Path integral in quantum mechanics based on S-6 Consider nonrelativistic quantum mechanics of one particle in one dimension with the hamiltonian:
|
|
- Sharyl Walters
- 6 years ago
- Views:
Transcription
1 Path integral in quantum mechanics based on S-6 Consider nonrelativistic quantum mechanics of one particle in one dimension with the hamiltonian: let s look at one piece first: P and Q obey: Probability amplitude for the particle to start at q at time t and end up at position q at time t is Campbell-Baker-Hausdorf formula complete set of momentum states where and are eigenstates of the position operator Q. In the Heisenberg picture: and we can define instantaneous eigenstates: Probability amplitude is then: to evaluate the transition amplitude: let s divide the time interval T = t - t into N+1 equal pieces insert N complete sets of position eigenstates = to evaluate the transition amplitude: let s divide the time interval T = t - t into N+1 equal pieces insert N complete sets of position eigenstates = - we find: important for general form of hamiltonians with terms containing both P and Q in our case, it doesn t make any difference 63 65
2 What is it good for? Consider e.g.: taking the limit we get: the result can be simply written using path integral as: should be understood as integration over all paths in phase space that start at and end at (with arbitrary initial and final momenta) In simple cases when hamiltonian is at most quadratic in momenta, the integral over p is gaussian and can be easily calculated: lagrangian Similarly: time ordering is crucial! prefactor can be absorbed into the definition of measure Time-ordered products appeared in LSZ formula for scattering amplitudes! Functional derivatives: Dirac delta function taking the limit we get: continuous generalization of they are defined to satisfy all the usual rules of derivatives (product rule,...) should be understood as integration over all paths in phase space that start at and end at (with arbitrary initial and final momenta) In simple cases when hamiltonian is at most quadratic in momenta: Consider modifying hamiltonian to: Then we have: And we find, e.g.: where is calculated by finding the stationary point of the p-integral by solving: for p and plugging the solution back to 67 69
3 thus we have: more examples: Similarly, for the replacement picks up the ground state as the final state in the limit. we can integrate over q and q which leads to a constant factor that can be absorbed into the normalization of the path integral. after we bring down as many qs and ps as we want we can set and return to the original hamiltonian: Thus, with the replacement the boundary conditions and we have: we don t have to care about Finally, we want both initial and final states to be ground states and take the limits and : Adding perturbations: we can simply write (suppressing the ): looks complicated, we will use the following trick instead: is the ground-state wave function eigenstate of H corresponding eigenvalue Finally, if perturbing hamiltonian depends only on q, and we want to calculate only time-ordered products of Qs, and if H is no more than quadratic in P and if the term quadratic in P does not involve Q, then the equation above can be written as: is wave function of n-th state let s replace with and take the limit : every state except the ground state is multiplied by a vanishing factor! 71 73
4 Path integral for harmonic oscillator based on S-7 Consider a harmonic oscillator: ground state to ground state transition amplitude is: external force it is convenient to change integration variables: then we get: a shift by a constant equivalent to thus going to lagrangian formulation (integrating over p) we get: and the transition amplitude is: using Furier-transformed variables: and setting for simplicity, we get But since, if there is no external force, a system in its ground state remain in its ground state. thus we have: and thus: E = E or, in terms of time-dependent variables: using inverse Fourier transformation where: 75 77
