DOCSLIB.ORG

Spinning Test Particles in Spacetime with Torsion Physics Department

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Spinning Test Particles in Spacetime with Torsion Physics Department 1239 Progress of Theoretical Physics, Vol. 86, No.6, December 1991 Spinning Test Particles in Spacetime with Torsion Koichi NOMURA, Takeshi SHIRAFUJI and Kenji HAYASHI* Physics Department, Saitama University, Urawa 338 * Institute of Physics, Kitasato University, Sagamihara 228 Downloaded from https://academic.oup.com/ptp/article/86/6/1239/1918293 by guest on 27 September 2021 (Received July 18, 1991) We use the Fock-Papapetrou method to derive equations of motion for a spinning test particle moving both in the background gravitational field and a torsion field. The resultant two equations, the equations of world line and spin can be written in a covariant manner (as is the same for Papapetrou's result in general relativity). The equation of spin reproduces all the known equations of spin precession due to torsion. § 1. Introduction In general relativity, the problem of motion of extended bodies has first been 1 studied by Fock ) and developed by Papapetrou,2) who derived the equations of world line and spin for a spinning test particle moving in a background gravitational field. By a test particle we mean that the field produced by this particle can be neglected. Consequently, the back-reaction due to emission of gravitational radiation is ignored. The test particle is assumed to be neutral and of sufficiently small size so that effects due to electromagnetic field and the coupling of its multipole moments to in­ 3 homogeneities of the gravitational field can be neglected. ) Accordingto the -Fock-Papapetrou method, all test particles move like a spinless mass point in the single-pole approximation. Namely, the world line is the familiar geodesics of the metric. The effects due to spin appear in the pole-dipole approxima­ tion. In effect, the world line of a spinning test particle deviates from the geodesics of the metric. In this paper we study the motion of an extended body such as a gyroscope or a wave packet moving both in the background gravitational field and a torsion field. At present we have two theoretical frameworks of gravitation constructed on the spacetime with torsion: Poincare gauge theory4)-7) and new general relativity.S) The former theory is based on the Riemann-Cartan spacetime characterized by curvature and torsion. The latter theory, on the other hand, is built on the Weizenbock spacetime which possesses absolute parallelism: The gravitation is then ascribed to the torsion of the spacetime, since the curvature is identically vanishing. These two theories give the same predictions as those of general relativity to all the experiments so far performed. The torsion is coupled to the intrinsic spin of fundamental particles (quarks and leptons), and it will give, if it exists, nontrivial effects on the motion of spin of these particles. Thus, observation of spin precession may possibly give us a clue to detect the torsion of the spacetime. In a previous paper,9) we applied the WKB method to the Dirac equation and the Proca equation, and derived the equations of world line and spin precession for massive particles with spin 1/2 and 1 in units of fz moving in 1240 K. Nomura, T. Shirajuji and K. Hayashi the background gravitational and torsion fields. In particular, we showed that the world line for these particles in free fall is the familiar geodesics of the metric, but that the equation of spin precession is not represented in a geometric manner. Namely, the spin vector. of these particles is neither Fermi-Walker transported relative to the asymmetric connection with torsion, nor parallel propagated with respect to the ordinary symmetric connection. 10 It has been suggested by Adamovicz and Trautmann ) using a spin-fluid model/I) on the other hand, that the spin vector of a spinning test particle is Fermi-Walker transported along the world line relative to the asymmetric connection. Our result Downloaded from https://academic.oup.com/ptp/article/86/6/1239/1918293 by guest on 27 September 2021 obtained by the WKB method does not agree with their suggestion. It seems highly desirable to apply the Fock-Papapetrou method to spinning test particles moving in the Riemann-Cartan spacetime. We shall arrange the content as follows: In § 2 a brief review is given of the spinor and tensor analyses in the spacetime with torsion in order to fix our notation. In § 3 the Fock-Papapetrou method is summarized, and the basic response equations are extended to those for a spinning body in the Riemann-Cartan spacetime. In §§ 4 and 5 the equations of motion are derived in the single-pole and pole-dipole approxima­ tions, respectively. In § 6 a possible way to get the equation of spin precession is suggested. The equation of world line is discussed in § 7. The final section is devoted to conclusion. Detailed proofs of basic relations are given in three appen­ dices. § 2. A brief review of the Riemann-Cartan spacetime Spacetime has a locally Lorentzian metric field gpv(x) and a global tetrad field ekV(x) with gpvekPe/= 7Jkl. Here !