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Identical Particles
8.06 Spring 2016 Lecture Notes 4. Identical particles Aram Harrow Last updated: May 19, 2016 Contents 1 Fermions and Bosons 1 1.1 Introduction and two-particle systems .......................... 1 1.2 N particles ......................................... 3 1.3 Non-interacting particles .................................. 5 1.4 Non-zero temperature ................................... 7 1.5 Composite particles .................................... 7 1.6 Emergence of distinguishability .............................. 9 2 Degenerate Fermi gas 10 2.1 Electrons in a box ..................................... 10 2.2 White dwarves ....................................... 12 2.3 Electrons in a periodic potential ............................. 16 3 Charged particles in a magnetic field 21 3.1 The Pauli Hamiltonian ................................... 21 3.2 Landau levels ........................................ 23 3.3 The de Haas-van Alphen effect .............................. 24 3.4 Integer Quantum Hall Effect ............................... 27 3.5 Aharonov-Bohm Effect ................................... 33 1 Fermions and Bosons 1.1 Introduction and two-particle systems Previously we have discussed multiple-particle systems using the tensor-product formalism (cf. Section 1.2 of Chapter 3 of these notes). But this applies only to distinguishable particles. In reality, all known particles are indistinguishable. In the coming lectures, we will explore the mathematical and physical consequences of this. First, consider classical many-particle systems. If a single particle has state described by position and momentum (~r; p~), then the state of N distinguishable particles can be written as (~r1; p~1; ~r2; p~2;:::; ~rN ; p~N ). The notation (·; ·;:::; ·) denotes an ordered list, in which different posi tions have different meanings; e.g. in general (~r1; p~1; ~r2; p~2)6 = (~r2; p~2; ~r1; p~1). 1 To describe indistinguishable particles, we can use set notation. -
Magnetic Materials I
5 Magnetic materials I Magnetic materials I ● Diamagnetism ● Paramagnetism Diamagnetism - susceptibility All materials can be classified in terms of their magnetic behavior falling into one of several categories depending on their bulk magnetic susceptibility χ. without spin M⃗ 1 χ= in general the susceptibility is a position dependent tensor M⃗ (⃗r )= ⃗r ×J⃗ (⃗r ) H⃗ 2 ] In some materials the magnetization is m / not a linear function of field strength. In A [ such cases the differential susceptibility M is introduced: d M⃗ χ = d d H⃗ We usually talk about isothermal χ susceptibility: ∂ M⃗ χ = T ( ∂ ⃗ ) H T Theoreticians define magnetization as: ∂ F⃗ H[A/m] M=− F=U−TS - Helmholtz free energy [4] ( ∂ H⃗ ) T N dU =T dS− p dV +∑ μi dni i=1 N N dF =T dS − p dV +∑ μi dni−T dS−S dT =−S dT − p dV +∑ μi dni i=1 i=1 Diamagnetism - susceptibility It is customary to define susceptibility in relation to volume, mass or mole: M⃗ (M⃗ /ρ) m3 (M⃗ /mol) m3 χ= [dimensionless] , χρ= , χ = H⃗ H⃗ [ kg ] mol H⃗ [ mol ] 1emu=1×10−3 A⋅m2 The general classification of materials according to their magnetic properties μ<1 <0 diamagnetic* χ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ B=μ H =μr μ0 H B=μo (H +M ) μ>1 >0 paramagnetic** ⃗ ⃗ ⃗ χ μr μ0 H =μo( H + M ) → μ r=1+χ μ≫1 χ≫1 ferromagnetic*** *dia /daɪə mæ ɡˈ n ɛ t ɪ k/ -Greek: “from, through, across” - repelled by magnets. We have from L2: 1 2 the force is directed antiparallel to the gradient of B2 F = V ∇ B i.e. -
5.1 Two-Particle Systems
