Spin and Relativistic Motion of Bound States
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The Particle Zoo
219 8 The Particle Zoo 8.1 Introduction Around 1960 the situation in particle physics was very confusing. Elementary particlesa such as the photon, electron, muon and neutrino were known, but in addition many more particles were being discovered and almost any experiment added more to the list. The main property that these new particles had in common was that they were strongly interacting, meaning that they would interact strongly with protons and neutrons. In this they were different from photons, electrons, muons and neutrinos. A muon may actually traverse a nucleus without disturbing it, and a neutrino, being electrically neutral, may go through huge amounts of matter without any interaction. In other words, in some vague way these new particles seemed to belong to the same group of by Dr. Horst Wahl on 08/28/12. For personal use only. particles as the proton and neutron. In those days proton and neutron were mysterious as well, they seemed to be complicated compound states. At some point a classification scheme for all these particles including proton and neutron was introduced, and once that was done the situation clarified considerably. In that Facts And Mysteries In Elementary Particle Physics Downloaded from www.worldscientific.com era theoretical particle physics was dominated by Gell-Mann, who contributed enormously to that process of systematization and clarification. The result of this massive amount of experimental and theoretical work was the introduction of quarks, and the understanding that all those ‘new’ particles as well as the proton aWe call a particle elementary if we do not know of a further substructure. -
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. -
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. -
A Tale of Three Equations: Breit, Eddington-Gaunt, and Two-Body
View metadata, citation and similarA Tale papers of at Three core.ac.uk Equations: Breit, Eddington-Gaunt, and Two-Body Dirac brought to you by CORE Peter Van Alstine provided by CERN Document Server 12474 Sunny Glenn Drive, Moorpark, Ca. 90125 Horace W. Crater The University of Tennessee Space Institute Tullahoma,Tennessee 37388 G.Breit’s original paper of 1929 postulates the Breit equation as a correction to an earlier defective equation due to Eddington and Gaunt, containing a form of interaction suggested by Heisenberg and Pauli. We observe that manifestly covariant electromagnetic Two-Body Dirac equations previously obtained by us in the framework of Relativistic Constraint Mechanics reproduce the spectral results of the Breit equation but through an interaction structure that contains that of Eddington and Gaunt. By repeating for our equation the analysis that Breit used to demonstrate the superiority of his equation to that of Eddington and Gaunt, we show that the historically unfamiliar interaction structures of Two-Body Dirac equations (in Breit-like form) are just what is needed to correct the covariant Eddington Gaunt equation without resorting to Breit’s version of retardation. 1 I. INTRODUCTION Three score and seven years ago, Gregory Breit extended Dirac’s spin-1/2 wave equation to a system of two charged particles [1]. He formed his equation by summing two free-particle Dirac Hamiltonians with an interaction obtained by substituting Dirac α~’s for velocities in the semi-relativistic electrodynamic interaction of Darwin:. α 1 EΨ= α~ ·p~ +β m +α~ ·p~ +β m − [1 − (α~ · α~ + α~ · r~ˆ α ·rˆ)] Ψ. -
Arxiv:Hep-Ph/9412261V1 8 Dec 1994
IPNO/TH 94-85 How to obtain a covariant Breit type equation from relativistic Constraint Theory J. Mourad Laboratoire de Mod`eles de Physique Math´ematique, Universit´ede Tours, Parc de Grandmont, F-37200 Tours, France H. Sazdjian Division de Physique Th´eorique∗, Institut de Physique Nucl´eaire, Universit´eParis XI, F-91406 Orsay Cedex, France arXiv:hep-ph/9412261v1 8 Dec 1994 October 1994 ∗Unit´ede Recherche des Universit´es Paris 11 et Paris 6 associ´ee au CNRS. 1 Abstract It is shown that, by an appropriate modification of the structure of the interaction po- tential, the Breit equation can be incorporated into a set of two compatible manifestly covariant wave equations, derived from the general rules of Constraint Theory. The com- plementary equation to the covariant Breit type equation determines the evolution law in the relative time variable. The interaction potential can be systematically calculated