}kl is the Minkowski metric, which we choose as 7Jkl =diag( -1, + 1, + 1, + 1).*) The Latin indices and the Greek ones refer to the Lorentz basis ek and the coordinate basis E p , respectively. The fundamental particles such as quarks and leptons are described by spinor fields if, on which the covariant derivative acts like (2'1) where (Jmn is the generator of the local Lorentz group, and Amnv is the Lorentz (or the spin) connection. In this paper we shall assume that matter fields are minimally coupled to the gravitational field and a torsion field: Namely, the interaction of matter fields with torsion is introduced through the covariant derivative, which satisfies the tetrad condition and the metric condition, (2·2) (2·3) with r;v being the affine connection coefficients. Here the r;v is not assumed to be *) We shall mainly follow the notation and convention of Ref. 7). Spinning Test Particles in Spacetime with Torsion 1241 symmetric with respect to fl and v, and accordingly the torsion tensor is defined by TA pv = rtv - rtp. The affine connection can then be expressed by (2·4) where {~v} is the Christoffel symbol and KApv denotes the contorsion tensor, (2·5) Downloaded from https://academic.oup.com/ptp/article/86/6/1239/1918293 by guest on 27 September 2021 which is anti symmetric with respect to It and fl. Putting (2·4) in (2·2), we see that the Lorentz connection is represented as Amnv = L1mnv + Kmnv " (2·6) where L1mnv is the Ricci rotation coefficients for the spin connection of general relativ­ ity. We shall write by l7 p the covariant derivative with respect to gv} and L1mnv: It also satisfies the tetrad condition and the metric condition. The comml,ltator of the covariant derivative reads (2·7) . with the curvature tensor given by (2·8) As for the covariant derivative l7 v, its commutator is (2·9) where Rmnpv is the Riemann-Christoffel curvature tensor which is obtained from Fmnpv of (2·8) by replacing Amnp and rtv with L1mnp and gv}, respectively. The curvature tensor Fp(JPv is thus related to the Rp(JPv by (2·10) The torsion tensor has three irreducible components; the tensor part tAPV, *l the vector part Vp and the axial-vector part ap, in terms of which the contorsion tensor is expressed by (2·11) In general relativity, the starting point of the Fock-Papapetrou method is the response equation of the energy-momentum tensor density TPV of matter fields in the background gravitational field. The response equation takes the familiar form: 17vTPv =O. (2·12) *l'The tAPV is symmetric with respect to A and p. and traceless. Moreover it satisfies the cyclic identity. tAPV+ f pvA + tVAP=O. See Refs. 7) and 8) for details. 1242 K. Nomura, T. Shirafuji and K. Hayashi Here TI-'v is symmetric with respect to fL and v, and the covariant derivative acts on the tensor density Tl-'v=J=[iTl-'v like [7ATl-'v=j-g[7ATl-'v. In the theory of gravitation constructed on the spacetime with torsion, on the other hand, the starting point is the response equations of the energy-momentum tensor density and the spin tensor density of matter fieds in the background gravita­ tional and torsion fields. These equations can be derived from Noether's 14 theorem. ),15) Since the action of matter fields is invariant under general coordinate transformations and local Lorentz ones, there are two sets of Noether's identity which 5 Downloaded from https://academic.oup.com/ptp/article/86/6/1239/1918293 by guest on 27 September 2021 are obtained by the well-known procedure. ) Using the field equations of matter fields in those identities which follow, we can get the following response equations:*) (2·13) (2'14) where the energy-momentum and spin tensor densities of matter fields are defined by smnv=_2 oLM (2·15) oAmnv' Here LM=j-gLM is the Lagrangian density of matter fields. Derivation of (2'13) and (2·14) is given in Appendix A: It should be stressed that these response equations are valid irrespectively of whether matter fields are coupled to the gravitational field and the torsion field minimally or not. We note that TI-'v is not symmetric with respect to fL and v: It is reduced to the canonical energy-momentum tensor in special relativistic limit, when the interaction is minimal coupling. § 3. Papapetrou's method 2 3.1. A brief review of Papapetrou's method ) Let us consider an extended body, whose motion sweeps a world tube in spacetime.
