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Elementary Particles: an Introduction
Elementary Particles: An Introduction By Dr. Mahendra Singh Deptt. of Physics Brahmanand College, Kanpur What is Particle Physics? • Study the fundamental interactions and constituents of matter? • The Big Questions: – Where does mass come from? – Why is the universe made mostly of matter? – What is the missing mass in the Universe? – How did the Universe begin? Fundamental building blocks of which all matter is composed: Elementary Particles *Pre-1930s it was thought there were just four elementary particles electron proton neutron photon 1932 positron or anti-electron discovered, followed by many other particles (muon, pion etc) We will discover that the electron and photon are indeed fundamental, elementary particles, but protons and neutrons are made of even smaller elementary particles called quarks Four Fundamental Interactions Gravitational Electromagnetic Strong Weak Infinite Range Forces Finite Range Forces Exchange theory of forces suggests that to every force there will be a mediating particle(or exchange particle) Force Exchange Particle Gravitational Graviton Not detected so for EM Photon Strong Pi mesons Weak Intermediate vector bosons Range of a Force R = c Δt c: velocity of light Δt: life time of mediating particle Uncertainity relation: ΔE Δt=h/2p mc2 Δt=h/2p R=h/2pmc So R α 1/m If m=0, R→∞ As masses of graviton and photon are zero, range is infinite for gravitational and EM interactions Since pions and vector bosons have finite mass, strong and weak forces have finite range. Properties of Fundamental Interactions Interaction -
Chapter 5 the Relativistic Point Particle
Chapter 5 The Relativistic Point Particle To formulate the dynamics of a system we can write either the equations of motion, or alternatively, an action. In the case of the relativistic point par- ticle, it is rather easy to write the equations of motion. But the action is so physical and geometrical that it is worth pursuing in its own right. More importantly, while it is difficult to guess the equations of motion for the rela- tivistic string, the action is a natural generalization of the relativistic particle action that we will study in this chapter. We conclude with a discussion of the charged relativistic particle. 5.1 Action for a relativistic point particle How can we find the action S that governs the dynamics of a free relativis- tic particle? To get started we first think about units. The action is the Lagrangian integrated over time, so the units of action are just the units of the Lagrangian multiplied by the units of time. The Lagrangian has units of energy, so the units of action are L2 ML2 [S]=M T = . (5.1.1) T 2 T Recall that the action Snr for a free non-relativistic particle is given by the time integral of the kinetic energy: 1 dx S = mv2(t) dt , v2 ≡ v · v, v = . (5.1.2) nr 2 dt 105 106 CHAPTER 5. THE RELATIVISTIC POINT PARTICLE The equation of motion following by Hamilton’s principle is dv =0. (5.1.3) dt The free particle moves with constant velocity and that is the end of the story. -
Theory of More Than Everything1
Universally of Marineland Alimentary Gastronomy Universe of Murray Gell-Mann Elementary My Dear Watson Unified Theory of My Elementary Participles .. ^ n THEORY OF MORE THAN EVERYTHING1 V. Gates, Empty Kangaroo, M. Roachcock, and W.C. Gall 2 Compartment of Physiques and Astrology Universally of Marineland, Alleged kraP, MD ABSTRACT We derive a theory which, after spontaneous, dynamical, and ad hoc symmetry breaking, and after elimination of all fields except a set of zero measure, produces 10-dimensional superstring theory. Since the latter is a theory of only everything, our theory describes much more than everything, and includes also anything, something, and nothing. (More text should go here. So sue me.) 1Work supported by little or no evidence. 2Address after September 1, 1988: ITP, SHIITE, Roc(e)ky Brook, NY Uniformity of Modern Elementary Particle Physics Unintelligibility of Many Elementary Particle Physicists Universal City of Movieland Alimony Parties You Truth is funnier than fiction | A no-name moose1] Publish or parish | J.C. Polkinghorne Gimme that old minimal supergravity. Gimme that old minimal supergravity. It was good enough for superstrings. 1 It's good enough for me. ||||| Christian Physicist hymn1 2 ] 2. CONCLUSIONS The standard model has by now become almost standard. However, there are at least 42 constants which it doesn't explain. As is well known, this requires a 42 theory with at least @0 times more particles in its spectrum. Unfortunately, so far not all of these 42 new levels of complexity have been discovered; those now known are: (1) grandiose unification | SU(5), SO(10), E6,E7,E8,B12, and Niacin; (2) supersummitry2]; (3) supergravy2]; (4) supursestrings3−5]. -
Gravitational Field of Massive Point Particle in General Relativity
