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         Mechanics Of Particles:     more books (100)
  1. Dynamics of Bubbles, Drops and Rigid Particles (Fluid Mechanics and Its Applications) by Z. Zapryanov, S. Tabakova, 1998-12-31
  2. Quantum Mechanics, Determinism, Causality and Particles: An International Collection of Contributions in Honour of Louis de Broglie on the Occasion of ... Physics and Applied Mathematics)
  3. Advanced Quantum Mechanics and Particle Physics Volume 1 by John A. Eisele, 1964
  4. Mechanics of Particles and Rigid Bodies. by J Prescott, 1947
  5. The Dynamics of Fluidized Particles (Cambridge Monographs on Mechanics) by Roy Jackson, 2000-09-04
  6. Modern Quantum Mechanics with Applications to Elementary Particle Physics by J.A. Eisele, 1969-08
  7. Meshfree & Particle Based Approaches in Computational Mechanics
  8. Hadronic Mechanics and Nonpotential Interactions: Mathematics by International Conference on Hadronic Mechanics and Nonpotential Interactions (5th : 1990 : University of Northern Iowa), 1992-09
  9. Nonlinear Dynamics Aspects of Particle Accelerators: Proceedings of the Joint Us-Cern School on Particle Accelerat (Lecture Notes in Physics)
  10. Proceedings of the Fifth International Conference on Hadronic Mechanics and Nonpotential Interactions Held at the University of Northern Iowa, Cedar: Held ... Iowa, Cedar Falls, Iowa, August 13-17, 1990 by International Conference on Hadronic Mechanics and Nonpotential Interactions (5th : 1990 : University of Northern Iowa), 1992-09
  11. Elementary-Particle Physics (<i>Physics Through the 1990s:</i> A Series) by Elementary-Particle Physics Panel, Physics Survey Committee, et all 1986-01-01
  12. Schaum's Outline of Quantum Mechanics (Schaum's) by Eliahu Zaarur, Phinik Reuven, 1998-04-30
  13. Wave-Particle Duality
  14. Statistical mechanics of charged particles (Monographs in statistical physics and thermodynamics) by Radu Balescu, 1963

61. KryssTal : An Introduction To Quantum Mechanics
separately worked out the mathematics of Quantum mechanics. Using this new theory,scientists could understand the behaviour of atoms and subatomic particles.
An Introduction to
Quantum Mechanics
A beginners' (non-mathematical) guide to the strange world of the atom
Part One - The Story of The Atom
In the essay on Relativity , I stated that the Theory of Relativity was one of the two most important ideas of 20th Century science. Relativity is a deviation from Newtonian Mechanics (also known as common sense!). The deviations were not discovered until this Century because they are only noticeable at high speeds and under very intense gravitational fields. There is another 20th Century idea that also violates Newtonian Mechanics. This is called Quantum Mechanics. In this essay I will give a taste of the strange and fascinating world of the atom. I will try to keep it general and simple because these ideas are even more weird than Relativity (if that is possible). The Ancient Greeks proposed that matter could not be divided indefinitely. They speculated that matter was made up of units called atoms . The word comes from a Greek word meaning single item or portion . They assumed that atoms were solid, different characteristics of substances being determined by the different shapes that atoms had. This atomic idea never really became popular.

62. Accretion Disk Polar Jetting, Binary Star Formation And Nemesis
Let s dissect the mechanics of a collapsing dust cloud. As particles in motionrelative to one another are gravitationally attracted to one another , they
On The Mechanics of Star Formation
Binary Stars and Nemesis
So. Charleston, WV, USA, May 12, 1996
The Hubble Telescope has provided perplexing images of star formation . Gone is the old theory of a simple collapse of a space cloud into a solid ball. Gone also is the theory of a simple accretion disk gradually winding its way to forming star. We now see volumous jets of material mysteriously being "ejected" along the axis of the accretion disk at high velocity and extending light years out from the disk center. In addition massive quantities of material are seen as blobs at some distance from the visible outer ends of the jets. What can explain these newly observed phenomena? The star is not ejecting material but simply holding material in orbit. Outflows could be buildups of otherwise incoming particles retarded by impulses from the outer reaches of the "jetting" orbital particles. Illumination is provided by heating due to collisions between particles. Observed beading within a jet may be due clumping within the original dust cloud. The accretion disk is not a barrier to material reaching the star's core. The disk is there as a storehouse for the angular momentum imbalance of the incoming dust cloud. It varies in size, shape, and orbital plane throughout star formation as dictated by dust particles within its influence. What happens as a star system's development matures? We know from our own solar system that an accretion disk can condense into planets which orbit the star in the plane of the original accretion disk. But what happens to any polar ejectate material in a highly eccentric elliptical polar orbit? It is possible that it may also condense into a sizeable body. Depending on its size it would be another star ranging from very bright to dark or it could even be a large planet.

