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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. -
Effective Field Theories, Reductionism and Scientific Explanation Stephan
To appear in: Studies in History and Philosophy of Modern Physics Effective Field Theories, Reductionism and Scientific Explanation Stephan Hartmann∗ Abstract Effective field theories have been a very popular tool in quantum physics for almost two decades. And there are good reasons for this. I will argue that effec- tive field theories share many of the advantages of both fundamental theories and phenomenological models, while avoiding their respective shortcomings. They are, for example, flexible enough to cover a wide range of phenomena, and concrete enough to provide a detailed story of the specific mechanisms at work at a given energy scale. So will all of physics eventually converge on effective field theories? This paper argues that good scientific research can be characterised by a fruitful interaction between fundamental theories, phenomenological models and effective field theories. All of them have their appropriate functions in the research process, and all of them are indispens- able. They complement each other and hang together in a coherent way which I shall characterise in some detail. To illustrate all this I will present a case study from nuclear and particle physics. The resulting view about scientific theorising is inherently pluralistic, and has implications for the debates about reductionism and scientific explanation. Keywords: Effective Field Theory; Quantum Field Theory; Renormalisation; Reductionism; Explanation; Pluralism. ∗Center for Philosophy of Science, University of Pittsburgh, 817 Cathedral of Learning, Pitts- burgh, PA 15260, USA (e-mail: [email protected]) (correspondence address); and Sektion Physik, Universit¨at M¨unchen, Theresienstr. 37, 80333 M¨unchen, Germany. 1 1 Introduction There is little doubt that effective field theories are nowadays a very popular tool in quantum physics. -
What Is the Dirac Equation?
What is the Dirac equation? M. Burak Erdo˘gan ∗ William R. Green y Ebru Toprak z July 15, 2021 at all times, and hence the model needs to be first or- der in time, [Tha92]. In addition, it should conserve In the early part of the 20th century huge advances the L2 norm of solutions. Dirac combined the quan- were made in theoretical physics that have led to tum mechanical notions of energy and momentum vast mathematical developments and exciting open operators E = i~@t, p = −i~rx with the relativis- 2 2 2 problems. Einstein's development of relativistic the- tic Pythagorean energy relation E = (cp) + (E0) 2 ory in the first decade was followed by Schr¨odinger's where E0 = mc is the rest energy. quantum mechanical theory in 1925. Einstein's the- Inserting the energy and momentum operators into ory could be used to describe bodies moving at great the energy relation leads to a Klein{Gordon equation speeds, while Schr¨odinger'stheory described the evo- 2 2 2 4 lution of very small particles. Both models break −~ tt = (−~ ∆x + m c ) : down when attempting to describe the evolution of The Klein{Gordon equation is second order, and does small particles moving at great speeds. In 1927, Paul not have an L2-conservation law. To remedy these Dirac sought to reconcile these theories and intro- shortcomings, Dirac sought to develop an operator1 duced the Dirac equation to describe relativitistic quantum mechanics. 2 Dm = −ic~α1@x1 − ic~α2@x2 − ic~α3@x3 + mc β Dirac's formulation of a hyperbolic system of par- tial differential equations has provided fundamental which could formally act as a square root of the Klein- 2 2 2 models and insights in a variety of fields from parti- Gordon operator, that is, satisfy Dm = −c ~ ∆ + 2 4 cle physics and quantum field theory to more recent m c . -
Dirac Equation - Wikipedia
Dirac equation - Wikipedia https://en.wikipedia.org/wiki/Dirac_equation Dirac equation From Wikipedia, the free encyclopedia In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it 1 describes all spin-2 massive particles such as electrons and quarks for which parity is a symmetry. It is consistent with both the principles of quantum mechanics and the theory of special relativity,[1] and was the first theory to account fully for special relativity in the context of quantum mechanics. It was validated by accounting for the fine details of the hydrogen spectrum in a completely rigorous way. The equation also implied the existence of a new form of matter, antimatter, previously unsuspected and unobserved and which was