Load more

Recommended publications
  • Spinning Test Particle in Four-Dimensional Einstein–Gauss–Bonnet Black Holes
    universe Communication Spinning Test Particle in Four-Dimensional Einstein–Gauss–Bonnet Black Holes Yu-Peng Zhang 1,2, Shao-Wen Wei 1,2 and Yu-Xiao Liu 1,2,3* 1 Joint Research Center for Physics, Lanzhou University and Qinghai Normal University, Lanzhou 730000 and Xining 810000, China; [email protected] (Y.-P.Z.); [email protected] (S.-W.W.) 2 Institute of Theoretical Physics & Research Center of Gravitation, Lanzhou University, Lanzhou 730000, China 3 Key Laboratory for Magnetism and Magnetic of the Ministry of Education, Lanzhou University, Lanzhou 730000, China * Correspondence: [email protected] Received: 27 June 2020; Accepted: 27 July 2020; Published: 28 July 2020 Abstract: In this paper, we investigate the motion of a classical spinning test particle in a background of a spherically symmetric black hole based on the novel four-dimensional Einstein–Gauss–Bonnet gravity [D. Glavan and C. Lin, Phys. Rev. Lett. 124, 081301 (2020)]. We find that the effective potential of a spinning test particle in this background could have two minima when the Gauss–Bonnet coupling parameter a is nearly in a special range −8 < a/M2 < −2 (M is the mass of the black hole), which means a particle can be in two separate orbits with the same spin-angular momentum and orbital angular momentum, and the accretion disc could have discrete structures. We also investigate the innermost stable circular orbits of the spinning test particle and find that the corresponding radius could be smaller than the cases in general relativity. Keywords: Gauss–Bonnet; innermost stable circular orbits; spinning test particle 1.
    [Show full text]
  • A620 Dynamics 2017 L
    Dynamics and how to use the orbits of stars to do interesting things ! chapter 3 of S+G- parts of Ch 11 of MWB (Mo, van den Bosch, White)! READ S&G Ch 3 sec 3.1, 3.2, 3 .4! we are skipping over epicycles ! 1! A Guide to the Next Few Lectures ! •The geometry of gravitational potentials : methods to derive gravitational ! potentials from mass distributions, and visa versa.! •Potentials define how stars move ! consider stellar orbit shapes, and divide them into orbit classes.! •The gravitational field and stellar motion are interconnected : ! the Virial Theorem relates the global potential energy and kinetic energy of !the system. ! ! • Collisions?! • The Distribution Function (DF) : ! the DF specifies how stars are distributed throughout the system and ! !with what velocities.! For collisionless systems, the DF is constrained by a continuity equation :! ! the Collisionless Boltzmann Equation ! •This can be recast in more observational terms as the Jeans Equation. ! The Jeans Theorem helps us choose DFs which are solutions to the continuity equations! 2! A Reminder of Newtonian Physics sec 3.1 in S&G! eq 3.1! φ (x) is the potential! eq 3.2! eq 3.3! Gauss's thm ∫ !φ •ds2=4πGM! the Integral of the normal component! over a closed surface =4πG x mass within eq 3.4! that surface! ρ(x) is the mass density3! distribution! Conservation of Energy and Angular Momentum ! Angular momentum L! 4! Some Basics - M. Whittle! • The gravitational potential energy is a scalar field ! • its gradient gives the net gravitational force (per unit mass) which is a vector field : see S&G pg 113 ! Poissons eq inside the mass distribution ! Outside the mass dist ! 5! Poisson's Eq+ Definition of Potential Energy (W) ! ρ(x) is the density dist! S+G pg 112-113! Potential energy W ! 6! Derivation of Poisson's Eq ! see S+G pg112 for detailed derivation or web page 'Poisson's equation' ! 7! More Newton-Spherical Systems Newtons 1st theorem: a body inside a spherical shell has no net gravitational force from that shell; e.g.
