A Peek Into Spin Physics
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Path Integrals in Quantum Mechanics
Path Integrals in Quantum Mechanics Dennis V. Perepelitsa MIT Department of Physics 70 Amherst Ave. Cambridge, MA 02142 Abstract We present the path integral formulation of quantum mechanics and demon- strate its equivalence to the Schr¨odinger picture. We apply the method to the free particle and quantum harmonic oscillator, investigate the Euclidean path integral, and discuss other applications. 1 Introduction A fundamental question in quantum mechanics is how does the state of a particle evolve with time? That is, the determination the time-evolution ψ(t) of some initial | i state ψ(t ) . Quantum mechanics is fully predictive [3] in the sense that initial | 0 i conditions and knowledge of the potential occupied by the particle is enough to fully specify the state of the particle for all future times.1 In the early twentieth century, Erwin Schr¨odinger derived an equation specifies how the instantaneous change in the wavefunction d ψ(t) depends on the system dt | i inhabited by the state in the form of the Hamiltonian. In this formulation, the eigenstates of the Hamiltonian play an important role, since their time-evolution is easy to calculate (i.e. they are stationary). A well-established method of solution, after the entire eigenspectrum of Hˆ is known, is to decompose the initial state into this eigenbasis, apply time evolution to each and then reassemble the eigenstates. That is, 1In the analysis below, we consider only the position of a particle, and not any other quantum property such as spin. 2 D.V. Perepelitsa n=∞ ψ(t) = exp [ iE t/~] n ψ(t ) n (1) | i − n h | 0 i| i n=0 X This (Hamiltonian) formulation works in many cases. -
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. -
Unification of Nature's Fundamental Forces
Unification of Nature’s Geoffrey B. West Fredrick M. Cooper Fundamental Forces Emil Mottola a continuing search Michael P. Mattis it was explicitly recognized at the time that basic research had an im- portant and seminal role to play even in the highly programmatic en- vironment of the Manhattan Project. Not surprisingly this mode of opera- tion evolved into the remarkable and unique admixture of pure, applied, programmatic, and technological re- search that is the hallmark of the present Laboratory structure. No- where in the world today can one find under one roof such diversity of talent dealing with such a broad range of scientific and technological challenges—from questions con- cerning the evolution of the universe and the nature of elementary parti- cles to the structure of new materi- als, the design and control of weapons, the mysteries of the gene, and the nature of AIDS! Many of the original scientists would have, in today’s parlance, identified themselves as nuclear or particle physicists. They explored the most basic laws of physics and continued the search for and under- standing of the “fundamental build- ing blocks of nature’’ and the princi- t is a well-known, and much- grappled with deep questions con- ples that govern their interactions. overworked, adage that the group cerning the consequences of quan- It is therefore fitting that this area of Iof scientists brought to Los tum mechanics, the structure of the science has remained a highly visi- Alamos to work on the Manhattan atom and its nucleus, and the devel- ble and active component of the Project constituted the greatest as- opment of quantum electrodynamics basic research activity at Los Alam- semblage of scientific talent ever (QED, the relativistic quantum field os. -
People and Things
