Chapter 9. Introduction to Relativistic Quantum Mechanics
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Unit 1 Old Quantum Theory
UNIT 1 OLD QUANTUM THEORY Structure Introduction Objectives li;,:overy of Sub-atomic Particles Earlier Atom Models Light as clectromagnetic Wave Failures of Classical Physics Black Body Radiation '1 Heat Capacity Variation Photoelectric Effect Atomic Spectra Planck's Quantum Theory, Black Body ~diation. and Heat Capacity Variation Einstein's Theory of Photoelectric Effect Bohr Atom Model Calculation of Radius of Orbits Energy of an Electron in an Orbit Atomic Spectra and Bohr's Theory Critical Analysis of Bohr's Theory Refinements in the Atomic Spectra The61-y Summary Terminal Questions Answers 1.1 INTRODUCTION The ideas of classical mechanics developed by Galileo, Kepler and Newton, when applied to atomic and molecular systems were found to be inadequate. Need was felt for a theory to describe, correlate and predict the behaviour of the sub-atomic particles. The quantum theory, proposed by Max Planck and applied by Einstein and Bohr to explain different aspects of behaviour of matter, is an important milestone in the formulation of the modern concept of atom. In this unit, we will study how black body radiation, heat capacity variation, photoelectric effect and atomic spectra of hydrogen can be explained on the basis of theories proposed by Max Planck, Einstein and Bohr. They based their theories on the postulate that all interactions between matter and radiation occur in terms of definite packets of energy, known as quanta. Their ideas, when extended further, led to the evolution of wave mechanics, which shows the dual nature of matter -
Stochastic Quantization of Fermionic Theories: Renormalization of the Massive Thirring Model
Instituto de Física Teórica IFT Universidade Estadual Paulista October/92 IFT-R043/92 Stochastic Quantization of Fermionic Theories: Renormalization of the Massive Thirring Model J.C.Brunelli Instituto de Física Teórica Universidade Estadual Paulista Rua Pamplona, 145 01405-900 - São Paulo, S.P. Brazil 'This work was supported by CNPq. Instituto de Física Teórica Universidade Estadual Paulista Rua Pamplona, 145 01405 - Sao Paulo, S.P. Brazil Telephone: 55 (11) 288-5643 Telefax: 55(11)36-3449 Telex: 55 (11) 31870 UJMFBR Electronic Address: [email protected] 47553::LIBRARY Stochastic Quantization of Fermionic Theories: 1. Introduction Renormalization of the Massive Thimng Model' The stochastic quantization method of Parisi-Wu1 (for a review see Ref. 2) when applied to fermionic theories usually requires the use of a Langevin system modified by the introduction of a kernel3 J. C. Brunelli (1.1a) InBtituto de Física Teórica (1.16) Universidade Estadual Paulista Rua Pamplona, 145 01405 - São Paulo - SP where BRAZIL l = 2Kah(x,x )8(t - ?). (1.2) Here tj)1 tp and the Gaussian noises rj, rj are independent Grassmann variables. K(xty) is the aforementioned kernel which ensures the proper equilibrium limit configuration for Accepted for publication in the International Journal of Modern Physics A. mas si ess theories. The specific form of the kernel is quite arbitrary but in what follows, we use K(x,y) = Sn(x-y)(-iX + ™)- Abstract In a number of cases, it has been verified that the stochastic quantization procedure does not bring new anomalies and that the equilibrium limit correctly reproduces the basic (jJsfsini g the Langevin approach for stochastic processes we study the renormalizability properties of the models considered4. -
Mesons Modeled Using Only Electrons and Positrons with Relativistic Onium Theory Ray Fleming [email protected]