    [Show full text]
  • Introduction. What Is a Classical Field Theory? Introduction This Course
    Classical Field Theory Introduction. What is a classical field theory? Introduction. What is a classical field theory? Introduction This course was created to provide information that can be used in a variety of places in theoretical physics, principally in quantum field theory, particle physics, electromagnetic theory, fluid mechanics and general relativity. Indeed, it is usually in such courses that the techniques, tools, and results we shall develop here are introduced { if only in bits and pieces. As explained below, it seems better to give a unified, systematic development of the tools of classical field theory in one place. If you want to be a theoretical/mathematical physicist you must see this stuff at least once. If you are just interested in getting a deeper look at some fundamental/foundational physics and applied mathematics ideas, this is a good place to do it. The most familiar classical* field theories in physics are probably (i) the \Newtonian theory of gravity", based upon the Poisson equation for the gravitational potential and Newton's laws, and (ii) electromagnetic theory, based upon Maxwell's equations and the Lorentz force law. Both of these field theories are exhibited below. These theories are, as you know, pretty well worked over in their own right. Both of these field theories appear in introductory physics courses as well as in upper level courses. Einstein provided us with another important classical field theory { a relativistic gravitational theory { via his general theory of relativity. This subject takes some investment in geometrical technology to adequately explain. It does, however, also typically get a course of its own.
    [Show full text]
  • Investigation of a Micro- and Nano-Particle In-Space Electrostatic Propulsion Concept
    Investigation of a Micro- and Nano-Particle In-Space Electrostatic Propulsion Concept by Louis D. Musinski A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Electrical Engineering) in The University of Michigan 2009 Doctoral Committee: Professor Brian E. Gilchrist, Co-Chair Professor Alec D. Gallimore, Co-Chair Professor Yogesh B. Gianchandani Associate Professor Michael J. Solomon Louis D. Musinski © ———————— 2009 All Rights Reserved Acknowledgements First, I would like to thank everyone who helped and supported me throughout my entire education. This work was made possible by everyone who has touched my life and includes people too numerous mention. To everyone who is not directly mentioned here, thank you! I would like to express gratitude for my advisor and co-chairman, Dr. Brian Gilchrist. His support over the last several years has been invaluable towards my development as an engineer and a person. The enthusiasm he exudes motivates people around him and I consider myself privileged to have worked with him. I look forward to future collaborations. I also thank my other co-chairman, Dr. Alec Gallimore, who has also provided significant guidance throughout my graduate school career. Dr. Michael Solomon and Dr. Yogesh Gianchandani, thank you for your help and guidance as members of my dissertation committee. I was fortunate to have been involved with a wonderful research group, nanoFET, which was composed of, not only excellent engineers, but also people that were a pleasure to work with. Thank you: Dr. Joanna Mirecki-Millunchick, Dr. Mark Burns, Dr. Michael Kiedar, Thomas Liu, Deshpremy Mukhija, Inkyu Eu, and David Liaw.