5 Comment: For even number of Qs we pair up Qs in all possible ways: is a Green s function for the equation of motion of the harmonic oscillator: you can evaluate it explicitly, treating the integral as a contour integral in the complex E-plane and using the residue theorem. Make sure you are careful about closing the contour in the correct half-plane for t > t and t < t and that you pick up the correct pole. in general: you should find: We can now easily generalized these results to a free field theory Let s calculate correlation functions of Q operators: for harmonic oscillator we find: For odd number of Qs there is always one f(t) left-over and the result is 0! Path integral for free field theory based on S-8 Hamiltonian density of a free field theory: similar to the hamiltonian of the harmonic oscillator dictionary between QM and QFT: classical field operator field classical source we repeat everything we did for the path integral in QM but now for fields; we divide space and time into small segments; take a field in each segment to be constant; the differences between fields in neighboring segments become derivatives; use the trick: multiplying by is equivalent to replacing with which we often don t write explicitly;... eventually we can integrate over momenta and 79 81
6 obtain path integral (functional integral) for our free field theory: path in the space of field configurations change of integration variables: Comments: lagrangian is manifestly Lorentz invariant and all the symmetries of a lagrangian are preserved by path integral a shift by a constant lagrangian seems to be more fundamental specification of a quantum field theory to evaluate we can closely follow the procedure we did for the harmonic oscillator: Fourier transformation: But 83 85
7 Thus we have: For even number of we pair up in all possible ways: where we used inverse Fourier transformation to go back to position-functions is the Feynman propagator, a Green s function for the Klien-Gordon equation: in general: integral over zero s component can be calculated explicitly by completing the contour and using the residue theorem, the three momentum integral can be calculated in terms of Bessel functions Wick s theorem Now we can calculate correlation functions: we find: For odd number of there is always one J left-over and the result is 0! 87
Summary of free theory: one particle state: vacuum state is annihilated by all a s: then, one particle state has normalization:
The LSZ reduction formula based on S-5 In order to describe scattering experiments we need to construct appropriate initial and final states and calculate scattering amplitude. Summary of free theory:
More informationREVIEW REVIEW. Quantum Field Theory II
Quantum Field Theory II PHYS-P 622 Radovan Dermisek, Indiana University Notes based on: M. Srednicki, Quantum Field Theory Chapters: 13, 14, 16-21, 26-28, 51, 52, 61-68, 44, 53, 69-74, 30-32, 84-86, 75,
More informationQuantum Field Theory II
Quantum Field Theory II PHYS-P 622 Radovan Dermisek, Indiana University Notes based on: M. Srednicki, Quantum Field Theory Chapters: 13, 14, 16-21, 26-28, 51, 52, 61-68, 44, 53, 69-74, 30-32, 84-86, 75,
More informationReview of scalar field theory. Srednicki 5, 9, 10
Review of scalar field theory Srednicki 5, 9, 10 2 The LSZ reduction formula based on S-5 In order to describe scattering experiments we need to construct appropriate initial and final states and calculate
More informationQuantum Field Theory II
Quantum Field Theory II PHYS-P 622 Radovan Dermisek, Indiana University Notes based on: M. Srednicki, Quantum Field Theory Chapters: 13, 14, 16-21, 26-28, 51, 52, 61-68, 44, 53, 69-74, 30-32, 84-86, 75,
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 informationBeta functions in quantum electrodynamics
Beta functions in quantum electrodynamics based on S-66 Let s calculate the beta function in QED: the dictionary: Note! following the usual procedure: we find: or equivalently: For a theory with N Dirac
More informationPart I. Many-Body Systems and Classical Field Theory
Part I. Many-Body Systems and Classical Field Theory 1. Classical and Quantum Mechanics of Particle Systems 3 1.1 Introduction. 3 1.2 Classical Mechanics of Mass Points 4 1.3 Quantum Mechanics: The Harmonic
More informationLSZ reduction for spin-1/2 particles
LSZ reduction for spin-1/2 particles based on S-41 In order to describe scattering experiments we need to construct appropriate initial and final states and calculate scattering amplitude. Summary of free
More informationREVIEW REVIEW. A guess for a suitable initial state: Similarly, let s consider a final state: Summary of free theory:
LSZ reduction for spin-1/2 particles based on S-41 In order to describe scattering experiments we need to construct appropriate initial and final states and calculate scattering amplitude. Summary of free