experimentally confirmed several years later. It also provided a theoretical justification for the introduction of several component wave functions in Pauli's phenomenological theory of spin; the wave functions in the Dirac theory are vectors of four complex numbers (known as bispinors), two of which resemble the Pauli wavefunction in the non-relativistic limit, in contrast to the Schrödinger equation which described wave functions of only one complex value. Moreover, in the limit of zero mass, the Dirac equation reduces to the Weyl equation. Although Dirac did not at first fully appreciate the importance of his results, the entailed explanation of spin as a consequence of the union of quantum mechanics and relativity—and the eventual discovery of the positron—represents one of the great triumphs of theoretical physics. -
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' ( ). -
Finite-Volume and Magnetic Effects on the Phase Structure of the Three-Flavor Nambu–Jona-Lasinio Model
PHYSICAL REVIEW D 99, 076001 (2019) Finite-volume and magnetic effects on the phase structure of the three-flavor Nambu–Jona-Lasinio model Luciano M. Abreu* Instituto de Física, Universidade Federal da Bahia, 40170-115 Salvador, Bahia, Brazil † Emerson B. S. Corrêa Faculdade de Física, Universidade Federal do Sul e Sudeste do Pará, 68505-080 Marabá, Pará, Brazil ‡ Cesar A. Linhares Instituto de Física, Universidade do Estado do Rio de Janeiro, 20559-900 Rio de Janeiro, Rio de Janeiro, Brazil Adolfo P. C. Malbouisson§ Centro Brasileiro de Pesquisas Físicas/MCTI, 22290-180 Rio de Janeiro, Rio de Janeiro, Brazil (Received 15 January 2019; revised manuscript received 27 February 2019; published 5 April 2019) In this work, we analyze the finite-volume and magnetic effects on the phase structure of a generalized version of Nambu–Jona-Lasinio model with three quark flavors. By making use of mean-field approximation and Schwinger’s proper-time method in a toroidal topology with antiperiodic conditions, we investigate the gap equation solutions under the change of the size of compactified coordinates, strength of magnetic field, temperature, and chemical potential. The ’t Hooft interaction contributions are also evaluated. The thermodynamic behavior is strongly affected by the combined effects of relevant variables. The findings suggest that the broken phase is disfavored due to both the increasing of temperature and chemical potential and the drop of the cubic volume of size L, whereas it is stimulated with the augmentation of magnetic field. In particular, the reduction of L (remarkably at L ≈ 0.5–3 fm) engenders a reduction of the constituent masses for u,d,s quarks through a crossover phase transition to the their corresponding current quark masses. -
Properties of Baryons in the Chiral Quark Model
Properties of Baryons in the Chiral Quark Model Tommy Ohlsson Teknologie licentiatavhandling Kungliga Tekniska Hogskolan¨ Stockholm 1997 Properties of Baryons in the Chiral Quark Model Tommy Ohlsson Licentiate Dissertation Theoretical Physics Department of Physics Royal Institute of Technology Stockholm, Sweden 1997 Typeset in LATEX Akademisk avhandling f¨or teknologie licentiatexamen (TeknL) inom ¨amnesomr˚adet teoretisk fysik. Scientific thesis for the degree of Licentiate of Engineering (Lic Eng) in the subject area of Theoretical Physics. TRITA-FYS-8026 ISSN 0280-316X ISRN KTH/FYS/TEO/R--97/9--SE ISBN 91-7170-211-3 c Tommy Ohlsson 1997 Printed in Sweden by KTH H¨ogskoletryckeriet, Stockholm 1997 Properties of Baryons in the Chiral Quark Model Tommy Ohlsson Teoretisk fysik, Institutionen f¨or fysik, Kungliga Tekniska H¨ogskolan SE-100 44 Stockholm SWEDEN E-mail: [email protected] Abstract In this thesis, several properties of baryons are studied using the chiral quark model. The chiral quark model is a theory which can be used to describe low energy phenomena of baryons. In Paper 1, the chiral quark model is studied using wave functions with configuration mixing. This study is motivated by the fact that the chiral quark model cannot otherwise break the Coleman–Glashow sum-rule for the magnetic moments of the octet baryons, which is experimentally broken by about ten standard deviations. Configuration mixing with quark-diquark components is also able to reproduce the octet baryon magnetic moments very accurately. In Paper 2, the chiral quark model is used to calculate the decuplet baryon ++ magnetic moments. The values for the magnetic moments of the ∆ and Ω− are in good agreement with the experimental results. -
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].