The Particle-Field Hamiltonian
|
|
- Scot Montgomery
- 6 years ago
- Views:
Transcription
1 The Particle-Field Hamiltonian For a fundamental understanding of the interaction of a particle with the electromagnetic field we need to know the total energy of the system consisting of particle and field. This energy remains conserved; the particle can borrow energy from the field (absorption) or it can donate energy to it (emission). The total energy corresponds to the classical Hamiltonian H, which constitutes the Hamiltonian operator Ĥ encountered in quantum mechanics. For particles consisting of many charges, the Hamiltonian soon becomes a very complex function: it depends on the mutual interaction between the charges, their kinetic energies and the exchange of energy with the external field. To understand the interaction between a particle and an electromagnetic field we first consider a single point-like particle with mass m and charge q. Later we generalize the situation to systems consisting of multiple charges and with finite size. The Hamiltonian for a single charge in an electromagnetic field is found by first deriving a Lagrangian function L(r, ṙ) which satisfies the Lagrange Euler equation ( ) d L L = 0, q = x, y, z. (1) dt q q Here, q = (x, y, z) and q = (ẋ, ẏ, ż) denote the coordinates of the charge and the velocities, respectively. 1 To determine L, we first consider the (non-relativistic) equation of motion for the charge F = d [mṙ] = q (E + ṙ B), () dt and replace E and B by the vector potential A and scalar potential φ according to E(r, t) = A(r, t) φ(r, t), t (3) B(r, t) = A(r, t). (4) Now we consider the vector-components of Eq. () separately. For the x-component we obtain [ d φ [mẋ] = q dt x + A ] [ ( x Ay + q ẏ t x A ) ( x Ax ż y z A )] z x = x [ qφ + q (A xẋ + A y ẏ + A z ż)] [ Ax q + ẋ A x t x + ẏ A y y + ż A ] z. (5) z Identifying the last expression in brackets with da x /dt (total differential) and rearranging terms, the equation above can be written as d dt [mẋ + qa x] x [ qφ + q (A xẋ + A y ẏ + A z ż)] = 0. (6) 1 It is a convention of the Hamiltonian formalism to designate the generalized coordinates by the symbol q. Here, it should not be confused with the charge q. 1
2 This equation has almost the form of the Lagrange Euler equation (1). Therefore, we seek a Lagrangian of the form L = qφ + q (A x ẋ + A y ẏ + A z ż) + f(x, ẋ), (7) with f/ x = 0. With this choice, the first term in Eq. (1) leads to ( ) d L = d [ qa x + f ]. (8) dt q dt ẋ This expression has to be identical with the first term in Eq. (6), which leads to f/ ẋ = mẋ. The solution f(x, ẋ) = mẋ / can be substituted into Eq. (7) and, after generalizing to all degrees of freedom, we finally obtain L = qφ + q (A x ẋ + A y ẏ + A z ż) + 1 m ( ẋ + ẏ + ż ), (9) which can be written as L = q φ + q v A + m v v. (10) To determine the Hamiltonian H we first calculate the canonical momentum p = (p x, p y, p z ) conjugate to the coordinate q = (x, y, z) according to p i = L/ q i. The canonical momentum turns out to be p = mv + qa, (11) which is the sum of mechanical momentum mv and field momentum qa. According to Hamiltonian mechanics, the Hamiltonian is derived from the Lagrangian according to H(q, p) = i [p i q i L(q, q)], (1) in which all the velocities q i have to be expressed in terms of the coordinates q i and conjugate momenta p i. This is easily done by using Eq. (11) as q i = p i /m qa i /m. Using this substitution in Eqs. (9) and (1) we finally obtain H = 1 m [p qa] + qφ. (13) This is the Hamiltonian of a free charge q with mass m in an external electromagnetic field. The first term renders the kinetic mechanical energy and the second term the potential energy of the charge. Notice that the derivation of L and H is independent of gauge, i.e. we did not imply any condition on A. Using Hamilton s canonical equations q i = H/ p i and ṗ i = H/ q i it is straightforward to show that the Hamiltonian in Eq. (13) reproduces the equations of motion stated in Eq. (). The Hamiltonian of Eq. (13) is not yet the total Hamiltonian H tot of the system charge + field since we did not include the energy of the electromagnetic field. Furthermore, if the charge is interacting with other charges, as in the case of an atom or a molecule, we must take into account the interaction between the charges. In general, the total Hamiltonian for a system of charges can be written as H tot = H particle + H rad + H int. (14) Careful, we re using the same symbol for the dipole moment and the canonical momentum!