All mesons modeled using only electrons and positrons with relativistic onium theory Ray Fleming [email protected] All mesons were investigated to determine if they can be modeled with the onium model discovered by Milne, Feynman, and Sternglass with only electrons and positrons. They discovered the relativistic positronium solution has the mass of a neutral pion and the relativistic onium mass increases in steps of me/α and me/2α per particle which is consistent with known mass quantization. Any pair of particles or resonances can orbit relativistically and particles and resonances can collocate to form increasingly complex resonances. Pions are positronium, kaons are pionium, D mesons are kaonium, and B mesons are Donium in the onium model. Baryons, which are addressed in another paper, have a non-relativistic nucleon combined with mesons. The results of this meson analysis shows that the compo- sition, charge, and mass of all mesons can be accurately modeled. Of the 220 mesons mod- eled, 170 mass estimates are within 5 MeV/c2 and masses of 111 of 121 D, B, charmonium, and bottomonium mesons are estimated to within 0.2% relative error. Since all mesons can be modeled using only electrons and positrons, quarks and quark theory are unnecessary. 1. Introduction 2. Method This paper is a report on an investigation to find Sternglass and Browne realized that a neutral pion whether mesons can be modeled as combinations of (π0), as a relativistic electron-positron pair, can orbit a only electrons and positrons using onium theory. A non-relativistic particle or resonance in what Browne companion paper on baryons is also available. -
Motion of the Reduced Density Operator
Motion of the Reduced Density Operator Nicholas Wheeler, Reed College Physics Department Spring 2009 Introduction. Quantum mechanical decoherence, dissipation and measurements all involve the interaction of the system of interest with an environmental system (reservoir, measurement device) that is typically assumed to possess a great many degrees of freedom (while the system of interest is typically assumed to possess relatively few degrees of freedom). The state of the composite system is described by a density operator ρ which in the absence of system-bath interaction we would denote ρs ρe, though in the cases of primary interest that notation becomes unavailable,⊗ since in those cases the states of the system and its environment are entangled. The observable properties of the system are latent then in the reduced density operator ρs = tre ρ (1) which is produced by “tracing out” the environmental component of ρ. Concerning the specific meaning of (1). Let n) be an orthonormal basis | in the state space H of the (open) system, and N) be an orthonormal basis s !| " in the state space H of the (also open) environment. Then n) N) comprise e ! " an orthonormal basis in the state space H = H H of the |(closed)⊗| composite s e ! " system. We are in position now to write ⊗ tr ρ I (N ρ I N) e ≡ s ⊗ | s ⊗ | # ! " ! " ↓ = ρ tr ρ in separable cases s · e The dynamics of the composite system is generated by Hamiltonian of the form H = H s + H e + H i 2 Motion of the reduced density operator where H = h I s s ⊗ e = m h n m N n N $ | s| % | % ⊗ | % · $ | ⊗ $ | m,n N # # $% & % &' H = I h e s ⊗ e = n M n N M h N | % ⊗ | % · $ | ⊗ $ | $ | e| % n M,N # # $% & % &' H = m M m M H n N n N i | % ⊗ | % $ | ⊗ $ | i | % ⊗ | % $ | ⊗ $ | m,n M,N # # % &$% & % &'% & —all components of which we will assume to be time-independent. -
Quantum Theory of the Hydrogen Atom
Quantum Theory of the Hydrogen Atom Chemistry 35 Fall 2000 Balmer and the Hydrogen Spectrum n 1885: Johann Balmer, a Swiss schoolteacher, empirically deduced a formula which predicted the wavelengths of emission for Hydrogen: l (in Å) = 3645.6 x n2 for n = 3, 4, 5, 6 n2 -4 •Predicts the wavelengths of the 4 visible emission lines from Hydrogen (which are called the Balmer Series) •Implies that there is some underlying order in the atom that results in this deceptively simple equation. 