63. Do Sub-atomic Particles Obey Newtons Laws Of Motion?
interference of waves. The theory which is able to describe the subatomicparticles is the Quantum mechanics. In Quantum mechanics
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Do sub-atomic particles obey Newtons Laws of motion? Asked by: Robyn Scott Answer In general, the behavior of the sub-atomic particles cannot be described by Netwon's Laws. The basic picture of the Newtonian mechanics can be described as follows. There are particles, with specified positions and velocities, interacting with each other by means of forces. There are several kinds of forces in Nature. These forces can act between two particles, and their strength and direction depend on the positions and the velocities of the particles. Second Newton's Law connects between the forces acting on a particle and the resulting acceleration. Knowledge of the positions and the velocities of all the relevant particles at a specific moment of time allows to predict the positions and the velocities at any other time. The laws which govern the behavior of the sub-atomic particles are completely different. It is impossible to assign a specific position and velocity to a particle. Each particle can be in a superposition of different states, which means that in some sense it is located at the same time in a whole region of space and has a whole range of velocities. If you measure the position (or the velocity) of the particle, you just get one of the values from that range, in random (possibly with different probabilities for each value). However, this is NOT because the particle actually HAD that position and you just hadn't known that, but the particle really HAD a whole range of positions the moment before the measurement. This is something strange and beautiful.

64. Wavicles And Quantum Mechanics
The color force is an example of a gauge theory, a theory using relativistic quantummechanics and principles of symmetry to summarize how particles with a
Wavicles One of the most important conceptions of matter and energy to come out of this century, besides the fact that they are interchangeable states of the same thing (expressed in Einstein's famous equation, E=mc ) is the fact that either state can act as either a particle or a wave. This was a very perplexing problem, and still remains so if we try to visualize what matterenergy looks like at the elementary level. A particle is localized in spacetime it can be assigned very distinct coordinates, and even thought of as stationary and static. A wave is not localized and cannot be static. The wave-particle duality is one of the best examples of the complementarity principle in quantum theory. An electron, for example, will either act like a particle or a wave, but never both at the same time. If we use a particle detector to see the electron, it will be a particle, and if we use a wave detector, it will be a wave. Somehow, we must think of the electron as being both , but in its ability to display both modes of mutually exclusive states of being, it is actually neither . The essence of what the electron really is must be something else entirely. Whatever that is, is quite impossible to visualize, and has been dubbed a wavicle.

65. Some Frequently Asked Questions About Virtual Particles
Then, the use of virtual particles as a communication channel is completelyconsistent with quantum mechanics and relativity. That s
[Physics FAQ] Original by Matt McIrvin 1994.
Some Frequently Asked Questions About Virtual Particles
  • What are virtual particles?
  • How can they be responsible for attractive forces?
  • Do they violate energy conservation?
  • Do they go faster than light? Do virtual particles contradict relativity or causality?
  • I hear physicists saying that the "quantum of the gravitational force" is something called a graviton. Doesn't general relativity say that gravity isn't a force at all?
    What are virtual particles?
    One of the first steps in the development of quantum mechanics was Max Planck's idea that a harmonic oscillator (classically, anything that wiggles like a mass bobbing on the end of an ideal spring) cannot have just any energy. Its possible energies come in a discrete set of equally spaced levels. An electromagnetic field wiggles in the same way when it possesses waves. Applying quantum mechanics to this oscillator reveals that it must also have discrete, evenly spaced energy levels. These energy levels are what we usually identify as different numbers of photons. The higher the energy level of a vibrational mode, the more photons there are. In this way, an electromagnetic wave acts as if it were made of particles. The electromagnetic field is a quantum field. Electromagnetic fields can do things other than vibration. For instance, the electric field produces an attractive or repulsive force between charged objects, which varies as the inverse square of distance. The force can change the momenta of the objects.
  • 66. Maple PowerTools - Classical Mechanics
    in intermediate to advanced Newtonian mechanics. Topics covered include inertialreference frames, kinematics and kinetics of mass particles, Newton s laws