3 Here, H rad is the Hamiltonian of the radiation field in the absence of the charges and H particle is the Hamiltonian of the system of charges (particle) in the absence of the electromagnetic field. The interaction between the two systems is described by the interaction Hamiltonian H int. Let us determine the individual contributions. The particle Hamiltonian H particle is determined by a sum of the kinetic energies p n p n /(m n ) of the N charges and the potential energies V (r m, r n ) between the charges (intramolecular potential), i.e. H particle = [ ] pn p n + V (r m, r n ), (15) m n,m n where the nth particle is specified by its charge q n, mass m n, and coordinate r n. Notice that V (r m, r n ) is determined in the absence of the external radiation field. This term is solely due to the Coulomb interaction between the charges. H rad is defined by integrating the electromagnetic energy density W of the radiation field over all space as 3 H rad = 1 [εo E + µ 1 o B ] dv, (16) where E = E and B = B. It should be noted that the inclusion of H rad is essential for a rigorous quantum electrodynamical treatment of light matter interactions. This term ensures that the system consisting of particles and fields is conservative; it permits the interchange of energy between the atomic states and the states of the radiation field. Spontaneous emission is a direct consequence of the inclusion of H rad and cannot be derived by semiclassical calculations in which H rad is not included. Finally, to determine H int we first consider each charge separately. Each charge contributes to H int a term that can be derived from Eq. (13) as H p p m = q q [p A + A p] + A A + qφ. (17) m m Here, we subtracted the kinetic energy of the charge from the classical particle field Hamiltonian since this term is already included in H particle. Using p A = A p and then summing the contributions of all N charges in the system we can write H int as 4 H int = n [ q n m n A(r n, t) p n + ] q n A(r n, t) A(r n, t) + q n φ(r n, t). m n In the next section we will show that H int can be expanded into a multipole series similar to our previous results for V E and V M. (18) Multipole expansion of the interaction Hamiltonian The Hamiltonian expressed in terms of the vector potential A and scalar potential φ is not unique. This is caused by the freedom of gauge, i.e. if the potentials are replaced by new potentials Ã, 3 This integration leads necessarily to an infinite result which caused difficulties in the development of the quantum theory of light. 4 In quantum mechanics, the canonical momentum p is converted to an operator according to p i (Jordan rule), which also turns H int into an operator. p and A commute only if the Coulomb gauge ( A = 0) is adopted. 3
4 φ according to A Ã + χ and φ φ χ/ t, (19) with χ(r, t) being an arbitrary gauge function, Maxwell s equations remain unaffected. This is easily seen by introducing the above substitutions in the definitions of A and φ (Eqs. (3) and (4)). To remove the ambiguity caused by the freedom of gauge we need to express H int in terms of the original fields E and B. To do this, we first expand the electric and magnetic fields in a Taylor series with origin r = 0 E(r) = E(0) + [r ]E(0) + 1! [r ] E(0) +, (0) B(r) = B(0) + [r ]B(0) + 1! [r ] B(0) +, (1) and introduce these expansions in the definitions for A and φ (Eqs. (3) and (4)). The task is now to find an expansion of A and φ in terms of E and B such that the left- and right-hand sides of Eqs. (3) and (4) are identical. These expansions have been determined by Barron and Gray as φ(r) = φ(0) i=0 r [r ] i (i+1)! E(0), A(r) = i=0 [r ] i B(0) r. () (i+) i! Inserting into the expression for H int in Eq. (18) leads to the so-called multipolar interaction Hamiltonian H int = q tot φ(0, t) p E(0, t) m B(0, t) [ Q ] E(0, t).. (3) in which we used the following definitions q tot = n q n, p = n q n r n, m = n (q n /m n ) r n p n, Q = n (q n /)r n r n q tot is the total charge of the system, p denotes the total electric dipole moment, m the total magnetic dipole moment, and Q the total electric quadrupole moment. If the system of charges is charge neutral, the first term in H int vanishes and we are left with an expansion which looks very much like the former expansion of the potential energy V E + V M. However, the two expansions are not identical! First, the new magnetic dipole moment is defined in terms of the canonical momenta p n and not by the mechanical momenta m n ṙ n. 5 Second, the expansion of H int contains a term nonlinear in B(0, t), which is non-existent in the expansion of V E + V M. The nonlinear term arises from the term A A of the Hamiltonian and is referred to as the diamagnetic term. It reads as (qn/8m n ) [r n B(0, t)]. (5) n Our previous expressions for V E and V M have been derived by neglecting retardation and assuming weak fields. In this limit, the nonlinear term in Eq. (5) can be neglected and the canonical momentum can be approximated by the mechanical momentum. 5 A gauge transformation also transforms the canonical momenta. Therefore, the canonical momenta p n are different from the original canonical momenta p n. (4) 4
