14.html

Next: 2.4 Variational principles Up: 2. Many-body Quantum Mechanics Previous: 2.2 The Born-Oppenheimer approximation Contents Subsections 2.3.1 Symmetries 2.3.2 Spin and statistics 2.3 Identical particles 2.3.1 Symmetries It is a consequence of quantum mechanics, usually expressed in the terms of the Heisenberg uncertainty principle that, in contrast to Newtonian mechanics, the trajectory of a particle is undefined. When dealing with identical particles this leads to complications, as illustrated in figure Figure 2.1: Indistinguishable particles in quantum mechanics: (left) initially there are two particles at A and B, later on two particles are found at C and D; (middle) but we cannot be certain whether the particles travelled from A to D and B to C or (right) from A to C and B to D, because they are identical. Consider a system of two identical particles represented by the wave-function and a particle-exchange operator which swaps the particles i.e. However, since the system must be unchanged by such an exchange of identical particles, the two states appearing in equation must be the same and hence differ only by a multiplicative complex constant;

83. BUBL LINK: 530.1 Quantum Mechanics, Quantum Physics particle physics, quantum physics DeweyClass 530.12 ResourceType documents Locationuk particles, Special Relativity and Quantum mechanics Collection of http://link.bubl.ac.uk/ISC6877

BUBL LINK Catalogue of selected Internet resources Home Search Subject Menus A-Z ... About 530.1 Quantum mechanics, quantum physics Titles Descriptions Brain Project: Quantum Physics Do Quantum Particles Have a Structure? Unified Reality Theory: The Evolution of Existence into Experience Introduction to Coupled-Cluster Theory ... Transactional Interpretation of Quantum Mechanics All links checked August 2001 Comments: bubl@bubl.ac.uk Brain Project: Quantum Physics Essays on issues relating to the nature of consciousness, including neuro-physiology, quantum physics and neural networks. Author: Stephen Jones Subjects: consciousness, neural networks, quantum physics DeweyClass: ResourceType: articles, essays Location: australia Do Quantum Particles Have a Structure? Discussion of the philosophical foundations and interpretation of quantum mechanics and the Special Theory of Relativity. Author: TS Natarajan, Indian Institute of Technology Subjects: quantum physics, relativity DeweyClass: ResourceType: article Location: usa Unified Reality Theory: The Evolution of Existence into Experience Online book, which attempts to provide a model of reality by exploring a relational matrix model of space-time, and an explanation of why physical reality behaves as it does, as well as answering questions about light, space, time and energy. Author: Steven E Kaufman Subjects: cosmology, quantum physics

84. STATISTICAL MECHANICS AND THERMODYNAMICS Statistical mechanics uses laws of probability to compute AVERAGE propertiesof both ordinary materials and of exotic elementary particles. http://cannon.sfsu.edu/~gmarcy/smgraphical.html

STATISTICAL MECHANICS AND THERMODYNAMICS PHYSICS 370 A GRAPHICAL TOUR A typical amount of material, let's say 1 gram, contains about 6x10^23 molecules: Avogadro's number. ...Continued Below... The water molecules shown above are randomly placed and randomly oriented. Statistical Mechanics uses laws of probability to compute AVERAGE properties of both ordinary materials and of exotic elementary particles. At the root of Stat. Mech. is the Heisenberg Uncertainty Principle. Macroscopic Quantities are: Energy, Entropy (S), Enthalpy (H), Gibbs Free Energy (G), etc. Pressure is the result of the molecules all imparting momentum to a wall. Stat. Mech. is an easy way to treat a very complicated quantum mechanical problem of many particles. The Fundamental Postulate of Statistical Mechanics. Each particle can have a different energy - often quantized, as shown below. Three Systems (Left, Middle, Right) each having available energy levels. The particles in each system are distributed differently among the energy levels. In the laboratory, we can't measure the indiviual energy levels of each particle. Instead we can only measure macroscopic quantities, such as N, V, T, P.

85. Prof. Jasprit Singh's Publications TABLE OF CONTENTS A JOLT FOR CLASSICAL PHYSICS. THE MATHEMATICAL FORMULATIONOF QUANTUM mechanics. particles IN SIMPLE POTENTIALS. THE TUNNELING PROBLEM. http://www.eecs.umich.edu/~singh/qm97.html

QUANTUM MECHANICS: Fundamentals and Applications to Technology Jasprit Singh Cover art and layout by Teresa Singh John Wiley and Sons, Inc. (1997) ISBN: 0-471-15758-9 Explore the relationship between quantum mechanics and information-age applications. Quantum Mechanics: Fundamentals and Applications to Technology is an excellent text for senior undergraduate and graduate level students, and a helpful reference for practicing scientists, engineers, and chemists in the semiconductor and electronic industries. TABLE OF CONTENTS A JOLT FOR CLASSICAL PHYSICS THE MATHEMATICAL FORMULATION OF QUANTUM MECHANICS PARTICLES IN SIMPLE POTENTIALS THE TUNNELING PROBLEM PARTICLES IN SPHERICALLY SYMMETRIC POTENTIALS PHYSICAL SYMMETRIES AND CONSERVATION LAWS IDENTICAL PARTICLES AND SECOND QUANTIZATION APPROXIMATION METHODS:TIME-INDEPENDENT PROBLEMS TIME-DEPENDENT PROBLEMS: APPROXIMATION METHODS COLLISIONS AND SCATTERING MAGNETIC EFFECTS APPENDICES MODERN CLASSICAL PHYSICS: A REVIEW IMPORTANT MATHEMATICAL FUNCTIONS DENSITY MATRIX DESCRIPTION Go to my Home page