More informationAttempts at relativistic QM
Attempts at relativistic QM based on S-1 A proper description of particle physics should incorporate both quantum mechanics and special relativity. However historically combining quantum mechanics and
More informationLoop corrections in Yukawa theory based on S-51
Loop corrections in Yukawa theory based on S-51 Similarly, the exact Dirac propagator can be written as: Let s consider the theory of a pseudoscalar field and a Dirac field: the only couplings allowed
More informationThe path integral for photons
The path integral for photons based on S-57 We will discuss the path integral for photons and the photon propagator more carefully using the Lorentz gauge: as in the case of scalar field we Fourier-transform
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 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 informationThe Feynman propagator of the scalar field
The Feynman propagator of the scalar field Let s evaluate the following expression: 0 φ(x 1 )φ(x 2 ) 0 2E p1 d 3 p 2 2E p2 0 a p1 a p 2 0 e i(p 2 x 2 p 1 x 1 ) (1) Using the following relation, we get:
More informationScattering amplitudes and the Feynman rules
Scattering amplitudes and the Feynman rules based on S-10 We have found Z( J ) for the phi-cubed theory and now we can calculate vacuum expectation values of the time ordered products of any number of
More informationParticle Physics Dr. Alexander Mitov Handout 2 : The Dirac Equation
Dr. A. Mitov Particle Physics 45 Particle Physics Dr. Alexander Mitov µ + e - e + µ - µ + e - e + µ - µ + e - e + µ - µ + e - e + µ - Handout 2 : The Dirac Equation Dr. A. Mitov Particle Physics 46 Non-Relativistic
More informationAdvanced Quantum Mechanics
Advanced Quantum Mechanics Rajdeep Sensarma! sensarma@theory.tifr.res.in Lecture #22 Path Integrals and QM Recap of Last Class Statistical Mechanics and path integrals in imaginary time Imaginary time
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 informationWeek 5-6: Lectures The Charged Scalar Field
Notes for Phys. 610, 2011. These summaries are meant to be informal, and are subject to revision, elaboration and correction. They will be based on material covered in class, but may differ from it by
More informationPhysics 218. Quantum Field Theory. Professor Dine. Green s Functions and S Matrices from the Operator (Hamiltonian) Viewpoint
Physics 28. Quantum Field Theory. Professor Dine Green s Functions and S Matrices from the Operator (Hamiltonian) Viewpoint Field Theory in a Box Consider a real scalar field, with lagrangian L = 2 ( µφ)
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 informationParticle Physics. Michaelmas Term 2011 Prof. Mark Thomson. Handout 2 : The Dirac Equation. Non-Relativistic QM (Revision)
Particle Physics Michaelmas Term 2011 Prof. Mark Thomson + e - e + - + e - e + - + e - e + - + e - e + - Handout 2 : The Dirac Equation Prof. M.A. Thomson Michaelmas 2011 45 Non-Relativistic QM (Revision)
More informationLecture notes for QFT I (662)
Preprint typeset in JHEP style - PAPER VERSION Lecture notes for QFT I (66) Martin Kruczenski Department of Physics, Purdue University, 55 Northwestern Avenue, W. Lafayette, IN 47907-036. E-mail: markru@purdue.edu
More informationPart III. Interacting Field Theory. Quantum Electrodynamics (QED)
November-02-12 8:36 PM Part III Interacting Field Theory Quantum Electrodynamics (QED) M. Gericke Physics 7560, Relativistic QM 183 III.A Introduction December-08-12 9:10 PM At this point, we have the
More informationIntroduction to particle physics Lecture 2
Introduction to particle physics Lecture 2 Frank Krauss IPPP Durham U Durham, Epiphany term 2009 Outline 1 Quantum field theory Relativistic quantum mechanics Merging special relativity and quantum mechanics
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 informationBy following steps analogous to those that led to (20.177), one may show (exercise 20.30) that in Feynman s gauge, = 1, the photon propagator is
20.2 Fermionic path integrals 74 factor, which cancels. But if before integrating over all gauge transformations, we shift so that 4 changes to 4 A 0, then the exponential factor is exp[ i 2 R ( A 0 4
More information1 Equal-time and Time-ordered Green Functions