5 The multipolar interaction Hamiltonian can easily be converted to an operator by simply applying Jordan s rule p i and replacing the fields E and B by the corresponding electric and magnetic field operators. However, this is beyond the present scope. Notice that the Hamiltonian H int in Eq. (3) is gauge independent. The gauge affects H int only when the latter is expressed in terms of A and φ but not when it is represented by the original fields E and B. The first term in the multipolar Hamiltonian of a charge neutral system is the dipole interaction, which is identical to the corresponding term in V E. In most circumstances, it is sufficiently accurate to reject the higher terms in the multipolar expansion. This is especially true for far-field interactions where the magnetic dipole and electric quadrupole interactions are roughly two orders of magnitude weaker than the electric dipole interaction. Therefore, standard selection rules for optical transitions are based on the electric dipole interaction. However, in strongly confined optical fields, as encountered in near-field optics, higher-order terms in the expansion of H int can become important and the standard selection rules can be violated. Finally, it should be noted that the multipolar form of H int can also be derived from Eq. (18) by a unitary transformation. This transformation, commonly referred to as the Power Zienau Woolley transformation, plays an important role in quantum optics. 5
Electric and Magnetic Forces in Lagrangian and Hamiltonian Formalism
Electric and Magnetic Forces in Lagrangian and Hamiltonian Formalism Benjamin Hornberger 1/26/1 Phy 55, Classical Electrodynamics, Prof. Goldhaber Lecture notes from Oct. 26, 21 Lecture held by Prof. Weisberger
More informationMaxwell s equations. based on S-54. electric field charge density. current density
Maxwell s equations based on S-54 Our next task is to find a quantum field theory description of spin-1 particles, e.g. photons. Classical electrodynamics is governed by Maxwell s equations: electric field
More informationThe Accelerator Hamiltonian in a Straight Coordinate System
Hamiltoninan Dynamics for Particle Accelerators, Lecture 2 The Accelerator Hamiltonian in a Straight Coordinate System Andy Wolski University of Liverpool, and the Cockcroft Institute, Daresbury, UK. Given
More informationFYS 3120: Classical Mechanics and Electrodynamics
FYS 3120: Classical Mechanics and Electrodynamics Formula Collection Spring semester 2014 1 Analytical Mechanics The Lagrangian L = L(q, q, t), (1) is a function of the generalized coordinates q = {q i
More information15. Hamiltonian Mechanics
University of Rhode Island DigitalCommons@URI Classical Dynamics Physics Course Materials 2015 15. Hamiltonian Mechanics Gerhard Müller University of Rhode Island, gmuller@uri.edu Creative Commons License
More informationM2A2 Problem Sheet 3 - Hamiltonian Mechanics
MA Problem Sheet 3 - Hamiltonian Mechanics. The particle in a cone. A particle slides under gravity, inside a smooth circular cone with a vertical axis, z = k x + y. Write down its Lagrangian in a) Cartesian,
More informationCurves in the configuration space Q or in the velocity phase space Ω satisfying the Euler-Lagrange (EL) equations,
Physics 6010, Fall 2010 Hamiltonian Formalism: Hamilton s equations. Conservation laws. Reduction. Poisson Brackets. Relevant Sections in Text: 8.1 8.3, 9.5 The Hamiltonian Formalism We now return to formal
More informationE = φ 1 A The dynamics of a particle with mass m and charge q is determined by the Hamiltonian
Lecture 9 Relevant sections in text: 2.6 Charged particle in an electromagnetic field We now turn to another extremely important example of quantum dynamics. Let us describe a non-relativistic particle
More informationNonlinear Single-Particle Dynamics in High Energy Accelerators
Nonlinear Single-Particle Dynamics in High Energy Accelerators Part 2: Basic tools and concepts Nonlinear Single-Particle Dynamics in High Energy Accelerators This course consists of eight lectures: 1.
More informationPhysics 5153 Classical Mechanics. Canonical Transformations-1
1 Introduction Physics 5153 Classical Mechanics Canonical Transformations The choice of generalized coordinates used to describe a physical system is completely arbitrary, but the Lagrangian is invariant
More informationPhysics 452 Lecture 33: A Particle in an E&M Field
Physics 452 Lecture 33: A Particle in an E&M Field J. Peatross In lectures 31 and 32, we considered the Klein-Gordon equation for a free particle. We would like to add a potential to the equation (since
More informationLegendre Transforms, Calculus of Varations, and Mechanics Principles
page 437 Appendix C Legendre Transforms, Calculus of Varations, and Mechanics Principles C.1 Legendre Transforms Legendre transforms map functions in a vector space to functions in the dual space. From
More informationClassical Field Theory
April 13, 2010 Field Theory : Introduction A classical field theory is a physical theory that describes the study of how one or more physical fields interact with matter. The word classical is used in
More informationGeneralized Coordinates, Lagrangians
Generalized Coordinates, Lagrangians Sourendu Gupta TIFR, Mumbai, India Classical Mechanics 2012 August 10, 2012 Generalized coordinates Consider again the motion of a simple pendulum. Since it is one
More informationMolecules in Magnetic Fields
Molecules in Magnetic Fields Trygve Helgaker Hylleraas Centre, Department of Chemistry, University of Oslo, Norway and Centre for Advanced Study at the Norwegian Academy of Science and Letters, Oslo, Norway