People and things An irresistible photograph: at a StAC Christmas party. Laboratory Director Pief Panofsky was presented with a CERN T-shirt, which he promptly put on. With him in the picture are (left to right) Roger Gear hart playing a seasonal master of ceremonies role, J. J. Murray and Ed Seppi. On people Elected vice-president of the Amer ican Physical Society for this year is Robert E. Marshak of Virginia Polytechnic Institute and State Uni versity. He succeeds Maurice Goldhaberr who becomes president elect. The new APS president is Arthur Schawlow of Stanford. In the same elections, Columbia theorist Malvin Ruder man was elected to serve for four years as councillor-at-large. Gisbert zu Pulitz, Scientific Director of the Darmstadt Heavy Ion Linear Accelerator Laboratory and Profes sor of Physics at the University of Heidelberg, has been elected as the new Chairman of the Association of German Research Centres (Ar- beitsgemeinschaft der Grossfor- schungseinrichtungen in der Bun- desrepublik Deutschland), succeed ing Herwig Schopper. The Associa the American Association for the tion includes the Julich and Karls Advancement of Science. ruhe nuclear research centres, LEP optimization DESY, and the Max Planck Institute for Plasma Physics as well as other The detail of the LEP electron-posi centres in the technical and biome tron storage ring project continues dical fields. to be studied so as to optimize the Moves at Brookhaven machine parameters from the point of view of performance and of cost. Nick Samios, former chairman of This optimization stays within the Brookhaven's Physics Department, description of Phase I of LEP which becomes the Laboratory's Deputy was agreed by the Member States Director for High Energy and Nu at the CERN Council meeting in clear Physics. -
5 the Dirac Equation and Spinors
5 The Dirac Equation and Spinors In this section we develop the appropriate wavefunctions for fundamental fermions and bosons. 5.1 Notation Review The three dimension differential operator is : ∂ ∂ ∂ = , , (5.1) ∂x ∂y ∂z We can generalise this to four dimensions ∂µ: 1 ∂ ∂ ∂ ∂ ∂ = , , , (5.2) µ c ∂t ∂x ∂y ∂z 5.2 The Schr¨odinger Equation First consider a classical non-relativistic particle of mass m in a potential U. The energy-momentum relationship is: p2 E = + U (5.3) 2m we can substitute the differential operators: ∂ Eˆ i pˆ i (5.4) → ∂t →− to obtain the non-relativistic Schr¨odinger Equation (with = 1): ∂ψ 1 i = 2 + U ψ (5.5) ∂t −2m For U = 0, the free particle solutions are: iEt ψ(x, t) e− ψ(x) (5.6) ∝ and the probability density ρ and current j are given by: 2 i ρ = ψ(x) j = ψ∗ ψ ψ ψ∗ (5.7) | | −2m − with conservation of probability giving the continuity equation: ∂ρ + j =0, (5.8) ∂t · Or in Covariant notation: µ µ ∂µj = 0 with j =(ρ,j) (5.9) The Schr¨odinger equation is 1st order in ∂/∂t but second order in ∂/∂x. However, as we are going to be dealing with relativistic particles, space and time should be treated equally. 25 5.3 The Klein-Gordon Equation For a relativistic particle the energy-momentum relationship is: p p = p pµ = E2 p 2 = m2 (5.10) · µ − | | Substituting the equation (5.4), leads to the relativistic Klein-Gordon equation: ∂2 + 2 ψ = m2ψ (5.11) −∂t2 The free particle solutions are plane waves: ip x i(Et p x) ψ e− · = e− − · (5.12) ∝ The Klein-Gordon equation successfully describes spin 0 particles in relativistic quan- tum field theory. -
The Critical Casimir Effect in Model Physical Systems