1 Equal-time and Time-ordered Green Functions Predictions for observables in quantum field theories are made by computing expectation values of products of field operators, which are called Green functions
More information( r) = 1 Z. e Zr/a 0. + n +1δ n', n+1 ). dt ' e i ( ε n ε i )t'/! a n ( t) = n ψ t = 1 i! e iε n t/! n' x n = Physics 624, Quantum II -- Exam 1
Physics 624, Quantum II -- Exam 1 Please show all your work on the separate sheets provided (and be sure to include your name) You are graded on your work on those pages, with partial credit where it is
More informationQFT PS7: Interacting Quantum Field Theory: λφ 4 (30/11/17) The full propagator in λφ 4 theory. Consider a theory of a real scalar field φ
QFT PS7: Interacting Quantum Field Theory: λφ 4 (30/11/17) 1 Problem Sheet 7: Interacting Quantum Field Theory: λφ 4 Comments on these questions are always welcome. For instance if you spot any typos or
More informationQuantum Mechanics-I Prof. Dr. S. Lakshmi Bala Department of Physics Indian Institute of Technology, Madras. Lecture - 21 Square-Integrable Functions
Quantum Mechanics-I Prof. Dr. S. Lakshmi Bala Department of Physics Indian Institute of Technology, Madras Lecture - 21 Square-Integrable Functions (Refer Slide Time: 00:06) (Refer Slide Time: 00:14) We
More informationWe will also need transformation properties of fermion bilinears:
We will also need transformation properties of fermion bilinears: Parity: some product of gamma matrices, such that so that is hermitian. we easily find: 88 And so the corresponding bilinears transform
More informationThe Klein-Gordon equation
Lecture 8 The Klein-Gordon equation WS2010/11: Introduction to Nuclear and Particle Physics The bosons in field theory Bosons with spin 0 scalar (or pseudo-scalar) meson fields canonical field quantization
More informationClassical field theory 2012 (NS-364B) Feynman propagator
Classical field theory 212 (NS-364B Feynman propagator 1. Introduction States in quantum mechanics in Schrödinger picture evolve as ( Ψt = Û(t,t Ψt, Û(t,t = T exp ı t dt Ĥ(t, (1 t where Û(t,t denotes the
More informationTwo particle elastic scattering at 1-loop
Two particle elastic scattering at 1-loop based on S-20 Let s use our rules to calculate two-particle elastic scattering amplitude in, theory in 6 dimensions including all one-loop corrections: For the
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 informationQFT. Unit 11: Cross Sections and Decay Rates
QFT Unit 11: Cross Sections and Decay Rates Decays and Collisions n When it comes to elementary particles, there are only two things that ever really happen: One particle decays into stuff Two particles
More informationQFT. Chapter 14: Loop Corrections to the Propagator
QFT Chapter 14: Loop Corrections to the Propagator Overview Here we turn to our next major topic: loop order corrections. We ll consider the effect on the propagator first. This has at least two advantages:
More informationQuantum Electrodynamics Test
MSc in Quantum Fields and Fundamental Forces Quantum Electrodynamics Test Monday, 11th January 2010 Please answer all three questions. All questions are worth 20 marks. Use a separate booklet for each
More information2 Quantization of the scalar field
22 Quantum field theory 2 Quantization of the scalar field Commutator relations. The strategy to quantize a classical field theory is to interpret the fields Φ(x) and Π(x) = Φ(x) as operators which satisfy
More information1 Quantum fields in Minkowski spacetime
1 Quantum fields in Minkowski spacetime The theory of quantum fields in curved spacetime is a generalization of the well-established theory of quantum fields in Minkowski spacetime. To a great extent,
More information2. As we shall see, we choose to write in terms of σ x because ( X ) 2 = σ 2 x.
Section 5.1 Simple One-Dimensional Problems: The Free Particle Page 9 The Free Particle Gaussian Wave Packets The Gaussian wave packet initial state is one of the few states for which both the { x } and
More informationlattice QCD and the hadron spectrum Jozef Dudek ODU/JLab
lattice QCD and the hadron spectrum Jozef Dudek ODU/JLab a black box? QCD lattice QCD observables (scattering amplitudes?) in these lectures, hope to give you a look inside the box 2 these lectures how
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 informationRepresentations of Lorentz Group
Representations of Lorentz Group based on S-33 We defined a unitary operator that implemented a Lorentz transformation on a scalar field: How do we find the smallest (irreducible) representations of the