More informationMicroscopic electrodynamics. Trond Saue (LCPQ, Toulouse) Microscopic electrodynamics Virginia Tech / 46
Microscopic electrodynamics Trond Saue (LCPQ, Toulouse) Microscopic electrodynamics Virginia Tech 2015 1 / 46 Maxwell s equations for electric field E and magnetic field B in terms of sources ρ and j The
More informationINTRODUCTION TO QUANTUM ELECTRODYNAMICS by Lawrence R. Mead, Prof. Physics, USM
INTRODUCTION TO QUANTUM ELECTRODYNAMICS by Lawrence R. Mead, Prof. Physics, USM I. The interaction of electromagnetic fields with matter. The Lagrangian for the charge q in electromagnetic potentials V
More informationMaxwell s equations. electric field charge density. current density
Maxwell s equations based on S-54 Our next task is to find a quantum field theory description of spin-1 particles, e.g. photons. Classical electrodynamics is governed by Maxwell s equations: electric field
More informationG : Statistical Mechanics
G25.2651: Statistical Mechanics Notes for Lecture 1 Defining statistical mechanics: Statistical Mechanics provies the connection between microscopic motion of individual atoms of matter and macroscopically
More informationClassical Mechanics and Electrodynamics
Classical Mechanics and Electrodynamics Lecture notes FYS 3120 Jon Magne Leinaas Department of Physics, University of Oslo December 2009 2 Preface These notes are prepared for the physics course FYS 3120,
More informationAn Exactly Solvable 3 Body Problem
An Exactly Solvable 3 Body Problem The most famous n-body problem is one where particles interact by an inverse square-law force. However, there is a class of exactly solvable n-body problems in which
More informationUnder evolution for a small time δt the area A(t) = q p evolves into an area
Physics 106a, Caltech 6 November, 2018 Lecture 11: Hamiltonian Mechanics II Towards statistical mechanics Phase space volumes are conserved by Hamiltonian dynamics We can use many nearby initial conditions
More informationClassical and Quantum Mechanics of a Charged Particle Moving in Electric and Magnetic Fields
Classical Mechanics Classical and Quantum Mechanics of a Charged Particle Moving in Electric and Magnetic Fields In this section I describe the Lagrangian and the Hamiltonian formulations of classical
More informationEnergy and Equations of Motion
Energy and Equations of Motion V. Tanrıverdi tanriverdivedat@googlemail.com Physics Department, Middle East Technical University, Ankara / TURKEY Abstract. From the total time derivative of energy an equation
More informationChapter 9. Electromagnetic Radiation
Chapter 9. Electromagnetic Radiation 9.1 Photons and Electromagnetic Wave Electromagnetic radiation is composed of elementary particles called photons. The correspondence between the classical electric
More informationthe EL equation for the x coordinate is easily seen to be (exercise)
Physics 6010, Fall 2016 Relevant Sections in Text: 1.3 1.6 Examples After all this formalism it is a good idea to spend some time developing a number of illustrative examples. These examples represent
More informationQUANTUM MECHANICS. Franz Schwabl. Translated by Ronald Kates. ff Springer
Franz Schwabl QUANTUM MECHANICS Translated by Ronald Kates Second Revised Edition With 122Figures, 16Tables, Numerous Worked Examples, and 126 Problems ff Springer Contents 1. Historical and Experimental
More informationMolecular Magnetic Properties
Molecular Magnetic Properties Trygve Helgaker Centre for Theoretical and Computational Chemistry (CTCC), Department of Chemistry, University of Oslo, Norway Raman Centre for Atomic, Molecular and Optical
More informationChapter 1. Principles of Motion in Invariantive Mechanics
Chapter 1 Principles of Motion in Invariantive Mechanics 1.1. The Euler-Lagrange and Hamilton s equations obtained by means of exterior forms Let L = L(q 1,q 2,...,q n, q 1, q 2,..., q n,t) L(q, q,t) (1.1)
More informationThe Principle of Least Action
The Principle of Least Action Anders Svensson Abstract In this paper, the principle of least action in classical mechanics is studied. The term is used in several different contexts, mainly for Hamilton
More informationWe start with some important background material in classical and quantum mechanics.
Chapter Basics We start with some important background material in classical and quantum mechanics.. Classical mechanics Lagrangian mechanics Compared to Newtonian mechanics, Lagrangian mechanics has the
More informationPhysics 221A Fall 1996 Notes 21 Hyperfine Structure in Hydrogen and Alkali Atoms
Physics 221A Fall 1996 Notes 21 Hyperfine Structure in Hydrogen and Alkali Atoms Hyperfine effects in atomic physics are due to the interaction of the atomic electrons with the electric and magnetic multipole
More informationSecond quantization: where quantization and particles come from?
110 Phys460.nb 7 Second quantization: where quantization and particles come from? 7.1. Lagrangian mechanics and canonical quantization Q: How do we quantize a general system? 7.1.1.Lagrangian Lagrangian
More information2 Canonical quantization
Phys540.nb 7 Canonical quantization.1. Lagrangian mechanics and canonical quantization Q: How do we quantize a general system?.1.1.lagrangian Lagrangian mechanics is a reformulation of classical mechanics.