in perturbation theory from Feynman diagrams. The normalization condition of the Breit wave function is determined. The wave equation is reduced, for general classes of poten- tial, to a single Pauli-Schr¨odinger type equation. As an application of the covariant Breit type equation, we exhibit massless pseudoscalar bound state solutions, corresponding to a particular class of confining potentials. PACS numbers : 03.65.Pm, 11.10.St, 12.39.Ki. 2 1 Introduction Historically, The Breit equation [1] represents the first attempt to describe the relativistic dynamics of two interacting fermion systems. It consists in summing the free Dirac hamiltonians of the two fermions and adding mutual and, eventually, external potentials. This equation, when applied to QED, with one-photon exchange diagram considered in the Coulomb gauge, and solved in the linearized approximation, provides the correct spectra to order α4 [1, 2] for various bound state problems. -
Quantum Mechanics
Quantum Mechanics Richard Fitzpatrick Professor of Physics The University of Texas at Austin Contents 1 Introduction 5 1.1 Intendedaudience................................ 5 1.2 MajorSources .................................. 5 1.3 AimofCourse .................................. 6 1.4 OutlineofCourse ................................ 6 2 Probability Theory 7 2.1 Introduction ................................... 7 2.2 WhatisProbability?.............................. 7 2.3 CombiningProbabilities. ... 7 2.4 Mean,Variance,andStandardDeviation . ..... 9 2.5 ContinuousProbabilityDistributions. ........ 11 3 Wave-Particle Duality 13 3.1 Introduction ................................... 13 3.2 Wavefunctions.................................. 13 3.3 PlaneWaves ................................... 14 3.4 RepresentationofWavesviaComplexFunctions . ....... 15 3.5 ClassicalLightWaves ............................. 18 3.6 PhotoelectricEffect ............................. 19 3.7 QuantumTheoryofLight. .. .. .. .. .. .. .. .. .. .. .. .. .. 21 3.8 ClassicalInterferenceofLightWaves . ...... 21 3.9 QuantumInterferenceofLight . 22 3.10 ClassicalParticles . .. .. .. .. .. .. .. .. .. .. .. .. .. .. 25 3.11 QuantumParticles............................... 25 3.12 WavePackets .................................. 26 2 QUANTUM MECHANICS 3.13 EvolutionofWavePackets . 29 3.14 Heisenberg’sUncertaintyPrinciple . ........ 32 3.15 Schr¨odinger’sEquation . 35 3.16 CollapseoftheWaveFunction . 36 4 Fundamentals of Quantum Mechanics 39 4.1 Introduction .................................. -
Solvable Two-Body Dirac Equation As a Potential Model of Light Mesons?
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 4 (2008), 048, 19 pages Solvable Two-Body Dirac Equation as a Potential Model of Light Mesons? Askold DUVIRYAK Institute for Condensed Matter Physics of National Academy of Sciences of Ukraine, 1 Svientsitskii Str., UA–79011 Lviv, Ukraine E-mail: [email protected] Received October 29, 2007, in final form May 07, 2008; Published online May 30, 2008 Original article is available at http://www.emis.de/journals/SIGMA/2008/048/ Abstract. The two-body Dirac equation with general local potential is reduced to the pair of ordinary second-order differential equations for radial components of a wave function. The class of linear + Coulomb potentials with complicated spin-angular structure is found, for which the equation is exactly solvable. On this ground a relativistic potential model of light mesons is constructed and the mass spectrum is calculated. It is compared with experimental data. Key words: two body Dirac equation; Dirac oscillator; solvable model; Regge trajectories 2000 Mathematics Subject Classification: 81Q05; 34A05 1 Introduction It is well known that light meson spectra are structured into Regge trajectories which are approximately linear and degenerated due to a weak dependence of masses of resonances on their spin state. These features are reproduced well within the exactly solvable simple relativistic oscillator model (SROM) [1, 2, 3] describing two Klein–Gordon particles harmonically bound. Actually, mesons consist of quarks, i.e., spin-1/2 constituents. Thus potential models based on the Dirac equation are expected to be more appropriate for mesons spectroscopy. In recent years the two body Dirac equations (2BDE) in different formulations1 with various confining potentials are used as a relativistic constituent quark models [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26]. -