University of California Los Angeles The Critical Casimir Effect in Model Physical Systems A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Physics by Jonathan Ariel Bergknoff 2012 ⃝c Copyright by Jonathan Ariel Bergknoff 2012 Abstract of the Dissertation The Critical Casimir Effect in Model Physical Systems by Jonathan Ariel Bergknoff Doctor of Philosophy in Physics University of California, Los Angeles, 2012 Professor Joseph Rudnick, Chair The Casimir effect is an interaction between the boundaries of a finite system when fluctua- tions in that system correlate on length scales comparable to the system size. In particular, the critical Casimir effect is that which arises from the long-ranged thermal fluctuation of the order parameter in a system near criticality. Recent experiments on the Casimir force in binary liquids near critical points and 4He near the superfluid transition have redoubled theoretical interest in the topic. It is an unfortunate fact that exact models of the experi- mental systems are mathematically intractable in general. However, there is often insight to be gained by studying approximations and toy models, or doing numerical computations. In this work, we present a brief motivation and overview of the field, followed by explications of the O(2) model with twisted boundary conditions and the O(n ! 1) model with free boundary conditions. New results, both analytical and numerical, are presented. ii The dissertation of Jonathan Ariel Bergknoff is approved. Giovanni Zocchi Alex Levine Lincoln Chayes Joseph Rudnick, Committee Chair University of California, Los Angeles 2012 iii To my parents, Hugh and Esther Bergknoff iv Table of Contents 1 Introduction :::::::::::::::::::::::::::::::::::::: 1 1.1 The Casimir Effect . -
Some Polarized Target Experiments for Elementary Particle Physics
SOME POLARIZED TARGET EXPERIMENTS FOR ELEMENTARY PARTICLE PHYSICS R.H. DALITZ Department of Theoretical Physics, Oxford INTRODUCTION At the present stage in elementary particle physics, there are three major areas of phenomena where the use of polarized targets may lead to especially illuminating information. a. High energy scattering, in the energy range where the dominant contributions to the processes observed may arise from Reggion ex change. The features of particular interest in these situjtions will be discussed at this meeting by Dr R.J.N. Phillips 1 • b. Tests of time-reversal (T) invariance for electrom~gnetic pro cesses, following the suggestion by Bernstein et al 2) that the CF-violation observed in the weak decay of kaons arises from the existence of an electromagnetic interaction which strongly viola tes T-invariance. Evidence for this hypothetical T-violating elec tromagnetic interaction can be sought most directly in the study of electromagnetic processes from a polarized proton target, for example in the study of the electro-excitation process e + N-+ e + N* , as discussed by Christ and Lee 3J. The experi~ents proposed will be discussed at this meeting by Dr M. Jacob 4J. c. Hadron spectroscopy. Very many resonant states have been obser ved for masonic and baryonic states. These hadronic states are be lieved to correspond to the patterns appropriate to unitary multi plets of states, and these unitary multiplets appear to be grouped in supermultiplet patterns. In our attempt to classify and under stand all these hadronic states, our first need is for the deter mination of the spin and parity for each state. -
Murray Gell-Mann Hadrons, Quarks And
A Life of Symmetry Dennis Silverman Department of Physics and Astronomy UC Irvine Biographical Background Murray Gell-Mann was born in Manhattan on Sept. 15, 1929, to Jewish parents from the Austro-Hungarian empire. His father taught German to Americans. Gell-Mann was a child prodigy interested in nature and math. He started Yale at 15 and graduated at 18 with a bachelors in Physics. He then went to graduate school at MIT where he received his Ph. D. in physics at 21 in 1951. His thesis advisor was the famous Vicky Weisskopf. His life and work is documented in remarkable detail on videos that he recorded on webofstories, which can be found by just Google searching “webofstories Gell-Mann”. The Young