More informationREVIEW. Hamilton s principle. based on FW-18. Variational statement of mechanics: (for conservative forces) action Equivalent to Newton s laws!
Hamilton s principle Variational statement of mechanics: (for conservative forces) action Equivalent to Newton s laws! based on FW-18 REVIEW the particle takes the path that minimizes the integrated difference
More informationHomework 3. 1 Goldstein Part (a) Theoretical Dynamics September 24, The Hamiltonian is given by
Theoretical Dynamics September 4, 010 Instructor: Dr. Thomas Cohen Homework 3 Submitted by: Vivek Saxena 1 Goldstein 8.1 1.1 Part (a) The Hamiltonian is given by H(q i, p i, t) = p i q i L(q i, q i, t)
More informationClassical Mechanics and Electrodynamics
Classical Mechanics and Electrodynamics Lecture notes FYS 3120 Jon Magne Leinaas Department of Physics, University of Oslo 2 Preface FYS 3120 is a course in classical theoretical physics, which covers
More informationList of Comprehensive Exams Topics
List of Comprehensive Exams Topics Mechanics 1. Basic Mechanics Newton s laws and conservation laws, the virial theorem 2. The Lagrangian and Hamiltonian Formalism The Lagrange formalism and the principle
More informationPhysical Dynamics (SPA5304) Lecture Plan 2018
Physical Dynamics (SPA5304) Lecture Plan 2018 The numbers on the left margin are approximate lecture numbers. Items in gray are not covered this year 1 Advanced Review of Newtonian Mechanics 1.1 One Particle
More informationPart II. Classical Dynamics. Year
Part II Year 28 27 26 25 24 23 22 21 20 2009 2008 2007 2006 2005 28 Paper 1, Section I 8B Derive Hamilton s equations from an action principle. 22 Consider a two-dimensional phase space with the Hamiltonian
More informationCP1 REVISION LECTURE 3 INTRODUCTION TO CLASSICAL MECHANICS. Prof. N. Harnew University of Oxford TT 2017
CP1 REVISION LECTURE 3 INTRODUCTION TO CLASSICAL MECHANICS Prof. N. Harnew University of Oxford TT 2017 1 OUTLINE : CP1 REVISION LECTURE 3 : INTRODUCTION TO CLASSICAL MECHANICS 1. Angular velocity and
More informationPhys 622 Problems Chapter 5
1 Phys 622 Problems Chapter 5 Problem 1 The correct basis set of perturbation theory Consider the relativistic correction to the electron-nucleus interaction H LS = α L S, also known as the spin-orbit
More informationRelativistic Dynamics
Chapter 4 Relativistic Dynamics The most important example of a relativistic particle moving in a potential is a charged particle, say an electron, moving in an electromagnetic field, which might be that
More informationLectures on basic plasma physics: Hamiltonian mechanics of charged particle motion
Lectures on basic plasma physics: Hamiltonian mechanics of charged particle motion Department of applied physics, Aalto University March 8, 2016 Hamiltonian versus Newtonian mechanics Newtonian mechanics:
More informationQuantum Field Theory Notes. Ryan D. Reece
Quantum Field Theory Notes Ryan D. Reece November 27, 2007 Chapter 1 Preliminaries 1.1 Overview of Special Relativity 1.1.1 Lorentz Boosts Searches in the later part 19th century for the coordinate transformation
More informationVector Fields. It is standard to define F µν = µ ϕ ν ν ϕ µ, so that the action may be written compactly as
Vector Fields The most general Poincaré-invariant local quadratic action for a vector field with no more than first derivatives on the fields (ensuring that classical evolution is determined based on the
More informationLecture I: Constrained Hamiltonian systems
Lecture I: Constrained Hamiltonian systems (Courses in canonical gravity) Yaser Tavakoli December 15, 2014 1 Introduction In canonical formulation of general relativity, geometry of space-time is given
More information221A Lecture Notes Electromagnetic Couplings
221A Lecture Notes Electromagnetic Couplings 1 Classical Mechanics The coupling of the electromagnetic field with a charged point particle of charge e is given by a term in the action (MKSA system) S int
More informationLecture 5. Alexey Boyarsky. October 21, Legendre transformation and the Hamilton equations of motion
Lecture 5 Alexey Boyarsky October 1, 015 1 The Hamilton equations of motion 1.1 Legendre transformation and the Hamilton equations of motion First-order equations of motion. In the Lagrangian formulation,
More informationPHY 5246: Theoretical Dynamics, Fall November 16 th, 2015 Assignment # 11, Solutions. p θ = L θ = mr2 θ, p φ = L θ = mr2 sin 2 θ φ.