2 1 The Bohr Atom n 1913: Niels Bohr uses quantum theory to explain the origin of the line spectrum of hydrogen 1. The electron in a hydrogen atom can exist only in discrete orbits 2. The orbits are circular paths about the nucleus at varying radii 3. Each orbit corresponds to a particular energy 4. Orbit energies increase with increasing radii 5. The lowest energy orbit is called the ground state 6. After absorbing energy, the e- jumps to a higher energy orbit (an excited state) 7. When the e- drops down to a lower energy orbit, the energy lost can be given off as a quantum of light 8. The energy of the photon emitted is equal to the difference in energies of the two orbits involved 3 Mohr Bohr n Mathematically, Bohr equated the two forces acting on the orbiting electron: coulombic attraction = centrifugal accelleration 2 2 2 -(Z/4peo)(e /r ) = m(v /r) n Rearranging and making the wild assumption: mvr = n(h/2p) n e- angular momentum can only have certain quantified values in whole multiples of h/2p 4 2 Hydrogen Energy Levels n Based on this model, Bohr arrived at a simple equation to calculate the electron energy levels in hydrogen: 2 En = -RH(1/n ) for n = 1, 2, 3, 4, . -
Configuration Interaction Study of the Ground State of the Carbon Atom
Configuration Interaction Study of the Ground State of the Carbon Atom María Belén Ruiz* and Robert Tröger Department of Theoretical Chemistry Friedrich-Alexander-University Erlangen-Nürnberg Egerlandstraße 3, 91054 Erlangen, Germany In print in Advances Quantum Chemistry Vol. 76: Novel Electronic Structure Theory: General Innovations and Strongly Correlated Systems 30th July 2017 Abstract Configuration Interaction (CI) calculations on the ground state of the C atom are carried out using a small basis set of Slater orbitals [7s6p5d4f3g]. The configurations are selected according to their contribution to the total energy. One set of exponents is optimized for the whole expansion. Using some computational techniques to increase efficiency, our computer program is able to perform partially-parallelized runs of 1000 configuration term functions within a few minutes. With the optimized computer programme we were able to test a large number of configuration types and chose the most important ones. The energy of the 3P ground state of carbon atom with a wave function of angular momentum L=1 and ML=0 and spin eigenfunction with S=1 and MS=0 leads to -37.83526523 h, which is millihartree accurate. We discuss the state of the art in the determination of the ground state of the carbon atom and give an outlook about the complex spectra of this atom and its low-lying states. Keywords: Carbon atom; Configuration Interaction; Slater orbitals; Ground state *Corresponding author: e-mail address: [email protected] 1 1. Introduction The spectrum of the isolated carbon atom is the most complex one among the light atoms. The ground state of carbon atom is a triplet 3P state and its low-lying excited states are singlet 1D, 1S and 1P states, more stable than the corresponding triplet excited ones 3D and 3S, against the Hund’s rule of maximal multiplicity. -
Quantum Field Theory*
Quantum Field Theory y Frank Wilczek Institute for Advanced Study, School of Natural Science, Olden Lane, Princeton, NJ 08540 I discuss the general principles underlying quantum eld theory, and attempt to identify its most profound consequences. The deep est of these consequences result from the in nite number of degrees of freedom invoked to implement lo cality.Imention a few of its most striking successes, b oth achieved and prosp ective. Possible limitation s of quantum eld theory are viewed in the light of its history. I. SURVEY Quantum eld theory is the framework in which the regnant theories of the electroweak and strong interactions, which together form the Standard Mo del, are formulated. Quantum electro dynamics (QED), b esides providing a com- plete foundation for atomic physics and chemistry, has supp orted calculations of physical quantities with unparalleled precision. The exp erimentally measured value of the magnetic dip ole moment of the muon, 11 (g 2) = 233 184 600 (1680) 10 ; (1) exp: for example, should b e compared with the theoretical prediction 11 (g 2) = 233 183 478 (308) 10 : (2) theor: In quantum chromo dynamics (QCD) we cannot, for the forseeable future, aspire to to comparable accuracy.Yet QCD provides di erent, and at least equally impressive, evidence for the validity of the basic principles of quantum eld theory. Indeed, b ecause in QCD the interactions are stronger, QCD manifests a wider variety of phenomena characteristic of quantum eld theory. These include esp ecially running of the e ective coupling with distance or energy scale and the phenomenon of con nement. -