Murray Gell-Mann Gell-Mann’s Academic Career (from Wikipedia) He was a postdoctoral fellow at the Institute for Advanced Study in 1951, and a visiting research professor at the University of Illinois at Urbana– Champaign from 1952 to 1953. He was a visiting associate professor at Columbia University and an associate professor at the University of Chicago in 1954-55, where he worked with Fermi. After Fermi’s death, he moved to the California Institute of Technology, where he taught from 1955 until he retired in 1993. Web of Stories video of Gell-Mann on Fermi and Weisskopf. The Weak Interactions: Feynman and Gell-Mann •Feynman and Gell-Mann proposed in 1957 and 1958 the theory of the weak interactions that acted with a current like that of the photon, minus a similar one that included parity violation. -
Relativistic Quantum Mechanics 1
Relativistic Quantum Mechanics 1 The aim of this chapter is to introduce a relativistic formalism which can be used to describe particles and their interactions. The emphasis 1.1 SpecialRelativity 1 is given to those elements of the formalism which can be carried on 1.2 One-particle states 7 to Relativistic Quantum Fields (RQF), which underpins the theoretical 1.3 The Klein–Gordon equation 9 framework of high energy particle physics. We begin with a brief summary of special relativity, concentrating on 1.4 The Diracequation 14 4-vectors and spinors. One-particle states and their Lorentz transforma- 1.5 Gaugesymmetry 30 tions follow, leading to the Klein–Gordon and the Dirac equations for Chaptersummary 36 probability amplitudes; i.e. Relativistic Quantum Mechanics (RQM). Readers who want to get to RQM quickly, without studying its foun- dation in special relativity can skip the first sections and start reading from the section 1.3. Intrinsic problems of RQM are discussed and a region of applicability of RQM is defined. Free particle wave functions are constructed and particle interactions are described using their probability currents. A gauge symmetry is introduced to derive a particle interaction with a classical gauge field. 1.1 Special Relativity Einstein’s special relativity is a necessary and fundamental part of any Albert Einstein 1879 - 1955 formalism of particle physics. We begin with its brief summary. For a full account, refer to specialized books, for example (1) or (2). The- ory oriented students with good mathematical background might want to consult books on groups and their representations, for example (3), followed by introductory books on RQM/RQF, for example (4). -
1. Dirac Equation for Spin ½ Particles 2
Advanced Particle Physics: III. QED III. QED for “pedestrians” 1. Dirac equation for spin ½ particles 2. Quantum-Electrodynamics and Feynman rules 3. Fermion-fermion scattering 4. Higher orders Literature: F. Halzen, A.D. Martin, “Quarks and Leptons” O. Nachtmann, “Elementarteilchenphysik” 1. Dirac Equation for spin ½ particles ∂ Idea: Linear ansatz to obtain E → i a relativistic wave equation w/ “ E = p + m ” ∂t r linear time derivatives (remove pr = −i ∇ negative energy solutions). Eψ = (αr ⋅ pr + β ⋅ m)ψ ∂ ⎛ ∂ ∂ ∂ ⎞ ⎜ ⎟ i ψ = −i⎜α1 ψ +α2 ψ +α3 ψ ⎟ + β mψ ∂t ⎝ ∂x1 ∂x2 ∂x3 ⎠ Solutions should also satisfy the relativistic energy momentum relation: E 2ψ = (pr 2 + m2 )ψ (Klein-Gordon Eq.) U. Uwer 1 Advanced Particle Physics: III. QED This is only the case if coefficients fulfill the relations: αiα j + α jαi = 2δij αi β + βα j = 0 β 2 = 1 Cannot be satisfied by scalar coefficients: Dirac proposed αi and β being 4×4 matrices working on 4 dim. vectors: ⎛ 0 σ ⎞ ⎛ 1 0 ⎞ σ are Pauli α = ⎜ i ⎟ and β = ⎜ ⎟ i 4×4 martices: i ⎜ ⎟ ⎜ ⎟ matrices ⎝σ i 0 ⎠ ⎝0 − 1⎠ ⎛ψ ⎞ ⎜ 1 ⎟ ⎜ψ ⎟ ψ = 2 ⎜ψ ⎟ ⎜ 3 ⎟ ⎜ ⎟ ⎝ψ 4 ⎠ ⎛ ∂ r ⎞ i⎜ β ψ + βαr ⋅ ∇ψ ⎟ − m ⋅1⋅ψ = 0 ⎝ ∂t ⎠ ⎛ 0 ∂ r ⎞ i⎜γ ψ + γr ⋅ ∇ψ ⎟ − m ⋅1⋅ψ = 0 ⎝ ∂t ⎠ 0 i where γ = β and γ = βα i , i = 1,2,3 Dirac Equation: µ i γ ∂µψ − mψ = 0 ⎛ψ ⎞ ⎜ 1 ⎟ ⎜ψ 2 ⎟ Solutions ψ describe spin ½ (anti) particles: ψ = ⎜ψ ⎟ ⎜ 3 ⎟ ⎜ ⎟ ⎝ψ 4 ⎠ Extremely 4 ⎛ µ ∂ ⎞ compressed j = 1...4 : ∑ ⎜∑ i ⋅ (γ ) jk µ − mδ jk ⎟ψ k description k =1⎝ µ ∂x ⎠ U. -