5.1 Two-Particle Systems We encountered a two-particle system in dealing with the addition of angular momentum. Let's treat such systems in a more formal way. The w.f. for a two-particle system must depend on the spatial coordinates of both particles as @Ψ well as t: Ψ(r1; r2; t), satisfying i~ @t = HΨ, ~2 2 ~2 2 where H = + V (r1; r2; t), −2m1r1 − 2m2r2 and d3r d3r Ψ(r ; r ; t) 2 = 1. 1 2 j 1 2 j R Iff V is independent of time, then we can separate the time and spatial variables, obtaining Ψ(r1; r2; t) = (r1; r2) exp( iEt=~), − where E is the total energy of the system. Let us now make a very fundamental assumption: that each particle occupies a one-particle e.s. [Note that this is often a poor approximation for the true many-body w.f.] The joint e.f. can then be written as the product of two one-particle e.f.'s: (r1; r2) = a(r1) b(r2). Suppose furthermore that the two particles are indistinguishable. Then, the above w.f. is not really adequate since you can't actually tell whether it's particle 1 in state a or particle 2. This indeterminacy is correctly reflected if we replace the above w.f. by (r ; r ) = a(r ) (r ) (r ) a(r ). 1 2 1 b 2 b 1 2 The `plus-or-minus' sign reflects that there are two distinct ways to accomplish this. Thus we are naturally led to consider two kinds of identical particles, which we have come to call `bosons' (+) and `fermions' ( ). -
Advances in Theoretical & Computational Physics
ISSN: 2639-0108 Research Article Advances in Theoretical & Computational Physics Supreme Theory of Everything Ulaanbaatar Tarzad *Corresponding author Ulaanbaatar Tarzad, Department of Physics, School of Applied Sciences, Department of Physics, School of Applied Sciences, Mongolian Mongolian University of Science and Technology, Ulaanbaatar, Mongolia, University of Science and Technology E-mail: [email protected] Submitted: 27 Mar 2019; Accepted: 24 Apr 2019; Published: 06 June 2019 Abstract Not only universe, but everything has general characters as eternal, infinite, cyclic and wave-particle duality. Everything from elementary particles to celestial bodies, from electromagnetic wave to gravity is in eternal motions, which dissects only to circle. Since everything is described only by trigonometry. Without trigonometry and mathematical circle, the science cannot indicate all the beauty of harmonic universe. Other method may be very good, but it is not perfect. Some part is very nice, another part is problematic. General Theory of Relativity holds that gravity is geometric. Quantum Mechanics describes all particles by wave function of trigonometry. In this paper using trigonometry, particularly mathematics circle, a possible version of the unification of partial theories, evolution history and structure of expanding universe, and the parallel universes are shown. Keywords: HRD, Trigonometry, Projection of Circle, Singularity, The reality of universe describes by geometry, because of that not Celestial Body, Black Hole and Parallel Universes. only gravity is geometrical, but everything is it and nothing is linear. One of the important branches of geometry is trigonometry dealing Introduction with circle and triangle. For this reason, it is easier to describe nature Today scientists describe the universe in terms of two basic partial of universe by mathematics circle. -
Scientific and Related Works of Chen Ning Yang
Scientific and Related Works of Chen Ning Yang [42a] C. N. Yang. Group Theory and the Vibration of Polyatomic Molecules. B.Sc. thesis, National Southwest Associated University (1942). [44a] C. N. Yang. On the Uniqueness of Young's Differentials. Bull. Amer. Math. Soc. 50, 373 (1944). [44b] C. N. Yang. Variation of Interaction Energy with Change of Lattice Constants and Change of Degree of Order. Chinese J. of Phys. 5, 138 (1944). [44c] C. N. Yang. Investigations in the Statistical Theory of Superlattices. M.Sc. thesis, National Tsing Hua University (1944). [45a] C. N. Yang. A Generalization of the Quasi-Chemical Method in the Statistical Theory of Superlattices. J. Chem. Phys. 13, 66 (1945). [45b] C. N. Yang. The Critical Temperature and Discontinuity of Specific Heat of a Superlattice. Chinese J. Phys. 6, 59 (1945). [46a] James Alexander, Geoffrey Chew, Walter Salove, Chen Yang. Translation of the 1933 Pauli article in Handbuch der Physik, volume 14, Part II; Chapter 2, Section B. [47a] C. N. Yang. On Quantized Space-Time. Phys. Rev. 72, 874 (1947). [47b] C. N. Yang and Y. Y. Li. General Theory of the Quasi-Chemical Method in the Statistical Theory of Superlattices. Chinese J. Phys. 7, 59 (1947). [48a] C. N. Yang. On the Angular Distribution in Nuclear Reactions and Coincidence Measurements. Phys. Rev. 74, 764 (1948). 2 [48b] S. K. Allison, H. V. Argo, W. R. Arnold, L. del Rosario, H. A. Wilcox and C. N. Yang. Measurement of Short Range Nuclear Recoils from Disintegrations of the Light Elements. Phys. Rev. 74, 1233 (1948). [48c] C. -
