Maxwell's Equations in Differential Form, and Uniform Plane Waves In
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An Introduction to Quantum Field Theory
AN INTRODUCTION TO QUANTUM FIELD THEORY By Dr M Dasgupta University of Manchester Lecture presented at the School for Experimental High Energy Physics Students Somerville College, Oxford, September 2009 - 1 - - 2 - Contents 0 Prologue....................................................................................................... 5 1 Introduction ................................................................................................ 6 1.1 Lagrangian formalism in classical mechanics......................................... 6 1.2 Quantum mechanics................................................................................... 8 1.3 The Schrödinger picture........................................................................... 10 1.4 The Heisenberg picture............................................................................ 11 1.5 The quantum mechanical harmonic oscillator ..................................... 12 Problems .............................................................................................................. 13 2 Classical Field Theory............................................................................. 14 2.1 From N-point mechanics to field theory ............................................... 14 2.2 Relativistic field theory ............................................................................ 15 2.3 Action for a scalar field ............................................................................ 15 2.4 Plane wave solution to the Klein-Gordon equation ........................... -
Chapter 3 Beam Dynamics II - Longitudinal
Chapter 3 Beam Dynamics II - Longitudinal Sequences of Gaps Transit Time Factor Phase Stability Chapter 3 Longitudinal Beam Dynamics 1 The Faraday Cage Protons are confined in a conducting box, at low energy. Assume they can bounce off the walls with no energy loss. Move the switch from position A to B. The potential on the box rises from 0 to 1 MV. What is the proton energy now? Chapter 3 Longitudinal Beam Dynamics 2 The Linac Drift Tube A linear accelerator (linac) is comprised of a succession of drift tubes. These drift tubes have holes in their ends so the particles can enter and exit, and when particles are inside the drift tube, a Faraday Cage, the potential of the drift tube may vary without changing the energy of the particle. Acceleration takes place when a charged particle is subjected to a field. The field inside the Faraday cage is not affected by the potential outside. (Aside from fields generated by the protons themselves, the field inside the Faraday cage is zero.) The drift tubes are arranged in a sequence with a passage through their middle for the particles to pass. The field in the gap between the drift tubes accelerates the particles. Chapter 3 Longitudinal Beam Dynamics 3 Some Actual Linac Configurations We will look at: Sloan-Lawrence Structure (Ising, Wideroe) Alvarez Structure RFQ Structure Coupled-Cavity Structure Chapter 3 Longitudinal Beam Dynamics 4 Some Kinematics For simplicity, we will assume the particles are non-relativistic. The normalized velocity is 2T = m c2 T is the kinetic energy of the particle, mc2 is the rest mass, 938 MeV for protons. -
Electromagnetism As Quantum Physics
Electromagnetism as Quantum Physics Charles T. Sebens California Institute of Technology May 29, 2019 arXiv v.3 The published version of this paper appears in Foundations of Physics, 49(4) (2019), 365-389. https://doi.org/10.1007/s10701-019-00253-3 Abstract One can interpret the Dirac equation either as giving the dynamics for a classical field or a quantum wave function. Here I examine whether Maxwell's equations, which are standardly interpreted as giving the dynamics for the classical electromagnetic field, can alternatively be interpreted as giving the dynamics for the photon's quantum wave function. I explain why this quantum interpretation would only be viable if the