    [Show full text]
  • A Second Order Differential Equation for a Point Charged Particle
    A SECOND ORDER DIFFERENTIAL EQUATION FOR A POINT CHARGED PARTICLE Ricardo Gallego Torrom´e Departamento de Matem´atica Universidade Federal de S˜ao Carlos, Brazil Email: [email protected] Abstract. A model for the dynamics of a classical point charged particle interacting with higher order jet fields is introduced. In this model, the dy- namics of the charged particle is described by an implicit ordinary second order differential equation. Such equation is free of run-away and pre-accelerated so- lutions of Dirac’s type. The theory is Lorentz invariant, compatible with the first law of Newton and Larmor’s power radiation formula. Few implications of the new equation in the phenomenology of non-neutral plasmas is considered. 1. Introduction To find a consistent theory of classical electrodynamics of point charged particles interacting with its own radiation field is one of the most notable open problems in classical field theory. Historically, the investigation of the radiation reaction of point charged particles lead to the relativistic Abraham-Lorentz-Dirac equation [7] (in short, the ALD equation). However, the theoretical problems and paradoxes associated with ALD inclined many authors to support the view that the classical electrodynamics of point charged particles should not be based on the coupled sys- tem ALD /Maxwell system of equations. Attempts to find a consistent theory of classical electrodynamics of point particles include reduction of order schemes for the ALD equation [17], re-normalization group schemes [24], higher order deriva- tive field theories [2, 20], Feynman-Wheeler’s absorber theory of electrodynamics [9], models where the observable mass is variable with time [1, 18], non-linear elec- trodynamic theories [3] and dissipative force models [22, 14].
    [Show full text]
  • Physical Setting/Physics Core Curriculum Was Reviewed by Many Teachers and Administrators Across the State
    PhysicalPhysical Setting/Setting/ PhysicsPhysics Core Curriculum THE UNIVERSITY OF THE STATE OF NEW YORK THE STATE EDUCATION DEPARTMENT http://www.emsc.nysed.gov THE UNIVERSITY OF THE STATE OF NEW YORK Regents of The University CARL T. HAYDEN, Chancellor, A.B., J.D. ............................................................................Elmira ADELAIDE L. SANFORD, Vice Chancellor, B.A., M.A., P.D. .................................................Hollis DIANE O’NEILL MCGIVERN, B.S.N., M.A., Ph.D. ..............................................................Staten Island SAUL B. COHEN, B.A., M.A., Ph.D. .....................................................................................New Rochelle JAMES C. DAWSON, A.A., B.A., M.S., Ph.D. .......................................................................Peru ROBERT M. BENNETT, B.A., M.S. ........................................................................................Tonawanda ROBERT M. JOHNSON, B.S., J.D. .........................................................................................Huntington ANTHONY S. BOTTAR, B.A., J.D. .........................................................................................North Syracuse MERRYL H. TISCH, B.A., M.A. ............................................................................................New York ENA L. FARLEY, B.A., M.A., Ph.D. .....................................................................................Brockport GERALDINE D. CHAPEY, B.A., M.A., Ed.D...........................................................................Belle
    [Show full text]
  • Lorentz-Violating Gravitoelectromagnetism
    Physics & Astronomy - Prescott College of Arts & Sciences 9-13-2010 Lorentz-Violating Gravitoelectromagnetism Quentin G. Bailey Embry-Riddle Aeronautical University, [email protected] Follow this and additional works at: https://commons.erau.edu/pr-physics Part of the Elementary Particles and Fields and String Theory Commons Scholarly Commons Citation Bailey, Q. G. (2010). Lorentz-Violating Gravitoelectromagnetism. Physical Review D, 82(6). https://doi.org/ 10.1103/PhysRevD.82.065012 © 2010 American Physical Society. This article is also available from the publisher at https://doi.org/ 10.1103/ PhysRevD.82.065012 This Article is brought to you for free and open access by the College of Arts & Sciences at Scholarly Commons. It has been accepted for inclusion in Physics & Astronomy - Prescott by an authorized administrator of Scholarly Commons. For more information, please contact [email protected]. PHYSICAL REVIEW D 82, 065012 (2010) Lorentz-violating gravitoelectromagnetism Quentin G. Bailey* Physics Department, Embry-Riddle Aeronautical University, 3700 Willow Creek Road, Prescott, Arizona 86301, USA (Received 16 May 2010; published 13 September 2010) The well-known analogy between a special limit of general relativity and electromagnetism is explored in the context of the Lorentz-violating standard-model extension. An analogy is developed for the minimal standard-model extension that connects a limit of the CPT-even component of the electromagnetic sector to the gravitational sector. We show that components of the post-Newtonian metric can be directly obtained from solutions to the electromagnetic sector. The method is illustrated with specific examples including static and rotating sources. Some unconventional effects that arise for Lorentz-violating electrostatics and magnetostatics have an analog in Lorentz-violating post-Newtonian gravity.