Identical Particles in Classical Physics One Can Distinguish Between
Identical particles In classical physics one can distinguish between identical particles in such a way as to leave the dynamics unaltered. Therefore, the exchange of particles of identical particles leads to di®erent con¯gurations. In quantum mechanics, i.e., in nature at the microscopic levels identical particles are indistinguishable. A proton that arrives from a supernova explosion is the same as the proton in your glass of water.1 Messiah and Greenberg2 state the principle of indistinguishability as \states that di®er only by a permutation of identical particles cannot be distinguished by any obser- vation whatsoever." If we denote the operator which interchanges particles i and j by the permutation operator Pij we have ^ y ^ hªjOjªi = hªjPij OPijjÃi ^ which implies that Pij commutes with an arbitrary observable O. In particular, it commutes with the Hamiltonian. All operators are permutation invariant. Law of nature: The wave function of a collection of identical half-odd-integer spin particles is completely antisymmetric while that of integer spin particles or bosons is completely symmetric3. De¯ning the permutation operator4 by Pij ª(1; 2; ¢ ¢ ¢ ; i; ¢ ¢ ¢ ; j; ¢ ¢ ¢ N) = ª(1; 2; ¢ ¢ ¢ ; j; ¢ ¢ ¢ ; i; ¢ ¢ ¢ N) we have Pij ª(1; 2; ¢ ¢ ¢ ; i; ¢ ¢ ¢ ; j; ¢ ¢ ¢ N) = § ª(1; 2; ¢ ¢ ¢ ; i; ¢ ¢ ¢ ; j; ¢ ¢ ¢ N) where the upper sign refers to bosons and the lower to fermions. Emphasize that the symmetry requirements only apply to identical particles. For the hydrogen molecule if I and II represent the coordinates and spins of the two protons and 1 and 2 of the two electrons we have ª(I;II; 1; 2) = ¡ ª(II;I; 1; 2) = ¡ ª(I;II; 2; 1) = ª(II;I; 2; 1) 1On the other hand, the elements on earth came from some star/supernova in the distant past. -
The Conventionality of Parastatistics
The Conventionality of Parastatistics David John Baker Hans Halvorson Noel Swanson∗ March 6, 2014 Abstract Nature seems to be such that we can describe it accurately with quantum theories of bosons and fermions alone, without resort to parastatistics. This has been seen as a deep mystery: paraparticles make perfect physical sense, so why don't we see them in nature? We consider one potential answer: every paraparticle theory is physically equivalent to some theory of bosons or fermions, making the absence of paraparticles in our theories a matter of convention rather than a mysterious empirical discovery. We argue that this equivalence thesis holds in all physically admissible quantum field theories falling under the domain of the rigorous Doplicher-Haag-Roberts approach to superselection rules. Inadmissible parastatistical theories are ruled out by a locality- inspired principle we call Charge Recombination. Contents 1 Introduction 2 2 Paraparticles in Quantum Theory 6 ∗This work is fully collaborative. Authors are listed in alphabetical order. 1 3 Theoretical Equivalence 11 3.1 Field systems in AQFT . 13 3.2 Equivalence of field systems . 17 4 A Brief History of the Equivalence Thesis 20 4.1 The Green Decomposition . 20 4.2 Klein Transformations . 21 4.3 The Argument of Dr¨uhl,Haag, and Roberts . 24 4.4 The Doplicher-Roberts Reconstruction Theorem . 26 5 Sharpening the Thesis 29 6 Discussion 36 6.1 Interpretations of QM . 44 6.2 Structuralism and Haecceities . 46 6.3 Paraquark Theories . 48 1 Introduction Our most fundamental theories of matter provide a highly accurate description of subatomic particles and their behavior. -
Talk M.Alouani