86. AMOLF Scientific Highlights 2001 hallmark of wave nature. According to quantum mechanics, microscopicparticles behave as both as waves and as particles (duality). http://www.amolf.nl/main/highlights/highlights_2001/2001_08.html

Scientific highlight of August 2001 "PHASE-FLUCTUATING 3D BOSE-EINSTEIN CONDENSATES IN ELONGATED TRAPS" D. S. Petrov, G.V. Shlyapnikov, and J.T.M. Walraven Physical Review Letters Explanation for all by Harm Geert Muller Article online Request a copy of this article Fluctuations in a cloud of ultracold atoms by Harm Geert Muller Bose-Einstein condensation is the effect that all atoms at very low temperatures get into the same 'quantum state' of lowest energy. This makes the 'condensate' behave more like a collective than as a collection of individual particles. Theoretical description of a cigar-shaped condensate at AMOLF has revealed that such a condensate has difficulty 'communicating' disturbances along its long axis, making it in fact behave as a collection of independent condensates (a so called quasi-condensate) then as a unified one. Long write-up The phenomenon that two waves not necessarily have to enhance each other is called interference, and is really the hallmark of wave nature. According to quantum mechanics, microscopic particles behave as both as waves and as particles ( duality How can it be that our perception of the macroscopic objects like marbles, is so different from the behavior of microscopic particles such as atoms? After all, the marbles are supposed to be built out of atoms. The reason is that interference effects are very susceptible to disturbance. The wave-like nature only reveals itself in statistics from the behavior of a particle, and it requires a large number of repetitions of the same experiment to get a reliable impression on how often that particle moves to one place or another. All repetitions of an experiment with a marble are likely to be performed under slightly different conditions (so they are in fact not repetitions at all). This can happen for instance because the air through which a marble moved has some tiny turbulences, or because we are technically unable to prepare the motion of all the atoms inside the marble with absolute precision.

87. Buy The Best-Selling Book Schaum's Outline Of Quantum Mechanics Buy the BestSelling Book Schaum's Outline of Quantum mechanics Master quantum mechanics with Schaumsthe high-performance study guide. It will help you cut study time, hone problem-solving skills, and achieve your personal best on exams! http://redirect-west.inktomi.com/click?u=http://www.shop-mcgraw-hill.com/mcgrawh

88. Astronomy 301: Quantum Mechanics http://hoku.as.utexas.edu/~gebhardt/a301f02/lect13.html

Quantum Mechanics The Building Block of the Universe In the early part of the 1900s, physics underwent a revolution. It was during that time that we begun to understand the atom. The properties of the atom directly helps us to understand about the Universe; how it was formed and evolved. Thus, to understand the largest structure known, we must study the smallest structures known. In 1905, Einstein showed that light behaves like a particle, called photons. In 1911, Rutherford showed that the atom was mainly empty space. In 1913, Bohr discovered that electron energies in atoms are quantized. This was the birth of QUANTUM MECHANICS. To probe these small regions requires extreme energies to break up the atom. Thus, until we were able to build giant accelerators were we able to really understand what the atom was made of. The picutre below shows an overview of Fermilab. From these atom smashers, we have found hundreds of particles. We now have a model that explains each of these particles. It is called the STANDARD MODEL; the physicist Gell-Mann was the creator of this scheme to explain the particles. The most important properties of a particle are its mass, charge and spin.

89. Wilhelm Fushchych: Biography On the basis of these results, the mathematical foundations of quantum mechanicsfor particles with variable mass and spin were constructed. http://www.imath.kiev.ua/~appmath/wifbio.html

Wilhelm Fushchych Biography Wilhelm Fushchych was born on December 18, 1936 in the village Siltze of the Zakarpattya (Transcarpathian) Region of Ukraine. He has graduated from the Uzhgorod University (1958) and finished the post-graduate course at the Institute of Mathematics (1963). He defended his Ph.D. thesis in 1964, and the doctor (Dr. Sc.) thesis in 1971. He worked at the Institute of Mathematics of the National Academy of Sciences of Ukraine since 1963, and was the head of the Department of Applied Research since 1978. He was elected a Corresponding Member of the National Academy of Sciences of Ukraine in 1987. The field of research interests of Wilhelm Fushchych included quantum field theory, representations of Lie groups and algebras, subgroup structure of Lie groups, group-theoretical analysis of differential equations, etc. However, the principal topic of his papers is symmetry in mathematical physics. Today this particular branch of mathematical physics is developing extremely quickly, and the papers and monographs by W. Fushchych have contributed considerably to this process. W. Fushchych has suggested equations of motion which have the symmetry intermediate between the Galilei and Poincaré groups. Mechanics based on these equations of motion provides the dependence of mass on velocity, but the relevant maximal velocity can exceed the velocity of light in vacuum. A new equation of motion for electromagnetic waves has also been suggested. For this equation the velocity of light in vacuum is not a constant but a nonlinear function of the strength of electric field.

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