    MapleNet Maple T.A. Toolboxes ... Contact Us
    This is a Maple course developed by Harald Kammerer in intermediate to advanced Newtonian mechanics Topics covered include inertial reference frames, kinematics and kinetics of mass particles, Newton's laws, conservation of energy, moments of inertia, rigid bodies, multiparticle systems and the Lagrangian equation. It assumes prior exposure to elementary physics and calculus. The text provides detailed explanations of the principles and their underlying mathematics and includes many worked examples. All principles and examples are illustrated with Maple diagrams and animations. Download the entire course or preview individual sections below Chapter 1: Introduction Introduction and Installation Instructions preview download Chapter 2: Kinematics of Mass Particles Mass Particles in Cartesian, Polar and Natural Coordinates

    67. Theoretical Mechanics Of Particles Continua: New & Used Books: Find The Lowest P
    Classical mechanics New used books Find the Lowest Price Classical mechanics Systems of Par Compare Prices, Classical mechanics Systemsof particles Hamiltonian Dynamics By Walter Greiner Paperback / October

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    68. 2. Some Basic Ideas About Quantum Mechanics
    In Quantum mechanics this neat distinction is blurred. Entities which we would normallythink of as particles (eg electrons) can behave like waves in certain
    2. Some Basic Ideas about Quantum Mechanics
    Modern physics is dominated by the concepts of Quantum Mechanics. This page aims to give a brief introduction to some of these ideas. Until the closing decades of the last century the physical world, as studied by experiment, could be explained according to the principles of classical (or Newtonian) mechanics: the physics of everyday life. By the turn of the century, however, the cracks were beginning to show and the disciplines of Relativity and Quantum Mechanics were developed to account for them. Relativity came first, and described the physics of very massive and very fast objects, then came Quantum Mechanics in the 1920's to describe the physics of very small objects. Neither of these theories provide an easy intuitive picture of the world, since they contradict the predictions of familiar Newtonian Mechanics in the regimes for which they were developed. Nevertheless, both schemes reproduce the Newtonian results when applied to the everyday world. In seeking to understand the physics of semiconductors at an atomic level we must start from a Quantum Mechanical viewpoint, since the entities with which we will be dealing (electrons, atoms, etc) are so very small....

    69. Molecular Mechanics
    Interactions determine the spatial distribution of atomlike particles and theirenergies. Note how these principles differ from those of quantum mechanics.
    Molecular Mechanics
    The "mechanical" molecular model was developed out of a need to describe molecular structures and properties in as practical a manner as possible. The range of applicability of molecular mechanics includes:
    • Molecules containing thousands of atoms.
    • Organics, oligonucleotides, peptides, and saccharides (metallo-organics and inorganics in some cases).
    • Vacuum, implicit, or explicit solvent environments.
    • Ground state only.
    • Thermodynamic and kinetic (via molecular dynamics ) properties.
    The great computational speed of molecular mechanics allows for its use in procedures such as molecular dynamics, conformational energy searching, and docking, that require large numbers of energy evaluations. Molecular mechanics methods are based on the following principles:
    • Nuclei and electrons are lumped into atom-like particles.
    • Atom-like particles are spherical (radii obtained from measurements or theory) and have a net charge (obtained from theory).
    • Interactions are based on springs and classical potentials.
    • Interactions must be preassigned to specific sets of atoms.