On the Possibility of Generating a 4-Neutron Resonance with a {\Boldmath $ T= 3/2$} Isospin 3-Neutron Force
On the possibility of generating a 4-neutron resonance with a T = 3/2 isospin 3-neutron force E. Hiyama Nishina Center for Accelerator-Based Science, RIKEN, Wako, 351-0198, Japan R. Lazauskas IPHC, IN2P3-CNRS/Universite Louis Pasteur BP 28, F-67037 Strasbourg Cedex 2, France J. Carbonell Institut de Physique Nucl´eaire, Universit´eParis-Sud, IN2P3-CNRS, 91406 Orsay Cedex, France M. Kamimura Department of Physics, Kyushu University, Fukuoka 812-8581, Japan and Nishina Center for Accelerator-Based Science, RIKEN, Wako 351-0198, Japan (Dated: September 26, 2018) We consider the theoretical possibility to generate a narrow resonance in the four neutron system as suggested by a recent experimental result. To that end, a phenomenological T = 3/2 three neutron force is introduced, in addition to a realistic NN interaction. We inquire what should be the strength of the 3n force in order to generate such a resonance. The reliability of the three-neutron force in the T = 3/2 channel is exmined, by analyzing its consistency with the low-lying T = 1 states of 4H, 4He and 4Li and the 3H+ n scattering. The ab initio solution of the 4n Schr¨odinger equation is obtained using the complex scaling method with boundary conditions appropiate to the four-body resonances. We find that in order to generate narrow 4n resonant states a remarkably attractive 3N force in the T = 3/2 channel is required. I. INTRODUCTION ergy axis (physical domain) it will have no significant impact on a physical process. The possibility of detecting a four-neutron (4n) structure of Following this line of reasoning, some calculations based any kind – bound or resonant state – has intrigued the nuclear on semi-realistic NN forces indicated null [13] or unlikely physics community for the last fifty years (see Refs. -
TO the POSSIBILITY of BOUND STATES BETWEEN TWO ELECTRONS Alexander A
WEPPP031 Proceedings of IPAC2012, New Orleans, Louisiana, USA TO THE POSSIBILITY OF BOUND STATES BETWEEN TWO ELECTRONS Alexander A. Mikhailichenko, Cornell University, LEPP, Ithaca, NY 14853, USA Abstract spins for reduction of minimal emittance restriction arisen from Eq. 1. In some sense it is an attempt to prepare the We analyze the possibility to compress dynamically the pure quantum mechanical state between just two polarized electron bunch so that the distance between electrons. What is important here is that the distance some electrons in the bunch comes close to the Compton between two electrons should be of the order of the wavelength, arranging a bound state, as the attraction by Compton wavelength. We attracted attention in [2,3] that the magnetic momentum-induced force at this distance attraction between two electrons determined by the dominates repulsion by the electrostatic force for the magnetic force of oppositely oriented magnetic moments. appropriately prepared orientation of the magnetic In this case the resulting spin is zero. Another possibility moments of the electron-electron pair. This electron pair considered below. behaves like a boson now, so the restriction for the In [4], it was suggested a radical explanation of minimal emittance of the beam becomes eliminated. structure of all elementary particles caused by magnetic Some properties of such degenerated electron gas attraction at the distances of the order of Compton represented also. wavelength. In [5], motion of charged particle in a field OVERVIEW of magnetic dipole was considered. In [4] and [5] the term in Hamiltonian responsible for the interaction Generation of beams of particles (electrons, positrons, between magnetic moments is omitted, however as it protons, muons) with minimal emittance is a challenging looks like problem in contemporary beam physics. -
Dirac-Exact Relativistic Methods: the Normalized Elimination of the Small Component Method† Dieter Cremer,∗ Wenli Zou and Michael Filatov