B2.IV Nuclear and Particle Physics
B2.IV Nuclear and Particle Physics A.J. Barr February 13, 2014 ii Contents 1 Introduction 1 2 Nuclear 3 2.1 Structure of matter and energy scales . 3 2.2 Binding Energy . 4 2.2.1 Semi-empirical mass formula . 4 2.3 Decays and reactions . 8 2.3.1 Alpha Decays . 10 2.3.2 Beta decays . 13 2.4 Nuclear Scattering . 18 2.4.1 Cross sections . 18 2.4.2 Resonances and the Breit-Wigner formula . 19 2.4.3 Nuclear scattering and form factors . 22 2.5 Key points . 24 Appendices 25 2.A Natural units . 25 2.B Tools . 26 2.B.1 Decays and the Fermi Golden Rule . 26 2.B.2 Density of states . 26 2.B.3 Fermi G.R. example . 27 2.B.4 Lifetimes and decays . 27 2.B.5 The flux factor . 28 2.B.6 Luminosity . 28 2.C Shell Model § ............................. 29 2.D Gamma decays § ............................ 29 3 Hadrons 33 3.1 Introduction . 33 3.1.1 Pions . 33 3.1.2 Baryon number conservation . 34 3.1.3 Delta baryons . 35 3.2 Linear Accelerators . 36 iii CONTENTS CONTENTS 3.3 Symmetries . 36 3.3.1 Baryons . 37 3.3.2 Mesons . 37 3.3.3 Quark flow diagrams . 38 3.3.4 Strangeness . 39 3.3.5 Pseudoscalar octet . 40 3.3.6 Baryon octet . 40 3.4 Colour . 41 3.5 Heavier quarks . 43 3.6 Charmonium . 45 3.7 Hadron decays . 47 Appendices 48 3.A Isospin § ................................ 49 3.B Discovery of the Omega § ...................... -
Generalized Lorentz Symmetry and Nonlinear Spinor Fields in a Flat Finslerian Space-Time
Proceedings of Institute of Mathematics of NAS of Ukraine 2004, Vol. 50, Part 2, 637–644 Generalized Lorentz Symmetry and Nonlinear Spinor Fields in a Flat Finslerian Space-Time George BOGOSLOVSKY † and Hubert GOENNER ‡ † Skobeltsyn Institute of Nuclear Physics, Moscow State University, Moscow, Russia E-mail: [email protected] ‡ Institute for Theoretical Physics, University of G¨ottingen, G¨ottingen, Germany E-mail: [email protected] The work is devoted to the generalization of the Dirac equation for a flat locally anisotropic, i.e. Finslerian space-time. At first we reproduce the corresponding metric and a group of the generalized Lorentz transformations, which has the meaning of the relativistic symmetry group of such event space. Next, proceeding from the requirement of the generalized Lorentz invariance we find a generalized Dirac equation in its explicit form. An exact solution of the nonlinear generalized Dirac equation is also presented. 1 Introduction In spite of the impressive successes of the unified gauge theory of strong, weak and electromag- netic interactions, known as the Standard Model, one cannot a priori rule out the possibility that Lorentz symmetry underlying the theory is an approximate symmetry of nature. This implies that at the energies already attainable today empirical evidence may be obtained in favor of violation of Lorentz symmetry. At the same time it is obvious that such effects might manifest themselves only as strongly suppressed effects of Planck-scale physics. Theoretical speculations about a possible violation of Lorentz symmetry continue for more than forty years and they are briefly outlined in [1]. -
Further Quantum Physics
Further Quantum Physics Concepts in quantum physics and the structure of hydrogen and helium atoms Prof Andrew Steane January 18, 2005 2 Contents 1 Introduction 7 1.1 Quantum physics and atoms . 7 1.1.1 The role of classical and quantum mechanics . 9 1.2 Atomic physics—some preliminaries . .... 9 1.2.1 Textbooks...................................... 10 2 The 1-dimensional projectile: an example for revision 11 2.1 Classicaltreatment................................. ..... 11 2.2 Quantum treatment . 13 2.2.1 Mainfeatures..................................... 13 2.2.2 Precise quantum analysis . 13 3 Hydrogen 17 3.1 Some semi-classical estimates . 17 3.2 2-body system: reduced mass . 18 3.2.1 Reduced mass in quantum 2-body problem . 19 3.3 Solution of Schr¨odinger equation for hydrogen . ..... 20 3.3.1 General features of the radial solution . 21 3.3.2 Precisesolution.................................. 