More informationTransient Phenomena in Quantum Bound States Subjected to a Sudden Perturbation
Symmetry, Integrability and Geometry: Methods and Applications Vol. (5), Paper 3, 9 pages Transient Phenomena in Quantum Bound States Subjected to a Sudden Perturbation Marcos MOSHINSKY and Emerson SADURNÍ
More informationEuclidean path integral formalism: from quantum mechanics to quantum field theory
: from quantum mechanics to quantum field theory Dr. Philippe de Forcrand Tutor: Dr. Marco Panero ETH Zürich 30th March, 2009 Introduction Real time Euclidean time Vacuum s expectation values Euclidean
More informationQuantum Quenches in Extended Systems
Quantum Quenches in Extended Systems Spyros Sotiriadis 1 Pasquale Calabrese 2 John Cardy 1,3 1 Oxford University, Rudolf Peierls Centre for Theoretical Physics, Oxford, UK 2 Dipartimento di Fisica Enrico
More informationThe Feynman Propagator and Cauchy s Theorem
The Feynman Propagator and Cauchy s Theorem Tim Evans 1 (1st November 2018) The aim of these notes is to show how to derive the momentum space form of the Feynman propagator which is (p) = i/(p 2 m 2 +
More informationWhat is a particle? Keith Fratus. July 17, 2012 UCSB
What is a particle? Keith Fratus UCSB July 17, 2012 Quantum Fields The universe as we know it is fundamentally described by a theory of fields which interact with each other quantum mechanically These
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 informationMassachusetts Institute of Technology Physics Department
Massachusetts Institute of Technology Physics Department Physics 8.32 Fall 2006 Quantum Theory I October 9, 2006 Assignment 6 Due October 20, 2006 Announcements There will be a makeup lecture on Friday,
More informationPlan for the rest of the semester. ψ a
Plan for the rest of the semester ϕ ψ a ϕ(x) e iα(x) ϕ(x) 167 Representations of Lorentz Group based on S-33 We defined a unitary operator that implemented a Lorentz transformation on a scalar field: and
More informationPRINCIPLES OF PHYSICS. \Hp. Ni Jun TSINGHUA. Physics. From Quantum Field Theory. to Classical Mechanics. World Scientific. Vol.2. Report and Review in
LONDON BEIJING HONG TSINGHUA Report and Review in Physics Vol2 PRINCIPLES OF PHYSICS From Quantum Field Theory to Classical Mechanics Ni Jun Tsinghua University, China NEW JERSEY \Hp SINGAPORE World Scientific
More informationRotations in Quantum Mechanics
Rotations in Quantum Mechanics We have seen that physical transformations are represented in quantum mechanics by unitary operators acting on the Hilbert space. In this section, we ll think about the specific
More informationQuantum Field Theory
Quantum Field Theory PHYS-P 621 Radovan Dermisek, Indiana University Notes based on: M. Srednicki, Quantum Field Theory 1 Attempts at relativistic QM based on S-1 A proper description of particle physics
More informationSrednicki Chapter 9. QFT Problems & Solutions. A. George. August 21, Srednicki 9.1. State and justify the symmetry factors in figure 9.
Srednicki Chapter 9 QFT Problems & Solutions A. George August 2, 22 Srednicki 9.. State and justify the symmetry factors in figure 9.3 Swapping the sources is the same thing as swapping the ends of the
More informationQFT. Unit 1: Relativistic Quantum Mechanics
QFT Unit 1: Relativistic Quantum Mechanics What s QFT? Relativity deals with things that are fast Quantum mechanics deals with things that are small QFT deals with things that are both small and fast What
More informationQuantum Theory and Group Representations
Quantum Theory and Group Representations Peter Woit Columbia University LaGuardia Community College, November 1, 2017 Queensborough Community College, November 15, 2017 Peter Woit (Columbia University)
More informationlattice QCD and the hadron spectrum Jozef Dudek ODU/JLab
lattice QCD and the hadron spectrum Jozef Dudek ODU/JLab the light meson spectrum relatively simple models of hadrons: bound states of constituent quarks and antiquarks the quark model empirical meson
More informationQuantum Mechanics: Fundamentals
Kurt Gottfried Tung-Mow Yan Quantum Mechanics: Fundamentals Second Edition With 75 Figures Springer Preface vii Fundamental Concepts 1 1.1 Complementarity and Uncertainty 1 (a) Complementarity 2 (b) The
More information3.15. Some symmetry properties of the Berry curvature and the Chern number.