Advanced Review Dirac-exact relativistic methods: the normalized elimination of the small component method† Dieter Cremer,∗ Wenli Zou and Michael Filatov Dirac-exact relativistic methods, i.e., 2- or 1-component methods which exactly reproduce the one-electron energies of the original 4-component Dirac method, have established a standard for reliable relativistic quantum chemical calculations targeting medium- and large-sized molecules. Their development was initiated and facilitated in the late 1990s by Dyall’s development of the normalized elimination of the small component (NESC). Dyall’s work has fostered the conversion of NESC and related (later developed) methods into routinely used, multipurpose Dirac- exact methods by which energies, first-order, and second-order properties can be calculated at computational costs, which are only slightly higher than those of nonrelativistic methods. This review summarizes the development of a generally applicable 1-component NESC algorithm leading to the calculation of reliable energies, geometries, electron density distributions, electric moments, electric field gradients, hyperfine structure constants, contact densities and Mossbauer¨ isomer shifts, nuclear quadrupole coupling constants, vibrational frequencies, infrared intensities, and static electric dipole polarizabilities. In addition, the derivation and computational possibilities of 2-component NESC methods are discussed and their use for the calculation of spin-orbit coupling (SOC) effects in connection with spin-orbit splittings and SOC-corrected -
Eigenvalues of the Breit Equation
Prog. Theor. Exp. Phys. 2013, 063B03 (13 pages) DOI: 10.1093/ptep/ptt041 Eigenvalues of the Breit equation Yoshio Yamaguchi1,† and Hikoya Kasari2,∗ 1Nishina Center, RIKEN, 350-0198 2-1 Hirosawa, Wako, Saitama, Japan 2Department of Physics, School of Science, Tokai University 259-1292 4-1-1 Kitakaname, Hiratsuka, Kanagawa, Japan ∗E-mail: [email protected] Received January 9, 2013; Accepted April 12, 2013; Published June 1, 2013 ............................................................................... Eigenvalues of the Breit equation Downloaded from e2 (α p + β m)αα δββ + δαα (−α p + β M)ββ − δαα δββ αβ = Eαβ , 1 1 2 2 r in which only the static Coulomb potential is considered, have been found. Here a detailed 1 discussion of the simple cases, S0, m = M, and m = M, is given, deriving the exact energy eigenvalues. An α2 expansion is used to find radial wave functions. The leading term is given by http://ptep.oxfordjournals.org/ the classical Coulomb wave function. The technique used here can be applied to other cases. ............................................................................... Subject Index B87 1. Introduction and the Breit equation at CERN LIBRARY on June 21, 2013 1.1. Introduction The Breit equation [1–3] has traditionally [4] been considered to be singular, and no attempt has ever been made to solve it. Instead, the Pauli approximation or a generalized Foldy–Wouthuysen trans- formation has been applied to derive the effective Hamiltonian Heff, consisting of the Schrödinger Hamiltonian 1 1 1 e2 H =− + p2 − 0 2 m M r and many other relativistic correction terms [4–6]. The perturbation method was then used to evaluate the eigenvalues (binding energies) in a power series with α as high as possible (or as high as necessary for experimental verification of quantum electrodynamics). -
On the Spectra of Atoms and Hadrons
TTP/05-22 Nov. 2005 On the spectra of atoms and hadrons Hartmut Pilkuhn Institut f¨ur Theoretische Teilchenphysik, Universit¨at, D-76128 Karlsruhe, Germany For relativistic closed systems, an operator is explained which has as stationary eigenvalues the squares of the total cms energies, while the wave function has only half as many components as the corresponding Dirac wave function. The operator’s time dependence is generalized to a Klein-Gordon equation. It ensures relativistic kinematics in radiative decays. The new operator is not hermitian. Energy levels of bound states are calculated by a variety of methods, which include relativity at least approximately. In atomic theory, the equation HψD = EψD is used, where H is the n-body Dirac-Breit Hamiltonian, and ψD is an n-electron Dirac spinor with 4n components. For half a century, great hopes were attached to the Bethe-Salpeter equation, for example in the calculation of positronium spectra [1]. Most of these methods have been adapted to hadrons, namely to mesons as quark-antiquark bound states and baryons as three-quark bound states. New methods such as nonrelativistic quantum electro- dynamics (NRQED) and numerical calculations on a space-time lattice have been added. In this note, the recent extension of another new method to radiative decays is presented. 2 2 Its time-independent form is similar to HψD = EψD, namely M ψ = E ψ, but ψ has only half as many components as ψD. It applies only to closed systems, where E denotes the total cms energy. For atoms, this implies a relativistic inclusion of the nucleus, which is of little practical importance.