electromagnetic field were sufficiently weak, then motivate a particular approach to introducing a wave function for the photon (following Good, 1957). This wave function ultimately turns out to be unsatisfactory because the probabilities derived from it do not always transform properly under Lorentz transformations. The fact that such a quantum interpretation of Maxwell's equations is unsatisfactory suggests that the electromagnetic field is more fundamental than the photon. Contents 1 Introduction2 arXiv:1902.01930v3 [quant-ph] 29 May 2019 2 The Weber Vector5 3 The Electromagnetic Field of a Single Photon7 4 The Photon Wave Function 11 5 Lorentz Transformations 14 6 Conclusion 22 1 1 Introduction Electromagnetism was a theory ahead of its time. It held within it the seeds of special relativity. Einstein discovered the special theory of relativity by studying the laws of electromagnetism, laws which were already relativistic.1 There are hints that electromagnetism may also have held within it the seeds of quantum mechanics, though quantum mechanics was not discovered by cultivating those seeds. -
2 Hilbert Spaces You Should Have Seen Some Examples Last Semester
2 Hilbert spaces You should have seen some examples last semester. The simplest (finite-dimensional) ex- C • A complex Hilbert space H is a complete normed space over whose norm is derived from an ample is Cn with its standard inner product. It’s worth recalling from linear algebra that if V is inner product. That is, we assume that there is a sesquilinear form ( , ): H H C, linear in · · × → an n-dimensional (complex) vector space, then from any set of n linearly independent vectors we the first variable and conjugate linear in the second, such that can manufacture an orthonormal basis e1, e2,..., en using the Gram-Schmidt process. In terms of this basis we can write any v V in the form (f ,д) = (д, f ), ∈ v = a e , a = (v, e ) (f , f ) 0 f H, and (f , f ) = 0 = f = 0. i i i i ≥ ∀ ∈ ⇒ ∑ The norm and inner product are related by which can be derived by taking the inner product of the equation v = aiei with ei. We also have n ∑ (f , f ) = f 2. ∥ ∥ v 2 = a 2. ∥ ∥ | i | i=1 We will always assume that H is separable (has a countable dense subset). ∑ Standard infinite-dimensional examples are l2(N) or l2(Z), the space of square-summable As usual for a normed space, the distance on H is given byd(f ,д) = f д = (f д, f д). • • ∥ − ∥ − − sequences, and L2(Ω) where Ω is a measurable subset of Rn. The Cauchy-Schwarz and triangle inequalities, • √ (f ,д) f д , f + д f + д , | | ≤ ∥ ∥∥ ∥ ∥ ∥ ≤ ∥ ∥ ∥ ∥ 2.1 Orthogonality can be derived fairly easily from the inner product. -
Chapter 3 Electromagnetic Waves & Maxwell's Equations
Chapter 3 Electromagnetic Waves & Maxwell’s Equations Part I Maxwell’s Equations Maxwell (13 June 1831 – 5 November 1879) was a Scottish physicist. Famous equations published in 1861 Maxwell’s Equations: Integral Form Gauss's law Gauss's law for magnetism Faraday's law of induction (Maxwell–Faraday equation) Ampère's law (with Maxwell's addition) Maxwell’s Equations Relation of the speed of light and electric and magnetic vacuum constants 1 c = 2.99792458 108 [m/s] 0 0 permittivity of free space, also called the electric constant permeability of free space, also called the magnetic constant Gauss’s Law For any closed surface enclosing total charge Qin, the net electric flux through the surface is This result for the electric flux is known as Gauss’s Law. Magnetic Gauss’s Law The net magnetic flux through any closed surface is equal to zero: As of today there is no evidence of magnetic monopoles See: Phys.Rev.Lett.85:5292,2000 Ampère's Law The magnetic field in space around an electric current is proportional to the electric current which serves as its source: B ds 0I I is the total current inside the loop. ds B i1 Direction of integration i 3 i2 Faraday’s Law The change of magnetic flux in a loop will induce emf, i.e., electric field B E ds dA A t Lenz's Law Claim: Direction of induced current must be so as to oppose the change; otherwise conservation of energy would be violated. Problem with Ampère's Law Maxwell realized that Ampere’s law is not valid when the current is discontinuous. -