Gravitational Field of Massive Point Particle in General Relativity P. P. Fiziev∗ Department of Theoretical Physics, Faculty of Physics, Sofia University, 5 James Bourchier Boulevard, Sofia 1164, Bulgaria. and The Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, 34014 Trieste, Italy. Utilizing various gauges of the radial coordinate we give a description of static spherically sym- metric space-times with point singularity at the center and vacuum outside the singularity. We show that in general relativity (GR) there exist a two-parameters family of such solutions to the Einstein equations which are physically distinguishable but only some of them describe the gravitational field of a single massive point particle with nonzero bare mass M0. In particular, we show that the widespread Hilbert’s form of Schwarzschild solution, which depends only on the Keplerian mass M < M0, does not solve the Einstein equations with a massive point particle’s stress-energy tensor as a source. Novel normal coordinates for the field and a new physical class of gauges are proposed, in this way achieving a correct description of a point mass source in GR. We also introduce a gravi- tational mass defect of a point particle and determine the dependence of the solutions on this mass − defect. The result can be described as a change of the Newton potential ϕN = GN M/r to a modi- M − 2 0 fied one: ϕG = GN M/ r + GN M/c ln M and a corresponding modification of the four-interval. In addition we give invariant characteristics of the physically and geometrically different classes of spherically symmetric static space-times created by one point mass. -
Ph 406: Elementary Particle Physics Problem Set 2 K.T
Ph 406: Elementary Particle Physics Problem Set 2 K.T. McDonald [email protected] Princeton University Due Monday, September 29, 2014 (updated September 20, 2016) 1. The reactions π±p → μ+μ− are thought to proceed via single- photon exchange according to the so-called Drell-Yan diagram. Use the quark model to predict the cross-section ratio σπ−p→μ+μ− . σπ+p→μ+μ− 2. Discuss the motion of an electron of charge −e and rest mass m that is at rest on average inside a plane electromagnetic wave which propagates in the +z direction of a rectangular coordinate system. Suppose the wave is linearly polarized along x, Ewave = xˆE0 cos(kz − ωt), Bwave = yˆE0 cos(kz − ωt), (1) where ω = kc is the angular frequency of the wave, k =2π/λ is the wave number, c is the speed of light in vacuum, and xˆ is a unit vector in the x direction. Consider only weak fields, for which the dimensionless field-strength parameter η 1, where eE η = 0 . (2) mωc First, ignore the longitudinal motion, and deduce the transverse motion, expressing its amplitude in terms of η and λ. Then, in a “macroscopic” view which averages over the “microscopic” motion, the time-average total energy of the electron can be regarded as mc2,wherem>mis the effective mass of the electron (considered as a quasiparticle in the quantum view). That is, the “background” electromagnetic field has “given” mass to the electron beyond that in zero field. This is an electromagnetic version of the Higgs (Kibble) mechanism.1 Also, deduce the form of the longitudinal motion for η 1. -
Fall 2009 PHYS 172: Modern Mechanics
PHYS 172: Modern Mechanics Fall 2009 Lecture 16 - Multiparticle Systems; Friction in Depth Read 8.6 Exam #2: Multiple Choice: average was 56/70 Handwritten: average was 15/30 Total average was 71/100 Remember, we are using an absolute grading scale where the values for each grade are listed on the syllabus Typically, exam grades (500/820 points) are lower than homework, recitation, clicker, & labs grades Clicker Question #1 You get in a parked car and start driving. Assume that you travel a distance D in 10 seconds, with a constant acceleration a. Define the system to be the car plus the driver (you), whose total mass is M and acceleration a. Choose the correct statement for this system: A. The final total kinetic energy is Ktotal= MaD. B. The final translational kinetic energy is Ktrans= MaD. C. The total energy of the system changes by +MaD. D. The total energy of the system changes by -MaD.. E. The total energy of the system does not change. Clicker Question Discussion You get in a parked car and start driving. Assume that you travel a distance D in 10 seconds, with a constant acceleration a. Define the system to be the car plus the driver (you), whose total mass is M and acceleration a. Choose the correct statement for this system: A. The final total kinetic energy is Ktotal= MaD.No, this is just the translational K. There is also relative K (wheels, etc.) B. The final translational kinetic energy is Ktrans= MaD. Correct, use point particle system; force is friction applied by the road on the tires. -
The Larmor Formula (Chapters 18-19)