    70. Quantum Mechanics History
    which extended the particlewave duality for light to all particles, in particular hisequation for the hydrogen atom and heralded the birth of wave mechanics.
    A history of Quantum Mechanics
    Mathematical Physics index History Topics Index
    It is hard to realise that the electron was only discovered a little over 100 years ago in 1897. That it was not expected is illustrated by a remark made by J J Thomson, the discoverer of the electron. He said I was told long afterwards by a distinguished physicist who had been present at my lecture that he thought I had been pulling their leg. The neutron was not discovered until 1932 so it is against this background that we trace the beginnings of quantum theory back to 1859. In 1859 Gustav Kirchhoff proved a theorem about blackbody radiation. A blackbody is an object that absorbs all the energy that falls upon it and, because it reflects no light, it would appear black to an observer. A blackbody is also a perfect emitter and Kirchhoff proved that the energy emitted E depends only on the temperature T and the frequency v of the emitted energy, i.e. E J T v He challenged physicists to find the function J In 1879 Josef Stefan proposed, on experimental grounds, that the total energy emitted by a hot body was proportional to the fourth power of the temperature. In the generality stated by Stefan this is false. The same conclusion was reached in 1884 by

    71. Springer-Verlag - Quantum Physics
    59,95 € Cart. Classical mechanics · Point particles and Relativity Greiner, W.,19.01.2004, ISBN 0387-95586-0, Softcover, Ready for shipping within 3 days.,10735,5-10111-0-0-0,00.html
    Please enable Javascript in your browser to browse this website. Select your subdiscipline Atoms, Molecules, Clusters Biophysics Condensed Matter Geophysics Mathematical Physics Quantum Physics Home Quantum Physics
    Select a discipline Biomedical Sciences Chemistry Computer Science Engineering Environmental Sciences Geosciences Law Life Sciences Materials Mathematics Medicine Statistics preloadImage('/sgw/cda/pageitems/designobject/cda_displaydesignobject/0,10885,5-0-17-900120-0,00.gif'); preloadImage('/sgw/cda/pageitems/designobject/cda_displaydesignobject/0,10885,5-0-17-900180-0,00.gif'); preloadImage('/sgw/cda/pageitems/designobject/cda_displaydesignobject/0,10885,5-0-17-900170-0,00.gif'); preloadImage('/sgw/cda/pageitems/designobject/cda_displaydesignobject/0,10885,5-0-17-900190-0,00.gif'); preloadImage('/sgw/cda/pageitems/designobject/cda_displaydesignobject/0,10885,5-0-17-900200-0,00.gif'); preloadImage('/sgw/cda/pageitems/designobject/cda_displaydesignobject/0,10885,5-0-17-900160-0,00.gif');
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    72. Springer-Verlag - Quantum Physics
    Classical mechanics · Point particles and Relativity Greiner, W., 19.01.2004, ISBN0387-95586-0, Softcover, In stock This item usually ships in 2-3 days.,10735,4-10111-0-0-0,00.html
    Please enable Javascript in your browser to browse this website. Select your subdiscipline Atoms, Molecules, Clusters Biophysics Condensed Matter Geophysics Nonlinear Physics Quantum Physics Home Quantum Physics
    Select a discipline Biomedical Sciences Chemistry Computer Science Engineering Environmental Sciences Geosciences Law Life Sciences Materials Mathematics Medicine Statistics preloadImage('/sgw/cda/pageitems/designobject/cda_displaydesignobject/0,10885,4-0-17-900120-0,00.gif'); preloadImage('/sgw/cda/pageitems/designobject/cda_displaydesignobject/0,10885,4-0-17-900180-0,00.gif'); preloadImage('/sgw/cda/pageitems/designobject/cda_displaydesignobject/0,10885,4-0-17-900170-0,00.gif'); preloadImage('/sgw/cda/pageitems/designobject/cda_displaydesignobject/0,10885,4-0-17-900190-0,00.gif'); preloadImage('/sgw/cda/pageitems/designobject/cda_displaydesignobject/0,10885,4-0-17-900200-0,00.gif');
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    73. [cond-mat/9411036] One-Dimensional Statistical Mechanics For Identical Particles
    0100 (13kb) OneDimensional Statistical mechanics for Identical particles The Calogero and Anyon Cases. Authors Alain DASNIÈRES
    Condensed Matter, abstract
    From: [ view email ] Date: Wed, 9 Nov 94 11:51:34 +0100 (13kb)
    One-Dimensional Statistical Mechanics for Identical Particles : The Calogero and Anyon Cases
    Authors: Orsay Fr-91406
    Comments: 17 pages
    Report-no: IPNO/TH 94-75 (September 1994)
    The thermodynamic of particles with intermediate statistics interpolating between Bose and Fermi statistics is adressed in the simple case where there is one quantum number per particle. Such systems are essentially one-dimensional. As an illustration, one considers the anyon model restricted to the lowest Landau level of a strong magnetic field at low temperature, the generalization of this model to several particles species, and the one dimensional Calogero model. One reviews a unified algorithm to compute the statistical mechanics of these systems. It is pointed out that Haldane's generalization of the Pauli principle can be deduced from the anyon model in a strong magnetic field at low temperature.
    Full-text: PostScript PDF , or Other formats
    References and citations for this submission:
    (autonomous citation navigation and analysis) Which authors of this paper are endorsers?