    [Show full text]
  • Gravitoelectromagnetism.Pdf
    GRAVITOELECTROMAGNETISM BAHRAM MASHHOON Department of Physics and Astronomy University of Missouri-Columbia Columbia, Missouri 65211, USA E-mail: [email protected] Gravitoelectromagnetism is briefly reviewed and some recent developments in this topic are discussed. The stress-energy content of the gravitoelectromagnetic field is described from different standpoints. In particular, the gravitational Poynting flux is analyzed and it is shown that there exists a steady flow of gravitational energy circulating around a rotating mass. 1 Introduction Gravitoelectromagnetism (GEM) is based upon the close formal analogy be- tween Newton’s law of gravitation and Coulomb’s law of electricity. The New- tonian theory of gravitation may thus be interpreted in terms of a gravito- electric field. Any field theory that would bring Newtonian gravitation and Lorentz invariance together in a consistent framework would necessarily con- tain a gravitomagnetic field as well. In general relativity, the non-Newtonian gravitomagentic field is due to mass current and has interesting physical prop- erties that are now becoming amenable to experimental observation. Developments in electrodynamics in the second half of the nineteenth cen- tury led Holzm¨uller [1] and Tisserand [2] to postulate a gravitomagnetic com- ponent for the gravitational influence of the Sun on the motion of planets. The magnitude of this ad hoc component could be adjusted so as to account for the excess perihelion motion of Mercury. However, Einstein’s general relativ- arXiv:gr-qc/0011014v1 3 Nov 2000 ity successfully accounted for the perihelion precession of Mercury by means of a post-Newtonian correction to the gravitoelectric influence of the Sun. The general relativistic effect of the rotation of the Sun on planetary orbits was first calculated by de Sitter [3] and later more generally by Thirring and Lense [4].
    [Show full text]
  • On the Motion of Test Particles in General Relativity
    408 L. I NFELD AND A. SCH ILD finds for the coordinates before, a series of inertia planes U', V', 8" ., each perpendicular to its companion, every pair defined by t= tp(n'Chs —P') cosy nshs+r sinpp nP(Chs —1) +r a parameter s and effectuating an acceleration or x=tp nshs cospp Chs sinpp PShs +r +r rotation dP=f(s)ds and dx=g(s)ds respectively. The y=tp nP(1 C—hs) r—cospp PShs+r sinip (n' —P'Chs) succession of elements U(s) and U'(s) would then Spa generate a collection of time tracks or space curves One verifies, that t' —x' —y' —s'=tp' —r' —sp' and that permitting to locate, for every s, a collection of points for P=O the formulae reduce to those of the rectilinear preserving rigid distances. A few examples have been hyperbolic acceleration found above. indicated by Herglotz. ' Putting tp —0 one finds the time tracks of a rigid One may take a constant separation plane U(= OI'Z) body occupying the instantaneous present space at the and a constant acceleration plane (OXT) and make the origin of time. This motion has not been described by uniform revolutions defined by P=o.u and y= ~u. One Herglotz. will have the superposition of a rotation and an acceler- ation along the axis of rotation Another conception of a rigid body has been put t = x pshaw, x= xpChg, =r cosy, s= r sinx, P/n = x/~ = u forward in early discussions, in which it was defined as y a collection of points, not time tracks, preserving This kind of motions has been called by Herglotz the constant distances among them.