The Lorentz Force
CLASSICAL CONCEPT REVIEW 14 The Lorentz Force We can find empirically that a particle with mass m and electric charge q in an elec- tric field E experiences a force FE given by FE = q E LF-1 It is apparent from Equation LF-1 that, if q is a positive charge (e.g., a proton), FE is parallel to, that is, in the direction of E and if q is a negative charge (e.g., an electron), FE is antiparallel to, that is, opposite to the direction of E (see Figure LF-1). A posi- tive charge moving parallel to E or a negative charge moving antiparallel to E is, in the absence of other forces of significance, accelerated according to Newton’s second law: q F q E m a a E LF-2 E = = 1 = m Equation LF-2 is, of course, not relativistically correct. The relativistically correct force is given by d g mu u2 -3 2 du u2 -3 2 FE = q E = = m 1 - = m 1 - a LF-3 dt c2 > dt c2 > 1 2 a b a b 3 Classically, for example, suppose a proton initially moving at v0 = 10 m s enters a region of uniform electric field of magnitude E = 500 V m antiparallel to the direction of E (see Figure LF-2a). How far does it travel before coming (instanta> - neously) to rest? From Equation LF-2 the acceleration slowing the proton> is q 1.60 * 10-19 C 500 V m a = - E = - = -4.79 * 1010 m s2 m 1.67 * 10-27 kg 1 2 1 > 2 E > The distance Dx traveled by the proton until it comes to rest with vf 0 is given by FE • –q +q • FE 2 2 3 2 vf - v0 0 - 10 m s Dx = = 2a 2 4.79 1010 m s2 - 1* > 2 1 > 2 Dx 1.04 10-5 m 1.04 10-3 cm Ϸ 0.01 mm = * = * LF-1 A positively charged particle in an electric field experiences a If the same proton is injected into the field perpendicular to E (or at some angle force in the direction of the field. -
Poynting Vector and Power Flow in Electromagnetic Fields
Poynting Vector and Power Flow in Electromagnetic Fields: Electromagnetic waves can transport energy as a result of their travelling or propagating characteristics. Starting from Maxwell's Equations: Together with the vector identity One can write In simple medium where and are constant, and Divergence theorem states, This equation is referred to as Poynting theorem and it states that the net power flowing out of a given volume is equal to the time rate of decrease in the energy stored within the volume minus the conduction losses. In the equation, the following term represents the rate of change of the stored energy in the electric and magnetic fields On the other hand, the power dissipation within the volume appears in the following form Hence the total decrease in power within the volume under consideration: Here (W/mt2) is called the Poynting vector and it represents the power density vector associated with the electromagnetic field. The integration of the Poynting vector over any closed surface gives the net power flowing out of the surface. Poynting vector for the time harmonic case: Using the convention, the instantaneous value of a quantity is the real part of the product of a phasor quantity and when is used as reference. Considering the following phasor: −푗훽푧 퐸⃗ (푧) = 푥̂퐸푥(푧) = 푥̂퐸0푒 The instantaneous field becomes: 푗푤푡 퐸⃗ (푧, 푡) = 푅푒{퐸⃗ (푧)푒 } = 푥̂퐸0푐표푠(휔푡 − 훽푧) when E0 is real. Let us consider two instantaneous quantities A and B such that where A and B are the phasor quantities. Therefore, Since A and B are periodic with period , the time average value of the product form AB, 푇 1 퐴퐵 = ∫ 퐴퐵푑푡 푎푣푒푟푎푔푒 푇 0 푇 1 퐴퐵 = ∫|퐴||퐵|푐표푠(휔푡 + 훼)푐표푠(휔푡 + 훽)푑푡 푎푣푒푟푎푔푒 푇 0 1 퐴퐵 = |퐴||퐵|푐표푠(훼 − 훽) 푎푣푒푟푎푔푒 2 For phasors, and , where * denotes complex conjugate. -
Electromagnetic Field Theory