From atoms to solids to nanostructures Mebarek Alouani IPCMS, 23 rue du Loess, UMR 7504, Strasbourg, France European Summer Campus 2012: Physics at the nanoscale, Strasbourg, France, July 01–07, 2012 1/107 Course content I. Magnetic moment and magnetic field II. No magnetism in classical mechanics III. Where does magnetism come from? IV. Crystal field, superexchange, double exchange V. Free electron model: Spontaneous magnetization VI. The local spin density approximation of the DFT VI. Beyond the DFT: LDA+U VII. Spin-orbit effects: Magnetic anisotropy, XMCD VIII. Bibliography European Summer Campus 2012: Physics at the nanoscale, Strasbourg, France, July 01–07, 2012 2/107 Course content I. Magnetic moment and magnetic field II. No magnetism in classical mechanics III. Where does magnetism come from? IV. Crystal field, superexchange, double exchange V. Free electron model: Spontaneous magnetization VI. The local spin density approximation of the DFT VI. Beyond the DFT: LDA+U VII. Spin-orbit effects: Magnetic anisotropy, XMCD VIII. Bibliography European Summer Campus 2012: Physics at the nanoscale, Strasbourg, France, July 01–07, 2012 3/107 Magnetic moments In classical electrodynamics, the vector potential, in the magnetic dipole approximation, is given by: μ μ × rˆ Ar()= 0 4π r2 where the magnetic moment μ is defined as: 1 μ =×Id('rl ) 2 ∫ It is shown that the magnetic moment is proportional to the angular momentum μ = γ L (γ the gyromagnetic factor) The projection along the quantification axis z gives (/2Bohrmaμ = eh m gneton) μzB= −gmμ Be European Summer Campus 2012: Physics at the nanoscale, Strasbourg, France, July 01–07, 2012 4/107 Magnetization and Field The magnetization M is the magnetic moment per unit volume In free space B = μ0 H because there is no net magnetization. -
Spontaneous Symmetry Breaking and Mass Generation As Built-In Phenomena in Logarithmic Nonlinear Quantum Theory
Vol. 42 (2011) ACTA PHYSICA POLONICA B No 2 SPONTANEOUS SYMMETRY BREAKING AND MASS GENERATION AS BUILT-IN PHENOMENA IN LOGARITHMIC NONLINEAR QUANTUM THEORY Konstantin G. Zloshchastiev Department of Physics and Center for Theoretical Physics University of the Witwatersrand Johannesburg, 2050, South Africa (Received September 29, 2010; revised version received November 3, 2010; final version received December 7, 2010) Our primary task is to demonstrate that the logarithmic nonlinearity in the quantum wave equation can cause the spontaneous symmetry break- ing and mass generation phenomena on its own, at least in principle. To achieve this goal, we view the physical vacuum as a kind of the funda- mental Bose–Einstein condensate embedded into the fictitious Euclidean space. The relation of such description to that of the physical (relativis- tic) observer is established via the fluid/gravity correspondence map, the related issues, such as the induced gravity and scalar field, relativistic pos- tulates, Mach’s principle and cosmology, are discussed. For estimate the values of the generated masses of the otherwise massless particles such as the photon, we propose few simple models which take into account small vacuum fluctuations. It turns out that the photon’s mass can be naturally expressed in terms of the elementary electrical charge and the extensive length parameter of the nonlinearity. Finally, we outline the topological properties of the logarithmic theory and corresponding solitonic solutions. DOI:10.5506/APhysPolB.42.261 PACS numbers: 11.15.Ex, 11.30.Qc, 04.60.Bc, 03.65.Pm 1. Introduction Current observational data in astrophysics are probing a regime of de- partures from classical relativity with sensitivities that are relevant for the study of the quantum-gravity problem [1,2]. -
The Discovery of Asymptotic Freedom
The Discovery of Asymptotic Freedom The 2004 Nobel Prize in Physics, awarded to David Gross, Frank Wilczek, and David Politzer, recognizes the key discovery that explained how quarks, the elementary constituents of the atomic nucleus, are bound together to form protons and neutrons. In 1973, Gross and Wilczek, working at Princeton, and Politzer, working independently at Harvard, showed that the attraction between quarks grows weaker as the quarks approach one another more closely, and correspondingly that the attraction grows stronger as the quarks are separated. This discovery, known as “asymptotic freedom,” established quantum chromodynamics (QCD) as the correct theory of the strong nuclear force, one of the four fundamental forces in Nature. At the time of the discovery, Wilczek was a 21-year-old graduate student working under Gross’s supervision at Princeton, while