PHY 5246: Theoretical Dynamics, Fall 215 November 16 th, 215 Assignment # 11, Solutions 1 Graded problems Problem 1 1.a) The Lagrangian is L = 1 2 m(ṙ2 +r 2 θ2 +r 2 sin 2 θ φ 2 ) V(r), (1) and the conjugate
More informationMSci EXAMINATION. Date: XX th May, Time: 14:30-17:00
MSci EXAMINATION PHY-415 (MSci 4242 Relativistic Waves and Quantum Fields Time Allowed: 2 hours 30 minutes Date: XX th May, 2010 Time: 14:30-17:00 Instructions: Answer THREE QUESTIONS only. Each question
More information06. Lagrangian Mechanics II
University of Rhode Island DigitalCommons@URI Classical Dynamics Physics Course Materials 2015 06. Lagrangian Mechanics II Gerhard Müller University of Rhode Island, gmuller@uri.edu Creative Commons License
More information2.3 Calculus of variations
2.3 Calculus of variations 2.3.1 Euler-Lagrange equation The action functional S[x(t)] = L(x, ẋ, t) dt (2.3.1) which maps a curve x(t) to a number, can be expanded in a Taylor series { S[x(t) + δx(t)]
More informationQuantization of scalar fields
Quantization of scalar fields March 8, 06 We have introduced several distinct types of fields, with actions that give their field equations. These include scalar fields, S α ϕ α ϕ m ϕ d 4 x and complex
More informationLagrangian and Hamiltonian Mechanics (Symon Chapter Nine)
Lagrangian and Hamiltonian Mechanics (Symon Chapter Nine Physics A301 Spring 2005 Contents 1 Lagrangian Mechanics 3 1.1 Derivation of the Lagrange Equations...................... 3 1.1.1 Newton s Second
More informationHamilton-Jacobi theory
Hamilton-Jacobi theory November 9, 04 We conclude with the crowning theorem of Hamiltonian dynamics: a proof that for any Hamiltonian dynamical system there exists a canonical transformation to a set of
More informationPhysics 221B Spring 2018 Notes 34 The Photoelectric Effect
Copyright c 2018 by Robert G. Littlejohn Physics 221B Spring 2018 Notes 34 The Photoelectric Effect 1. Introduction In these notes we consider the ejection of an atomic electron by an incident photon,
More informationMATHEMATICAL PHYSICS
MATHEMATICAL PHYSICS Third Year SEMESTER 1 015 016 Classical Mechanics MP350 Prof. S. J. Hands, Prof. D. M. Heffernan, Dr. J.-I. Skullerud and Dr. M. Fremling Time allowed: 1 1 hours Answer two questions
More informationImage by MIT OpenCourseWare.
8.07 Lecture 37: December 12, 2012 (THE LAST!) RADIATION Radiation: infinity. Electromagnetic fields that carry energy off to At large distances, E ~ and B ~ fall off only as 1=r, so the Poynting vector
More informationdf(x) = h(x) dx Chemistry 4531 Mathematical Preliminaries Spring 2009 I. A Primer on Differential Equations Order of differential equation
Chemistry 4531 Mathematical Preliminaries Spring 009 I. A Primer on Differential Equations Order of differential equation Linearity of differential equation Partial vs. Ordinary Differential Equations
More informationFrom Particles to Fields
From Particles to Fields Tien-Tsan Shieh Institute of Mathematics Academic Sinica July 25, 2011 Tien-Tsan Shieh (Institute of MathematicsAcademic Sinica) From Particles to Fields July 25, 2011 1 / 24 Hamiltonian
More informationHamiltonian. March 30, 2013
Hamiltonian March 3, 213 Contents 1 Variational problem as a constrained problem 1 1.1 Differential constaint......................... 1 1.2 Canonic form............................. 2 1.3 Hamiltonian..............................