50 Phys620.nb z M 3 at the K point z M 3 3 t ' sin 3 t ' sin (3.36) (3.362) Therefore, as long as M 3 3 t ' sin, the system is an topological insulator ( z flips sign). If M 3 3 t ' sin, z is always positive
More informationIntroduction to Path Integrals
Introduction to Path Integrals Consider ordinary quantum mechanics of a single particle in one space dimension. Let s work in the coordinate space and study the evolution kernel Ut B, x B ; T A, x A )
More informationTENTATIVE SYLLABUS INTRODUCTION
Physics 615: Overview of QFT Fall 2010 TENTATIVE SYLLABUS This is a tentative schedule of what we will cover in the course. It is subject to change, often without notice. These will occur in response to
More informationTime-Independent Perturbation Theory
4 Phys46.nb Time-Independent Perturbation Theory.. Overview... General question Assuming that we have a Hamiltonian, H = H + λ H (.) where λ is a very small real number. The eigenstates of the Hamiltonian
More informationRegularization Physics 230A, Spring 2007, Hitoshi Murayama
Regularization Physics 3A, Spring 7, Hitoshi Murayama Introduction In quantum field theories, we encounter many apparent divergences. Of course all physical quantities are finite, and therefore divergences
More information8.04: Quantum Mechanics Professor Allan Adams. Problem Set 7. Due Tuesday April 9 at 11.00AM
8.04: Quantum Mechanics Professor Allan Adams Massachusetts Institute of Technology Thursday April 4 Problem Set 7 Due Tuesday April 9 at 11.00AM Assigned Reading: E&R 6 all Li. 7 1 9, 8 1 Ga. 4 all, 6
More informationwhich implies that we can take solutions which are simultaneous eigen functions of
Module 1 : Quantum Mechanics Chapter 6 : Quantum mechanics in 3-D Quantum mechanics in 3-D For most physical systems, the dynamics is in 3-D. The solutions to the general 3-d problem are quite complicated,
More informationNon-locality in QFT due to Quantum Effects in Gravity
Non-locality in QFT due to Quantum Effects in Gravity Xavier Calmet Physics & Astronomy University of Sussex 1 Effective action for GR How can we describe general relativity quantum mechanically? Well
More informationTime evolution of states in quantum mechanics 1
Time evolution of states in quantum mechanics 1 The time evolution from time t 0 to t of a quantum mechanical state is described by a linear operator Û(t, t 0. Thus a ket at time t that started out at
More information6.730 Physics for Solid State Applications
6.730 Physics for Solid State Applications Lecture 19: Motion of Electronic Wavepackets Outline Review of Last Time Detailed Look at the Translation Operator Electronic Wavepackets Effective Mass Theorem
More informationNon-relativistic scattering
Non-relativistic scattering Contents Scattering theory 2. Scattering amplitudes......................... 3.2 The Born approximation........................ 5 2 Virtual Particles 5 3 The Yukawa Potential
More informationQuantization of Scalar Field
Quantization of Scalar Field Wei Wang 2017.10.12 Wei Wang(SJTU) Lectures on QFT 2017.10.12 1 / 41 Contents 1 From classical theory to quantum theory 2 Quantization of real scalar field 3 Quantization of
More informationHelicity conservation in Born-Infeld theory
Helicity conservation in Born-Infeld theory A.A.Rosly and K.G.Selivanov ITEP, Moscow, 117218, B.Cheryomushkinskaya 25 Abstract We prove that the helicity is preserved in the scattering of photons in the
More informationNumerical Methods in Quantum Field Theories
Numerical Methods in Quantum Field Theories Christopher Bell 2011 NSF/REU Program Physics Department, University of Notre Dame Advisors: Antonio Delgado, Christopher Kolda 1 Abstract In this paper, preliminary
More informationLecture-05 Perturbation Theory and Feynman Diagrams
Lecture-5 Perturbation Theory and Feynman Diagrams U. Robkob, Physics-MUSC SCPY639/428 September 3, 218 From the previous lecture We end up at an expression of the 2-to-2 particle scattering S-matrix S
More informationQFT Perturbation Theory
QFT Perturbation Theory Ling-Fong Li (Institute) Slide_04 1 / 43 Interaction Theory As an illustration, take electromagnetic interaction. Lagrangian density is The combination L = ψ (x ) γ µ ( i µ ea µ
More informationAdvanced Quantum Mechanics
Advanced Quantum Mechanics University of York Lecturer: Rex Godby Notes by Victor Naden Robinson Lecture 1: TDSE Lecture 2: TDSE Lecture 3: FMG Lecture 4: FMG Lecture 5: Ehrenfest s Theorem and the Classical
More information2.1 Green Functions in Quantum Mechanics
Chapter 2 Green Functions and Observables 2.1 Green Functions in Quantum Mechanics We will be interested in studying the properties of the ground state of a quantum mechanical many particle system. We
More informationConventions for fields and scattering amplitudes