Lecture 4 Electromagnetic Field Theory “Our thoughts and feelings have Dr. G. V. Nagesh Kumar Professor and Head, Department of EEE, electromagnetic reality. JNTUA College of Engineering Pulivendula Manifest wisely.” Topics 1. Biot Savart’s Law 2. Ampere’s Law 3. Curl 2 Releation between Electric Field and Magnetic Field On 21 April 1820, Ørsted published his discovery that a compass needle was deflected from magnetic north by a nearby electric current, confirming a direct relationship between electricity and magnetism. 3 Magnetic Field 4 Magnetic Field 5 Direction of Magnetic Field 6 Direction of Magnetic Field 7 Properties of Magnetic Field 8 Magnetic Field Intensity • The quantitative measure of strongness or weakness of the magnetic field is given by magnetic field intensity or magnetic field strength. • It is denoted as H. It is a vector quantity • The magnetic field intensity at any point in the magnetic field is defined as the force experienced by a unit north pole of one Weber strength, when placed at that point. • The magnetic field intensity is measured in • Newtons/Weber (N/Wb) or • Amperes per metre (A/m) or • Ampere-turns / metre (AT/m). 9 Magnetic Field Density 10 Releation between B and H 11 Permeability 12 Biot Savart’s Law 13 Biot Savart’s Law 14 Biot Savart’s Law : Distributed Sources 15 Problem 16 Problem 17 H due to Infinitely Long Conductor 18 H due to Finite Long Conductor 19 H due to Finite Long Conductor 20 H at Centre of Circular Cylinder 21 H at Centre of Circular Cylinder 22 H on the axis of a Circular Loop -
Electro Magnetic Fields Lecture Notes B.Tech
ELECTRO MAGNETIC FIELDS LECTURE NOTES B.TECH (II YEAR – I SEM) (2019-20) Prepared by: M.KUMARA SWAMY., Asst.Prof Department of Electrical & Electronics Engineering MALLA REDDY COLLEGE OF ENGINEERING & TECHNOLOGY (Autonomous Institution – UGC, Govt. of India) Recognized under 2(f) and 12 (B) of UGC ACT 1956 (Affiliated to JNTUH, Hyderabad, Approved by AICTE - Accredited by NBA & NAAC – ‘A’ Grade - ISO 9001:2015 Certified) Maisammaguda, Dhulapally (Post Via. Kompally), Secunderabad – 500100, Telangana State, India ELECTRO MAGNETIC FIELDS Objectives: • To introduce the concepts of electric field, magnetic field. • Applications of electric and magnetic fields in the development of the theory for power transmission lines and electrical machines. UNIT – I Electrostatics: Electrostatic Fields – Coulomb’s Law – Electric Field Intensity (EFI) – EFI due to a line and a surface charge – Work done in moving a point charge in an electrostatic field – Electric Potential – Properties of potential function – Potential gradient – Gauss’s law – Application of Gauss’s Law – Maxwell’s first law, div ( D )=ρv – Laplace’s and Poison’s equations . Electric dipole – Dipole moment – potential and EFI due to an electric dipole. UNIT – II Dielectrics & Capacitance: Behavior of conductors in an electric field – Conductors and Insulators – Electric field inside a dielectric material – polarization – Dielectric – Conductor and Dielectric – Dielectric boundary conditions – Capacitance – Capacitance of parallel plates – spherical co‐axial capacitors. Current density – conduction and Convection current densities – Ohm’s law in point form – Equation of continuity UNIT – III Magneto Statics: Static magnetic fields – Biot‐Savart’s law – Magnetic field intensity (MFI) – MFI due to a straight current carrying filament – MFI due to circular, square and solenoid current Carrying wire – Relation between magnetic flux and magnetic flux density – Maxwell’s second Equation, div(B)=0, Ampere’s Law & Applications: Ampere’s circuital law and its applications viz. -
Chapter 22 Magnetism
Chapter 22 Magnetism 22.1 The Magnetic Field 22.2 The Magnetic Force on Moving Charges 22.3 The Motion of Charged particles in a Magnetic Field 22.4 The Magnetic Force Exerted on a Current- Carrying Wire 22.5 Loops of Current and Magnetic Torque 22.6 Electric Current, Magnetic Fields, and Ampere’s Law Magnetism – Is this a new force? Bar magnets (compass needle) align themselves in a north-south direction. Poles: Unlike poles attract, like poles repel Magnet has NO effect on an electroscope and is not influenced by gravity Magnets attract only some objects (iron, nickel etc) No magnets ever repel non magnets Magnets have no effect on things like copper or brass Cut a bar magnet-you get two smaller magnets (no magnetic monopoles) Earth is like a huge bar magnet Figure 22–1 The force between two bar magnets (a) Opposite poles attract each other. (b) The force between like poles is repulsive. Figure 22–2 Magnets always have two poles When a bar magnet is broken in half two new poles appear. Each half has both a north pole and a south pole, just like any other bar magnet. Figure 22–4 Magnetic field lines for a bar magnet The field lines are closely spaced near the poles, where the magnetic field B is most intense. In addition, the lines form closed loops that leave at the north pole of the magnet and enter at the south pole. Magnetic Field Lines If a compass is placed in a magnetic field the needle lines up with the field. -