More information1 The Quantum Anharmonic Oscillator
1 The Quantum Anharmonic Oscillator Perturbation theory based on Feynman diagrams can be used to calculate observables in Quantum Electrodynamics, like the anomalous magnetic moment of the electron, and
More informationSketchy Notes on Lagrangian and Hamiltonian Mechanics
Sketchy Notes on Lagrangian and Hamiltonian Mechanics Robert Jones Generalized Coordinates Suppose we have some physical system, like a free particle, a pendulum suspended from another pendulum, or a field
More informationClassical Mechanics in Hamiltonian Form
Classical Mechanics in Hamiltonian Form We consider a point particle of mass m, position x moving in a potential V (x). It moves according to Newton s law, mẍ + V (x) = 0 (1) This is the usual and simplest
More informationTime-Dependent Statistical Mechanics 5. The classical atomic fluid, classical mechanics, and classical equilibrium statistical mechanics
Time-Dependent Statistical Mechanics 5. The classical atomic fluid, classical mechanics, and classical equilibrium statistical mechanics c Hans C. Andersen October 1, 2009 While we know that in principle
More informationClassical Mechanics Comprehensive Exam Solution
Classical Mechanics Comprehensive Exam Solution January 31, 011, 1:00 pm 5:pm Solve the following six problems. In the following problems, e x, e y, and e z are unit vectors in the x, y, and z directions,
More informationGauge Fixing and Constrained Dynamics in Numerical Relativity
Gauge Fixing and Constrained Dynamics in Numerical Relativity Jon Allen The Dirac formalism for dealing with constraints in a canonical Hamiltonian formulation is reviewed. Gauge freedom is discussed and
More informationPath Integral Quantization of the Electromagnetic Field Coupled to A Spinor
EJTP 6, No. 22 (2009) 189 196 Electronic Journal of Theoretical Physics Path Integral Quantization of the Electromagnetic Field Coupled to A Spinor Walaa. I. Eshraim and Nasser. I. Farahat Department of
More informationFREQUENCY-DEPENDENT MAGNETIZABILITY
FYS-3900 MASTER'S THESIS IN PHYSICS FREQUENCY-DEPENDENT MAGNETIZABILITY Anelli Marco November, 2010 Faculty of Science and Technology Department of Physics and Technology University of Tromsø 2 FYS-3900
More informationCONSTRAINTS: notes by BERNARD F. WHITING
CONSTRAINTS: notes by BERNARD F. WHITING Whether for practical reasons or of necessity, we often find ourselves considering dynamical systems which are subject to physical constraints. In such situations
More informationLECTURES ON QUANTUM MECHANICS
LECTURES ON QUANTUM MECHANICS GORDON BAYM Unitsersity of Illinois A II I' Advanced Bock Progrant A Member of the Perseus Books Group CONTENTS Preface v Chapter 1 Photon Polarization 1 Transformation of
More informationQuantum Field Theory II
Quantum Field Theory II T. Nguyen PHY 391 Independent Study Term Paper Prof. S.G. Rajeev University of Rochester April 2, 218 1 Introduction The purpose of this independent study is to familiarize ourselves
More information3 Quantization of the Dirac equation
3 Quantization of the Dirac equation 3.1 Identical particles As is well known, quantum mechanics implies that no measurement can be performed to distinguish particles in the same quantum state. Elementary
More informationRetarded Potentials and Radiation
Retarded Potentials and Radiation No, this isn t about potentials that were held back a grade :). Retarded potentials are needed because at a given location in space, a particle feels the fields or potentials
More informationAnalytical Mechanics for Relativity and Quantum Mechanics
Analytical Mechanics for Relativity and Quantum Mechanics Oliver Davis Johns San Francisco State University OXPORD UNIVERSITY PRESS CONTENTS Dedication Preface Acknowledgments v vii ix PART I INTRODUCTION:
More informationMathematical Methods of Physics I ChaosBook.org/ predrag/courses/phys Homework 1
PHYS 6124 Handout 6 23 August 2012 Mathematical Methods of Physics I ChaosBook.org/ predrag/courses/phys-6124-12 Homework 1 Prof. P. Goldbart School of Physics Georgia Tech Homework assignments are posted
More informationLecture 2. Contents. 1 Fermi s Method 2. 2 Lattice Oscillators 3. 3 The Sine-Gordon Equation 8. Wednesday, August 28
Lecture 2 Wednesday, August 28 Contents 1 Fermi s Method 2 2 Lattice Oscillators 3 3 The Sine-Gordon Equation 8 1 1 Fermi s Method Feynman s Quantum Electrodynamics refers on the first page of the first
More informationChapter 1. Hamilton s mechanics. 1.1 Path integrals
Chapter 1 Hamilton s mechanics William Rowan Hamilton was an Irish physicist/mathematician from Dublin. Born in 1806, he basically invented modern mechanics in his 60 years and laid the groundwork for
More information2.4 Parity transformation
2.4 Parity transformation An extremely simple group is one that has only two elements: {e, P }. Obviously, P 1 = P, so P 2 = e, with e represented by the unit n n matrix in an n- dimensional representation.
More informationPhysical Dynamics (PHY-304)
Physical Dynamics (PHY-304) Gabriele Travaglini March 31, 2012 1 Review of Newtonian Mechanics 1.1 One particle Lectures 1-2. Frame, velocity, acceleration, number of degrees of freedom, generalised coordinates.