Conventions for fields and scattering amplitudes Thomas DeGrand 1 1 Department of Physics, University of Colorado, Boulder, CO 80309 USA (Dated: September 21, 2017) Abstract This is a discussion of conventions
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 informationMechanics Physics 151
Mechanics Physics 151 Lecture 4 Continuous Systems and Fields (Chapter 13) What We Did Last Time Built Lagrangian formalism for continuous system Lagrangian L Lagrange s equation = L dxdydz Derived simple
More informationQuantum Field Theory 2 nd Edition
Quantum Field Theory 2 nd Edition FRANZ MANDL and GRAHAM SHAW School of Physics & Astromony, The University of Manchester, Manchester, UK WILEY A John Wiley and Sons, Ltd., Publication Contents Preface
More informationQuestioning Quantum Mechanics? Kurt Barry SASS Talk January 25 th, 2012
Questioning Quantum Mechanics? Kurt Barry SASS Talk January 25 th, 2012 2 Model of the Universe Fundamental Theory Low-Energy Limit Effective Field Theory Quantum Mechanics Quantum Mechanics is presently
More informationObservables from Correlation Functions
Observables from Correlation Functions In this chapter we learn how to compute physical quantities from correlation functions beyond leading order in the perturbative expansion. We will not discuss ultraviolet
More informationLecture 7 From Dirac equation to Feynman diagramms. SS2011: Introduction to Nuclear and Particle Physics, Part 2 2
Lecture 7 From Dirac equation to Feynman diagramms SS2011: Introduction to Nuclear and Particle Physics, Part 2 2 1 Dirac equation* The Dirac equation - the wave-equation for free relativistic fermions
More informationLight-Cone Quantization of Electrodynamics
Light-Cone Quantization of Electrodynamics David G. Robertson Department of Physics, The Ohio State University Columbus, OH 43210 Abstract Light-cone quantization of (3+1)-dimensional electrodynamics is
More informationContinuous quantum states, Particle on a line and Uncertainty relations
Continuous quantum states, Particle on a line and Uncertainty relations So far we have considered k-level (discrete) quantum systems. Now we turn our attention to continuous quantum systems, such as a
More informationLS coupling. 2 2 n + H s o + H h f + H B. (1) 2m
LS coupling 1 The big picture We start from the Hamiltonian of an atomic system: H = [ ] 2 2 n Ze2 1 + 1 e 2 1 + H s o + H h f + H B. (1) 2m n e 4πɛ 0 r n 2 4πɛ 0 r nm n,m Here n runs pver the electrons,
More information2 Feynman rules, decay widths and cross sections
2 Feynman rules, decay widths and cross sections 2.1 Feynman rules Normalization In non-relativistic quantum mechanics, wave functions of free particles are normalized so that there is one particle in
More informationQuantum Field Theory. and the Standard Model. !H Cambridge UNIVERSITY PRESS MATTHEW D. SCHWARTZ. Harvard University
Quantum Field Theory and the Standard Model MATTHEW D. Harvard University SCHWARTZ!H Cambridge UNIVERSITY PRESS t Contents v Preface page xv Part I Field theory 1 1 Microscopic theory of radiation 3 1.1
More informationQuantum Physics 2006/07
Quantum Physics 6/7 Lecture 7: More on the Dirac Equation In the last lecture we showed that the Dirac equation for a free particle i h t ψr, t = i hc α + β mc ψr, t has plane wave solutions ψr, t = exp
More informationHeisenberg-Euler effective lagrangians
Heisenberg-Euler effective lagrangians Appunti per il corso di Fisica eorica 7/8 3.5.8 Fiorenzo Bastianelli We derive here effective lagrangians for the electromagnetic field induced by a loop of charged
More informationFinal reading report: BPHZ Theorem
Final reading report: BPHZ Theorem Wang Yang 1 1 Department of Physics, University of California at San Diego, La Jolla, CA 92093 This reports summarizes author s reading on BPHZ theorem, which states
More informationElectric Dipole Paradox: Question, Answer, and Interpretation
Electric Dipole Paradox: Question, Answer, and Interpretation Frank Wilczek January 16, 2014 Abstract Non-vanishing electric dipole moments for the electron, neutron, or other entities are classic signals
More informationAdvanced Quantum Mechanics
Advanced Quantum Mechanics Rajdeep Sensarma sensarma@theory.tifr.res.in Quantum Dynamics Lecture #2 Recap of Last Class Schrodinger and Heisenberg Picture Time Evolution operator/ Propagator : Retarded
More informationPath integrals and the classical approximation 1 D. E. Soper 2 University of Oregon 14 November 2011
Path integrals and the classical approximation D. E. Soper University of Oregon 4 November 0 I offer here some background for Sections.5 and.6 of J. J. Sakurai, Modern Quantum Mechanics. Introduction There
More informationSection 11: Review. µ1 x < 0
Physics 14a: Quantum Mechanics I Section 11: Review Spring 015, Harvard Below are some sample problems to help study for the final. The practice final handed out is a better estimate for the actual length
More information