21 3.3.3 Meanradius...................................... 25 3.3.4 How to remember hydrogen . 25 3.3.5 Mainpoints.................................... 25 3.3.6 Appendix on series solution of hydrogen equation, off syllabus . 26 3 4 CONTENTS 4 Hydrogen-like systems and spectra 27 4.1 Hydrogen-like systems . 27 4.2 Spectroscopy ........................................ 29 4.2.1 Main points for use of grating spectrograph . ...... 29 4.2.2 Resolution...................................... 30 4.2.3 Usefulness of both emission and absorption methods . 30 4.3 The spectrum for hydrogen . 31 5 Introduction to fine structure and spin 33 5.1 Experimental observation of fine structure . ..... 33 5.2 TheDiracresult ..................................... 34 5.3 Schr¨odinger method to account for fine structure . 35 5.4 Physical nature of orbital and spin angular momenta . -
Bouncing Oil Droplets, De Broglie's Quantum Thermostat And
Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 28 August 2018 doi:10.20944/preprints201808.0475.v1 Peer-reviewed version available at Entropy 2018, 20, 780; doi:10.3390/e20100780 Article Bouncing oil droplets, de Broglie’s quantum thermostat and convergence to equilibrium Mohamed Hatifi 1, Ralph Willox 2, Samuel Colin 3 and Thomas Durt 4 1 Aix Marseille Université, CNRS, Centrale Marseille, Institut Fresnel UMR 7249,13013 Marseille, France; hatifi[email protected] 2 Graduate School of Mathematical Sciences, the University of Tokyo, 3-8-1 Komaba, Meguro-ku, 153-8914 Tokyo, Japan; [email protected] 3 Centro Brasileiro de Pesquisas Físicas, Rua Dr. Xavier Sigaud 150,22290-180, Rio de Janeiro – RJ, Brasil; [email protected] 4 Aix Marseille Université, CNRS, Centrale Marseille, Institut Fresnel UMR 7249,13013 Marseille, France; [email protected] Abstract: Recently, the properties of bouncing oil droplets, also known as ‘walkers’, have attracted much attention because they are thought to offer a gateway to a better understanding of quantum behaviour. They indeed constitute a macroscopic realization of wave-particle duality, in the sense that their trajectories are guided by a self-generated surrounding wave. The aim of this paper is to try to describe walker phenomenology in terms of de Broglie-Bohm dynamics and of a stochastic version thereof. In particular, we first study how a stochastic modification of the de Broglie pilot-wave theory, à la Nelson, affects the process of relaxation to quantum equilibrium, and we prove an H-theorem for the relaxation to quantum equilibrium under Nelson-type dynamics. -
Density Matrix Description of NMR
Density Matrix Description of NMR BCMB/CHEM 8190 Operators in Matrix Notation • It will be important, and convenient, to express the commonly used operators in matrix form • Consider the operator Iz and the single spin functions α and β - recall ˆ ˆ ˆ ˆ ˆ ˆ Ix α = 1 2 β Ix β = 1 2α Iy α = 1 2 iβ Iy β = −1 2 iα Iz α = +1 2α Iz β = −1 2 β α α = β β =1 α β = β α = 0 - recall the expectation value for an observable Q = ψ Qˆ ψ = ∫ ψ∗Qˆψ dτ Qˆ - some operator ψ - some wavefunction - the matrix representation is the possible expectation values for the basis functions α β α ⎡ α Iˆ α α Iˆ β ⎤ ⎢ z z ⎥ ⎢ ˆ ˆ ⎥ β ⎣ β Iz α β Iz β ⎦ ⎡ α Iˆ α α Iˆ β ⎤ ⎡1 2 α α −1 2 α β ⎤ ⎡1 2 0 ⎤ ⎡1 0 ⎤ ˆ ⎢ z z ⎥ 1 Iz = = ⎢ ⎥ = ⎢ ⎥ = ⎢ ⎥ ⎢ ˆ ˆ ⎥ 1 2 β α −1 2 β β 0 −1 2 2 0 −1 ⎣ β Iz α β Iz β ⎦ ⎣⎢ ⎦⎥ ⎣ ⎦ ⎣ ⎦ • This is convenient, as the operator is just expressed as a matrix of numbers – no need to derive it again, just store it in computer Operators in Matrix Notation • The matrices for Ix, Iy,and Iz are called the Pauli spin matrices ˆ ⎡ 0 1 2⎤ 1 ⎡0 1 ⎤ ˆ ⎡0 −1 2 i⎤ 1 ⎡0 −i ⎤ ˆ ⎡1 2 0 ⎤ 1 ⎡1 0 ⎤ Ix = ⎢ ⎥ = ⎢ ⎥ Iy = ⎢ ⎥ = ⎢ ⎥ Iz = ⎢ ⎥ = ⎢ ⎥ ⎣1 2 0 ⎦ 2 ⎣1 0⎦ ⎣1 2 i 0 ⎦ 2 ⎣i 0 ⎦ ⎣ 0 −1 2⎦ 2 ⎣0 −1 ⎦ • Express α , β , α and β as 1×2 column and 2×1 row vectors ⎡1⎤ ⎡0⎤ α = ⎢ ⎥ β = ⎢ ⎥ α = [1 0] β = [0 1] ⎣0⎦ ⎣1⎦ • Using matrices, the operations of Ix, Iy, and Iz on α and β , and the orthonormality relationships, are shown below ˆ 1 ⎡0 1⎤⎡1⎤ 1 ⎡0⎤ 1 ˆ 1 ⎡0 1⎤⎡0⎤ 1 ⎡1⎤ 1 Ix α = ⎢ ⎥⎢ ⎥ = ⎢ ⎥ = β Ix β = ⎢ ⎥⎢ ⎥ = ⎢ ⎥ = α 2⎣1 0⎦⎣0⎦ 2⎣1⎦ 2 2⎣1 0⎦⎣1⎦ 2⎣0⎦ 2 ⎡1⎤ ⎡0⎤ α α = [1 0]⎢ ⎥