More informationAppendix A. The Particle in a Box: A Demonstration of Quantum Mechanical Principles for a Simple, One-Dimensional, One-Electron Model System
Appendix A The Particle in a Box: A Demonstration of Quantum Mechanical Principles for a Simple, One-Dimensional, One-Electron Model System Real quantum mechanical systems have the tendency to become mathematically
More informationB = 0. E = 1 c. E = 4πρ
Photons In this section, we will treat the electromagnetic field quantum mechanically. We start by recording the Maxwell equations. As usual, we expect these equations to hold both classically and quantum
More informationHamiltonian Mechanics
Alain J. Brizard Saint Michael's College Hamiltonian Mechanics 1 Hamiltonian The k second-order Euler-Lagrange equations on con guration space q =(q 1 ; :::; q k ) d @ _q j = @q j ; (1) can be written
More information2 Quantization of the Electromagnetic Field
2 Quantization of the Electromagnetic Field 2.1 Basics Starting point of the quantization of the electromagnetic field are Maxwell s equations in the vacuum (source free): where B = µ 0 H, D = ε 0 E, µ
More informationField quantization in dielectric media and the generalized multipolar Hamiltonian
PHYSICAL REVIEW A VOLUME 54, NUMBER 3 SEPTEMBER 996 Field quantization in dielectric media and the generalized multipolar Hamiltonian B. J. Dalton,, E. S. Guerra, and P. L. Knight Optics Section, Blacett
More informationLecture 4. Alexey Boyarsky. October 6, 2015
Lecture 4 Alexey Boyarsky October 6, 2015 1 Conservation laws and symmetries 1.1 Ignorable Coordinates During the motion of a mechanical system, the 2s quantities q i and q i, (i = 1, 2,..., s) which specify
More informationThe Principle of Least Action
The Principle of Least Action In their never-ending search for general principles, from which various laws of Physics could be derived, physicists, and most notably theoretical physicists, have often made
More informationLagrangian Mechanics I
Lagrangian Mechanics I Wednesday, 11 September 2013 How Lagrange revolutionized Newtonian mechanics Physics 111 Newton based his dynamical theory on the notion of force, for which he gave a somewhat incomplete
More informationPhysics 221A Fall 2005 Homework 11 Due Thursday, November 17, 2005
Physics 221A Fall 2005 Homework 11 Due Thursday, November 17, 2005 Reading Assignment: Sakurai pp. 234 242, 248 271, Notes 15. 1. Show that Eqs. (15.64) follow from the definition (15.61) of an irreducible
More informationCHAPTER 1. SPECIAL RELATIVITY AND QUANTUM MECHANICS
CHAPTER 1. SPECIAL RELATIVITY AND QUANTUM MECHANICS 1.1 PARTICLES AND FIELDS The two great structures of theoretical physics, the theory of special relativity and quantum mechanics, have been combined
More information1 The postulates of quantum mechanics
1 The postulates of quantum mechanics The postulates of quantum mechanics were derived after a long process of trial and error. These postulates provide a connection between the physical world and the
More informationAs usual, these notes are intended for use by class participants only, and are not for circulation. Week 7: Lectures 13, 14.
As usual, these notes are intended for use by class participants only, and are not for circulation. Week 7: Lectures 13, 14 Majorana spinors March 15, 2012 So far, we have only considered massless, two-component
More informationLecture 27: Generalized Coordinates and Lagrange s Equations of Motion
Lecture 27: Generalize Coorinates an Lagrange s Equations of Motion Calculating T an V in terms of generalize coorinates. Example: Penulum attache to a movable support 6 Cartesian Coorinates: (X, Y, Z)
More informationSemi-Classical Theory of Radiative Transitions
Semi-Classical Theory of Radiative Transitions Massimo Ricotti ricotti@astro.umd.edu University of Maryland Semi-Classical Theory of Radiative Transitions p.1/13 Atomic Structure (recap) Time-dependent
More informationfor changing independent variables. Most simply for a function f(x) the Legendre transformation f(x) B(s) takes the form B(s) = xs f(x) with s = df
Physics 106a, Caltech 1 November, 2018 Lecture 10: Hamiltonian Mechanics I The Hamiltonian In the Hamiltonian formulation of dynamics each second order ODE given by the Euler- Lagrange equation in terms
More informationPAPER No. : 8 (PHYSICAL SPECTROSCOPY) MODULE No. : 5 (TRANSITION PROBABILITIES AND TRANSITION DIPOLE MOMENT. OVERVIEW OF SELECTION RULES)
Subject Chemistry Paper No and Title Module No and Title Module Tag 8 and Physical Spectroscopy 5 and Transition probabilities and transition dipole moment, Overview of selection rules CHE_P8_M5 TABLE
More informationMany-Body Problems and Quantum Field Theory
Philippe A. Martin Francois Rothen Many-Body Problems and Quantum Field Theory An Introduction Translated by Steven Goldfarb, Andrew Jordan and Samuel Leach Second Edition With 102 Figures, 7 Tables and
More informationKet space as a vector space over the complex numbers
Ket space as a vector space over the complex numbers kets ϕ> and complex numbers α with two operations Addition of two kets ϕ 1 >+ ϕ 2 > is also a ket ϕ 3 > Multiplication with complex numbers α ϕ 1 >
More informationChapter 11. Radiation. How accelerating charges and changing currents produce electromagnetic waves, how they radiate.
Chapter 11. Radiation How accelerating charges and changing currents produce electromagnetic waves, how they radiate. 11.1.1 What is Radiation? Assume a radiation source is localized near the origin. Total
More information