    [Show full text]
  • General Relativity University of Cambridge Part III Mathematical Tripos
    Preprint typeset in JHEP style - HYPER VERSION Michaelmas Term, 2019 General Relativity University of Cambridge Part III Mathematical Tripos David Tong Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge, CB3 OBA, UK http://www.damtp.cam.ac.uk/user/tong/gr.html [email protected] –1– Recommended Books and Resources There are many decent text books on general relativity. Here are a handful that I like: Sean Carroll, “Spacetime and Geometry” • A straightforward and clear introduction to the subject. Bob Wald, “General Relativity” • The go-to relativity book for relativists. Steven Weinberg, “Gravitation and Cosmology” • The go-to relativity book for particle physicists. Misner, Thorne and Wheeler, “Gravitation” • Extraordinary and ridiculous in equal measure, this book covers an insane amount of material but with genuinely excellent explanations. Now, is that track 1 or track 2? Tony Zee, “Einstein Gravity in a Nutshell” • Professor Zee likes a bit of a chat. So settle down, prepare yourself for more tangents than Tp(M), and enjoy this entertaining, but not particularly concise, meander through the subject. Nakahara, “Geometry, Topology and Physics” • Areallyexcellentbookthatwillsatisfyyourgeometricalandtopologicalneedsforthis course and much beyond. It is particularly useful for Sections 2 and 3 of these lectures where we cover di↵erential geometry. Anumberofexcellentlecturenotesareavailableontheweb,includingan early version of Sean Carroll’s book. Links can be found
    [Show full text]
  • List of Symbols
    Appendix A List of Symbols For convenience, many symbols frequently used in the text are listed below alphabetically. Some of the symbols may play a dual role, in which case the meaning should be obvious from the context. A number in parentheses refers to an Eq. number (or the text or footnote near that Eq. number) in which the symbol is defined or appears for the first time; in a few cases a page number, Table number, Sect. number, footnote (fn.) number, or Fig. number is given instead. Quantum number is abbreviated as q. n. Quantum numbers (atomic): Symbol Description Eq. No. j D l ˙ s Inner q. n. of an electron (3.13) J D L C S Inner (total angular momentum) q. n. of a state (5.51b) l P Azimuthal (orbital) q. n. of an electron (3.1) and (3.2) L D l Azimuthal (orbital) q. n. of a state (5.51c) ml Magnetic orbital q. n. of an electron (3.1) ms Magnetic spin q. n. of an electron (3.1) M Total magnetic q. n. (5.51d) n Principal q. n. of an electron (3.1) s P Spin q. n. of an electron (3.13) S D s Spin q. n. of a state (5.51e) Relativistic azimuthal q. n. of an electron (3.15a) and (3.15b) Relativistic magnetic q. n. of an electron (3.18a) and (3.18b) W.F. Huebner and W.D. Barfield, Opacity, Astrophysics and Space Science Library 402, 457 DOI 10.1007/978-1-4614-8797-5, © Springer Science+Business Media New York 2014 458 Appendix A: List of Symbols Quantum numbers (molecular)1: Symbol Description Eq.
    [Show full text]
  • Electric Potential Equipotential Surfaces; Potential Due to a Point
    Electric Potential Equipotential Surfaces; Potential due to a Point Charge and a Group of Point Charges; Potential due to an Electric Dipole; Potential due to a Charge Distribution; Relation between Electric Field and Electric Potential Energy 1. How to find the electric force on a particle 1 of charge q1 when the particle is placed near a particle 2 of charge q2. 2. A nagging question remains: How does particle 1 “know “of the presence of particle 2? 3. That is, since the particles do not touch, how can particle 2 push on particle 1—how can there be such an action at a distance? 4. One purpose of physics is to record observations about our world, such as the magnitude and direction of the push on particle 1. Another purpose is to provide a deeper explanation of what is recorded. 5. One purpose of this chapter is to provide such a deeper explanation to our nagging questions about electric force at a distance. We can answer those questions by saying that particle 2 sets up an electric field in the space surrounding itself. If we place particle 1 at any given point in that space, the particle “knows” of the presence of particle 2 because it is affected by the electric field that particle 2 has already set up at that point. Thus, particle 2 pushes on particle 1 not by touching it but by means of the electric field produced by particle 2. One purpose of physics is to record observations about our world, 1 Equipotential Surfaces: Potential energy: When an electrostatic force acts between two or more charged particles within a system of particles, we can assign an electric potential energy U to the system.
    [Show full text]