Electromagnetic Energy
Physics 142 Electromagnetic Enmergy Page !1 Electromagnetic Energy A child of five can understand this; send someone to fetch a child of five. — Groucho Marx Energy in the fields can move from place to place We have discussed the energy in an electrostatic field, such as that stored in a capacitor. We found that this energy can be thought of as distributed in space, with an energy per unit volume (energy density) at each point in space. We obtained a formula for this 1 2 quantity, ! ue = 2 ε0E (if there are no dielectric materials). This formula applies to any E- field. We also found that magnetic fields possess energy distributed in space, described 2 by the magnetic energy density ! um = B /2µ0 . If there are both electric and magnetic fields, the total electromagnetic energy density is the sum of ! ue and ! um . These specify how much electromagnetic field energy there is at any point in space. But we have not yet considered how this energy moves from place to place. Consider the energy flow in a flashlight. We know that energy moves from the battery to the bulb, where it is converted into heat and light. It is tempting to assume that this energy flows through the conductors, like water in a pipe. But if we look carefully we find that the electromagnetic energy density in the conductors is much too small to account for the amount of energy in transit. Nearly all of the energy gets to the bulb by flowing through space near the conductors. In a sense they guide the energy but do not carry much of it. -
Maxwell's Equations in Differential Form
M03_RAO3333_1_SE_CHO3.QXD 4/9/08 1:17 PM Page 71 CHAPTER Maxwell’s Equations in Differential Form 3 In Chapter 2 we introduced Maxwell’s equations in integral form. We learned that the quantities involved in the formulation of these equations are the scalar quantities, elec- tromotive force, magnetomotive force, magnetic flux, displacement flux, charge, and current, which are related to the field vectors and source densities through line, surface, and volume integrals. Thus, the integral forms of Maxwell’s equations,while containing all the information pertinent to the interdependence of the field and source quantities over a given region in space, do not permit us to study directly the interaction between the field vectors and their relationships with the source densities at individual points. It is our goal in this chapter to derive the differential forms of Maxwell’s equations that apply directly to the field vectors and source densities at a given point. We shall derive Maxwell’s equations in differential form by applying Maxwell’s equations in integral form to infinitesimal closed paths, surfaces, and volumes, in the limit that they shrink to points. We will find that the differential equations relate the spatial variations of the field vectors at a given point to their temporal variations and to the charge and current densities at that point. In this process we shall also learn two important operations in vector calculus, known as curl and divergence, and two related theorems, known as Stokes’ and divergence theorems. 3.1 FARADAY’S LAW We recall from the previous chapter that Faraday’s law is given in integral form by d E # dl =- B # dS (3.1) CC dt LS where S is any surface bounded by the closed path C.In the most general case,the elec- tric and magnetic fields have all three components (x, y,and z) and are dependent on all three coordinates (x, y,and z) in addition to time (t).