Polarized Target for the Measurement of the Gluon Contribution to the Nucleon Spin in the COMPASS Experiment
Polarized target for the measurement of the gluon contribution to the nucleon spin in the COMPASS experiment Nagoya University Thesis for the degree Doctor of Philosophy in Physics by Naoki TAKABAYASHI 1 2002 1 Supported by Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists. CONTENTS 1. Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1 2. Spin structure of the nucleon : : : : : : : : : : : : : : : : : : : : : 5 2.1 Deep inelastic scattering . 5 2.2 Cross section and structure functions in unpolarized DIS . 7 2.3 Cross section and structure functions in polarized DIS . 9 2.4 Structure functions and spin puzzle in Parton Model picture . 13 2.4.1 Parton distribution . 14 2.4.2 Spin puzzle in Parton Model picture . 16 2.5 Structure functions and spin puzzle in QCD evolved Parton Model . 18 2.5.1 Unpolarized structure functions . 18 2.5.2 Factorization schemes . 20 2.5.3 DGLAP equation and Scaling violation . 22 2.5.4 Spin-dependent structure functions and polarized DGLAP equation . 23 2.6 Interpretation of small ∆Σ. Is gluon contribution large? . 24 2.6.1 Sum rules . 25 2.6.2 Axial anomaly . 26 3. COMPASS experiment : : : : : : : : : : : : : : : : : : : : : : : : : 30 3.1 Open charm lepto-production via photon-gluon fusion . 30 3.2 Expectations . 35 3.2.1 Luminosity . 35 3.2.2 Reconstruction of the open charm events . 35 3.2.3 Statistical accuracy . 37 3.3 Comparison with the other experiments on ∆g . 40 3.3.1 High pT hadron pairs . 40 3.3.2 pp¯ collision . 41 3.3.3 Comparison of the expected precisions and the kine- matic ranges . -
Hadron Spin Dynamics∗
SLAC-PUB-9107 January 2002 Hadron Spin Dynamics¤ Stanley J. Brodsky Stanford Linear Accelerator Center, Stanford University Stanford, California 94309 e-mail: [email protected] Abstract Spin effects in exclusive and inclusive reactions provide an essential new dimension for testing QCD and unraveling hadron structure. Remarkable new experiments from SLAC, HERMES (DESY), and Jefferson Lab present many challenges to theory, including measurements at HERMES and SMC of the sin- gle spin asymmetries in ep ! e0¼X where the proton is polarized normal to the scattering plane. This type of single spin asymmetry may be due to the effects of rescattering of the outgoing quark on the spectators of the target proton, an effect usually neglected in conventional QCD analyses. Many aspects of spin, such as single-spin asymmetries and baryon magnetic moments are sensitive to the dynamics of hadrons at the amplitude level, rather than probability dis- tributions. I will illustrate the novel features of spin dynamics for relativistic systems by examining the explicit form of the light-front wavefunctions for the two-particle Fock state of the electron in QED, thus connecting the Schwinger anomalous magnetic moment to the spin and orbital momentum carried by its Fock state constituents and providing a transparent basis for understand- ing the structure of relativistic composite systems and their matrix elements in hadronic physics. I also present a survey of outstanding spin puzzles in QCD, particularly ANN in elastic pp scattering, the J=Ã ! ½¼ puzzle, and J=Ã polarization at the Tevatron. Concluding Theory Talk, presented at the 3rd Circum-Pan-Pacific Symposium On High Energy Spin Physics (SPIN 2001) Beijing, China 8–13 October 2001 ¤Work supported by the Department of Energy, contract DE–AC03–76SF00515.