The Larmor Formula (Chapters 18-19) T. Johnson 2017-02-28 Dispersive Media, Lecture 12 - Thomas Johnson 1 Outline • Brief repetition of emission formula • The emission from a single free particle - the Larmor formula • Applications of the Larmor formula – Harmonic oscillator – Cyclotron radiation – Thompson scattering – Bremstrahlung Next lecture: • Relativistic generalisation of Larmor formula – Repetition of basic relativity – Co- and contra-variant tensor notation and Lorentz transformations – Relativistic Larmor formula • The Lienard-Wiechert potentials – Inductive and radiative electromagnetic fields – Alternative derivation of the Larmor formula • Abraham-Lorentz force 2017-02-28 Dispersive Media, Lecture 12 - Thomas Johnson 2 Repetition: Emission formula • The energy emitted by a wave mode M (using antihermitian part of the propagator), when integrating over the δ-function in ω – the emission formula for UM ; the density of emission in k-space • Emission per frequency and solid angle '( (+) – Rewrite integral: �"� = �%���%Ω = �% ) ���%Ω '+ Here �/ is the unit vector in the �-direction 2017-02-28 Dispersive Media, Lecture 12 - Thomas Johnson 3 Repetition: Emission from multipole moments • Multipole moments are related to the Fourier transform of the current: Emission formula Emission formula (k-space power density) (integrated over solid angles) 2017-02-28 Dispersive Media, Lecture 12 - Thomas Johnson 4 Current from a single particle • Let’s calculate the radiation from a single particle – at X(t) with charge q. – The density, n, and current, J, from the particle: – or in Fourier space 5 � �, � = � 3 �� �68+9 3 �"� �8�:� �̇ � � � − � � = 65 5 = � 3 �� �68+9 1 + � � : �(�) + ⋯ �̇ � = 65 5 = −���� � + 3 �� �68+9 � � : �(�) �̇ � + ⋯ 65 Dipole: d=qX 2017-02-28 Dispersive Media, Lecture 12 - Thomas Johnson 5 Dipole current from single particle • Thus, the field from a single particle is approximately a dipole field • When is this approximation valid? – Assume oscillating motion: - The dipole approximation is based on the small term: Dipole approx. -
“WHAT IS an ELECTRON?” Contents 1. Quantum Mechanics in A
\WHAT IS AN ELECTRON?" OR PARTICLES AS REPRESENTATIONS ROK GREGORIC Abstract. In these informal notes, we attempt to offer some justification for the math- ematical physicist's answer: \A certain kind of representation.". This requires going through some reasonably basic physical ideas, such as the barest basics of quantum mechanics and special relativity. Contents 1. Quantum Mechanics in a Nutshell2 2. Symmetry7 3. Toward Particles 11 4. Special Relativity in a Nutshell 14 5. Spin 21 6. Particles in Relativistic Quantum Mechaniscs 25 0.1. What these notes are. This are higly informal notes on basic physics. Being a mathematician-in-training myself, I am afraid I am incapable of writing for any other audience. Thus these notes must fall into the unfortunate genre of \physics for math- ematicians", an idiosyncracy on the level of \music for the deaf" or \visual art for the blind" - surely possible in some fashion, but only after overcoming substantial technical obstacles, and even then likely to be accussed by the cognoscenti of \missing the point" of the original. And yet we beat on, boats against the current, . 0.2. How they came about. As usual, this write-up began its life in the form of a series of emails to my good friend Tom Gannon. Allow me to recount, only a tinge apocryphally, one major impetus for its existence: the 2020 New Year's party. Hosted at Tom's apartment, it featured a decent contingent of UT math department's grad student compartment. Among others in attendence were Ivan Tulli, a senior UT grad student working on mathematics inperceptably close to theoretical physics, and his wife Anabel, a genuine experimental physicist from another department, in town to visit her husband. -
PHYS 352 Electromagnetic Waves
Part 1: Fundamentals These are notes for the first part of PHYS 352 Electromagnetic Waves. This course follows on from PHYS 350. At the end of that course, you will have seen the full set of Maxwell's equations, which in vacuum are ρ @B~ r~ · E~ = r~ × E~ = − 0 @t @E~ r~ · B~ = 0 r~ × B~ = µ J~ + µ (1.1) 0 0 0 @t with @ρ r~ · J~ = − : (1.2) @t In this course, we will investigate the implications and applications of these results. We will cover • electromagnetic waves • energy and momentum of electromagnetic fields • electromagnetism and relativity • electromagnetic waves in materials and plasmas • waveguides and transmission lines • electromagnetic radiation from accelerated charges • numerical methods for solving problems in electromagnetism By the end of the course, you will be able to calculate the properties of electromagnetic waves in a range of materials, calculate the radiation from arrangements of accelerating charges, and have a greater appreciation of the theory of electromagnetism and its relation to special relativity. The spirit of the course is well-summed up by the \intermission" in Griffith’s book. After working from statics to dynamics in the first seven chapters of the book, developing the full set of Maxwell's equations, Griffiths comments (I paraphrase) that the full power of electromagnetism now lies at your fingertips, and the fun is only just beginning. It is a disappointing ending to PHYS 350, but an exciting place to start PHYS 352! { 2 { Why study electromagnetism? One reason is that it is a fundamental part of physics (one of the four forces), but it is also ubiquitous in everyday life, technology, and in natural phenomena in geophysics, astrophysics or biophysics. -
Chapter 11 the Relativistic Quantum Point Particle
Chapter 11 The Relativistic Quantum Point Particle To prepare ourselves for quantizing the string, we study the light-cone gauge quantization of the relativistic point particle. We set up the quantum theory by requiring that the Heisenberg operators satisfy the classical equations of motion. We show that the quantum states of the relativistic point particle coincide with the one-particle states of the quantum scalar field. Moreover, the Schr¨odinger equation for the particle wavefunctions coincides with the classical scalar field equations. Finally, we set up light-cone gauge Lorentz generators. 11.1 Light-cone point particle In this section we study the classical relativistic point particle using the light-cone gauge. This is, in fact, a much easier task than the one we faced in Chapter 9, where we examined the classical relativistic string in the light- cone gauge. Our present discussion will allow us to face the complications of quantization in the simpler context of the particle. Many of the ideas needed to quantize the string are also needed to quantize the point particle. The action for the relativistic point particle was studied in Chapter 5. Let’s begin our analysis with the expression given in equation (5.2.4), where an arbitrary parameter τ is used to parameterize the motion of the particle: τf dxµ dxν S = −m −ηµν dτ . (11.1.1) τi dτ dτ 249 250 CHAPTER 11. RELATIVISTIC QUANTUM PARTICLE In writing the above action, we have set c =1.Wewill also set =1when appropriate. Finally, the time parameter τ will be dimensionless, just as it was for the relativistic string. -
Neutron Star Equation of State Via Gravitational Wave Observations
Neutron star equation of state via gravitational wave observations C Markakis1, J S Read2, M Shibata3, K Uryu¯ 4, J D E Creighton1, J L Friedman1, and B D Lackey1 1 Department of Physics, University of Wisconsin–Milwaukee, P.O. Box 413, Milwaukee, WI 53201, USA 2 Max-Planck-Institut fur¨ Gravitationsphysik, Albert-Einstein-Institut, Golm, Germany 3 Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan 4 Department of Physics, University of the Ryukyus, 1 Senbaru, Nishihara, Okinawa 903-0213, Japan Abstract. Gravitational wave observations can potentially measure properties of neutron star equations of state by measuring departures from the point-particle limit of the gravitational waveform produced in the late inspiral of a neutron star binary. Numerical simulations of inspiraling neutron star binaries computed for equations of state with varying stiffness are compared. As the stars approach their final plunge and merger, the gravitational wave phase accumulates more rapidly if the neutron stars are more compact. This suggests that gravitational wave observations at frequencies around 1 kHz will be able to measure a compactness parameter and place stringent bounds on possible neutron star equations of state. Advanced laser interferometric gravitational wave observatories will be able to tune their frequency band to optimize sensitivity in the required frequency range to make sensitive measures of the late-inspiral phase of the coalescence. 1. Introduction In numerical simulations of the late inspiral and merger of binary neutron-star systems, one commonly specifies an equation of state for the matter, perform a numerical simulation and extract the gravitational waveforms produced in the inspiral. -
What Is a Photon? Foundations of Quantum Field Theory
What is a Photon? Foundations of Quantum Field Theory C. G. Torre June 16, 2018 2 What is a Photon? Foundations of Quantum Field Theory Version 1.0 Copyright c 2018. Charles Torre, Utah State University. PDF created June 16, 2018 Contents 1 Introduction 5 1.1 Why do we need this course? . 5 1.2 Why do we need quantum fields? . 5 1.3 Problems . 6 2 The Harmonic Oscillator 7 2.1 Classical mechanics: Lagrangian, Hamiltonian, and equations of motion . 7 2.2 Classical mechanics: coupled oscillations . 8 2.3 The postulates of quantum mechanics . 10 2.4 The quantum oscillator . 11 2.5 Energy spectrum . 12 2.6 Position, momentum, and their continuous spectra . 15 2.6.1 Position . 15 2.6.2 Momentum . 18 2.6.3 General formalism . 19 2.7 Time evolution . 20 2.8 Coherent States . 23 2.9 Problems . 24 3 Tensor Products and Identical Particles 27 3.1 Definition of the tensor product . 27 3.2 Observables and the tensor product . 31 3.3 Symmetric and antisymmetric tensors. Identical particles. 32 3.4 Symmetrization and anti-symmetrization for any number of particles . 35 3.5 Problems . 36 4 Fock Space 38 4.1 Definitions . 38 4.2 Occupation numbers. Creation and annihilation operators. 40 4.3 Observables. Field operators. 43 4.3.1 1-particle observables . 43 4.3.2 2-particle observables . 46 4.3.3 Field operators and wave functions . 47 4.4 Time evolution of the field operators . 49 4.5 General formalism . 51 3 4 CONTENTS 4.6 Relation to the Hilbert space of quantum normal modes .