    74. Wave-Particles And Quantum Mechanics
    was not quantized The deterministic laws of classical mechanics would be use Newton swords like the solid, massy, hard, impenetrable particles formed by

    Wave-Particle Duality
    Quantum Mechanics
    by Craig Rusbult, Ph.D. Foreword
    This page is adapted, with minor revisions and major excisions (...snip...), from Chapter 17 of a physics book I wrote in the late-1980s. It introduces basic concepts that show the strangeness of wave-particle duality and the mysteries of quantum mechanics.
    In this page, most links are inside-the-page and are fast; italicized links open a new page in a new window, so this page remains open in this window.
    This is the first of two pages. The second page Quantum Mechanics: Principles, Interpretations, and Speculations critically examines speculative metaphysical claims about quantum mechanics.
    The purpose of this chapter [about Wave-Particle Duality and Quantum Mechanics] is to help you become comfortable with the radical ideas of quantum mechanics, to help you combine creativity and critical thinking so you can be freely imaginative without being silly and illogical. Sections and explain wave-particle duality, and try to convince you that "yes, things really are strange." Section

    75. Background Information: Fluid Mechanics
    Simple Fluid mechanics background For volcanic clouds, the general fluid of studyis mass per unit volume, also affects the rate at which particles fall or
    Michigan Tech Home Remote Sensing Institute MTU Volcanoes Page Volcanic Clouds ... VAAC
    Topical Outline

    Main Topics




    Fluid mechanics
    Simple Fluid Mechanics background
    For volcanic clouds, the general "fluid" of study is the atmosphere. To determine how an ash particle falls out of the atmosphere (also called fallout), dimensionless numbers are used. These are explained below.
    Reynolds Number
    The atmosphere's density and viscosity decrease with altitude. Viscosity is the property of a fluid or gas which makes it resist flow. Fluids with low viscosity flow readily and vice versa. Atmospheric Density, which is defined as mass per unit volume, also affects the rate at which particles fall or settle through a fluid. These two properties are used to calculate the Reynolds number (Re) of a particle moving through the atmosphere. Reynolds number is a dimensionless number (i.e. it has no units) that is a measure of the type of flow through a fluid. In the case of falling particles, this describes the way that air flows around the particle. There are three basic types: laminar turbulent
    An equation for Reynolds number (Re): A review of fluid principles related to particle fall is fundamental to understanding clouds, which consist of particles with low terminal velocities.

    76. PhilSci Archive - Quantum Mechanics And Discontinuous Motion Of Particles
    particles. Keywords interpretation of quantum mechanics,particles,discontinuousmotion,dynamical collapse,discrete spacetime. Subjects
    About Browse Search Register ... Help
    Quantum mechanics and discontinuous motion of particles
    Shan, Gao (2002) Quantum mechanics and discontinuous motion of particles. Full text available as:
    - Requires a viewer, such as Adobe Acrobat Reader or other PDF viewer.
    We discuss a new realistic interpretation of quantum mechanics based on discontinuous motion of particles. The historical and logical basis of discontinuous motion of particles is given. It proves that if there exists only one kind of physical reality-particles, then the realistic motion of particles described by quantum mechanics should be discontinuous motion. We further denote that protective measurement may provide a direct method to confirm the existence of discontinuous motion of particles. Keywords: interpretation of quantum mechanics,particles,discontinuous motion,dynamical collapse,discrete space-time Subjects: Specific Sciences Physics Quantum Mechanics ID Code: Deposited By: Gao, Shan Deposited On: 27 August 2002
    Send feedback to:

    77. Objective Science - Quantum Mechanics And Dissidents By Eric Dennis
    Probability in dBB emerges just as in classical statistical mechanics, as an expressionof our lack of knowledge of initial positions for the particles, not as
    There is a misconception, of some currency, that Bell's results close the door on all realist versions of quantum mechanics. This is ironic because these very results were motivated by Bell's surprise and profound appreciation upon discovering such a version already in the literature. This was David Bohm's completion of an idea that started with Louis de Broglie. It has emerged as a powerful and precise alternative to the fuzziness of standard theory. Part 2
    Quantum Mechanics and Dissidents
    By Eric Dennis Click here for Part 1
    [] The failure of Little's "theory of elementary waves" (TEW), must not be taken to support the sophistry connected with the standard interpretation of quantum mechanics, from the idea that entities lose their attributes until we observe them to the supposed victory of indeterminism in physics. In fact, a politically disinclined group of dissidentsincluding Einstein, Schrodinger, David Bohm, and John Bellmaintained their commitment to realism against the idealist and positivist tendencies of the physics establishment [11]. There is a misconception, of some currency, that Bell's results close the door on all realist versions of quantum mechanics. This is ironic because these very results were motivated by Bell's surprise and profound appreciation upon discovering such a version already in the literature. This was David Bohm's completion of an idea that started with Louis de Broglie. It has emerged as a powerful and precise alternative to the fuzziness of standard theory.

    78. The Particles Of Quantum Mechanics
    Upsilon, 9.460, 0, bb, 0. Force Carrying particles. Force, Particle, Actson, particles affected. Electromagnetic, Photon, Electric Charge, Chargedparticles.
    (spin 1/2)
    name mass (MeV/c charge electron electron nutrino muon mu nutrino tau tau nutrino
    (spin 1/2)
    flavor mass (GeV/c charge up down strange charm bottom top
    (Includes Baryons and Mesons
    Baryons name mass (GeV/c charge quark structure spin nucleons proton uud neutron udd hyperions lambda uds omega sss sigma uus sigma o uds sigma dds Xi o uss Xi dss Others Lambda-c udc Sigma-c uuc
    Mesons name mass (GeV/c charge quark structure spin Pion u d Pion o u u or d u Pion d u Kaon s u Kaon o s d D o c u D c d J/Psi c c B b u B o b d Upsilon b b
    Force Carrying Particles Force Particle Acts on Particles affected Electromagnetic Photon Electric Charge Charged particles Gravitational Graviton All particles Strong Gluon Color Charge Weak W W Z o Flavor Charge Name mass (GeV/c charge spin Photon Graviton Gluon W W Z o This page was written in HTML by Joran Kreiss Monday 4 August 1997, 3:24:18 p.m. PST Modified Saturday 23 January 1999

    79. Todd's Quantum Intro
    some other atomic properties that are quantized, which gives quantum mechanics itsname. in 1704 Isaac Newton explained that light was made of tiny particles.
    Intro to Quantum Mechanics Sigh
    So please read on, and take a dip in an ocean of information that I find completely invigorating!
    If the above picture is your idea of an atom, with electrons looping around the nucleus, you are about 70 years out of date. It's time to open your eyes to the modern world of quantum mechanics! The picture below shows some plots of where you would most likely find an electron in a hydrogen atom (the nucleus is at the center of each plot). What is quantum mechanics? Simply put, quantum mechanics is the study of matter and radiation at an atomic level. Why was quantum mechanics developed? In the early 20th century some experiments produced results which could not be explained by classical physics (the science developed by Galileo Galilei, Isaac Newton, etc.). For instance, it was well known that electrons orbited the nucleus of an atom. However, if they did so in a manner which resembled the planets orbiting the sun, classical physics predicted that the electrons would spiral in and crash into the nucleus within a fraction of a second. Obviously that doesn't happen, or life as we know it would not exist. (Chemistry depends upon the interaction of the electrons in atoms, and life depends upon chemistry). That incorrect prediction, along with some other experiments that classical physics could not explain, showed scientists that something new was needed to explain science at the atomic level. If classical physics is wrong, why do we still use it?

    80. 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
    Next: 2.4 Variational principles Up: 2. Many-body Quantum Mechanics Previous: 2.2 The Born-Oppenheimer approximation Contents

    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;

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