Politzer was a 23-year-old graduate student at Harvard. Currently Gross is the Director of the Kavli Institute for Theoretical Physics at the University of California at Santa Barbara, and Wilczek is the Herman Feshbach Professor of Physics at MIT. Politzer is Professor of Theoretical Physics at Caltech; he joined the Caltech faculty in 1976. Of the four fundamental forces --- the others besides the strong nuclear force are electromagnetism, the weak nuclear force (responsible for the decay of radioactive nuclei), and gravitation --- the strong force was by far the most poorly understood in the early 1970s. It had been suggested in 1964 by Caltech physicist Murray Gell-Mann that protons and neutrons contain more elementary objects, which he called quarks. Yet isolated quarks are never seen, indicating that the quarks are permanently bound together by powerful nuclear forces. -
Cheehok - Quantum Mechanics, Elementary.Pmd1 10/8/2010, 10:37 AM October 1, 2010 8:53 WSPC - Proceedings Trim Size: 9.75In X 6.5In 00B-Preface
Proceedings of the Conference in Honour of Murray Gell-Mann's 80th Birthday Quantum Mechanics, Elementary Particles, Quantum Cosmology and Complexity ■Editors -^ H Fritzsch - K K Phua . Co-editors BE Baaquie - AH Chan N-P Chang • SACheong LCKwek • CHOh World Scientific This page is intentionally left blank This page is intentionally left blank Proceedings of the Conference in Honour of Murray Gell-Mann’s 80th Birthday Quantum Mechanics, Elementary Particles, Quantum Cosmology and Complexity 8043tp.indd 1 10/19/10 1:44 PM Proceedings of the Conference in Honour of Murray Gell-Mann’s 80th Birthday Quantum Mechanics, Elementary Particles, Quantum Cosmology and Complexity Nanyang Technological University, Singapore, 24–26 February 2010 Editors H Fritzsch (University of Munich, Germany) K K Phua (Nanyang Technological University, Singapore) Co-editors B E Baaquie (National University of Singapore, Singapore) A H Chan (National University of Singapore, Singapore) N-P Chang (City University of New York, USA) S A Cheong (Nanyang Technological University, Singapore) L C Kwek (National University of Singapore, Singapore) C H Oh (National University of Singapore, Singapore) World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI 8043tp.indd 2 10/19/10 1:44 PM Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. QUANTUM MECHANICS, ELEMENTARY PARTICLES, QUANTUM COSMOLOGY AND COMPLEXITY Proceedings of the Conference in Honour of Murray Gell-Mann’s 80th Birthday Copyright © 2011 by World Scientific Publishing Co. -
Standard Model of Particle Physics, Or Beyond?
Standard Model of Particle Physics, or Beyond? Mariano Quir´os High Energy Phys. Inst., BCN (Spain) ICTP-SAIFR, November 13th, 2019 Outline The outline of this colloquium is I Standard Model: reminder I Electroweak interactions I Strong interactions I The Higgs sector I Experimental successes I Theoretical and observational drawbacks I Beyond the Standard Model I Supersymmetry I Large extra dimensions I Warped extra dimensions/composite Higgs I Concluding remarks Disclaimer: I will not discuss any technical details. With my apologies to my theorist (and experimental) colleagues The Standard Model: reminder I The knowledge of the Standard Model of strong and electroweak interactions requires (as any other physical theory) the knowledge of I The elementary particles or fields (the characters of the play) I How particles interact (their behavior) The characters of the play I Quarks: spin-1/2 fermions I Leptons: spin-1/2 fermions I Higgs boson: spin-0 boson I Carriers of the interactions: spin-1 (gauge) bosons I All these particles have already been discovered and their mass, spin, and charge measured \More in detail the characters of the play" - Everybody knows the Periodic Table of the Elements - Compare elementary particles with some (of course composite) very heavy nuclei What are the interactions between the elementary building blocks of the Standard Model? I Interactions are governed by a symmetry principle I The more symmetric the theory the more couplings are related (the less of them they are) and the more predictive it is Strong interactions: