or we could divide the total time T into N steps, with δ = T/N. Then and then we could insert the identity everywhere along the path.
|
|
- Derick Stephens
- 6 years ago
- Views:
Transcription
1 D. L. Rubin September, 011 These notes are based on Sakurai,.4, Gottfried and Yan,.7, Shankar 8 & 1, and Richard MacKenzie s Vietnam School of Physics lecture notes arxiv:quanth/ v1 1 Path Integral Suppose we have the propagator Kx f, t f, x 0, t 0 x f, t f x 0, t 0 x f e iht f t 0 / x 0 We can just as easily take two steps Kx f, t f, x 0, t 0 x f, t f x 0, t 0 x f e iht f t 1 / e iht 1 t 0 / x 0 or we could divide the total time T into N steps, with δ T/N. Then Kx f, t f, x 0, t 0 x f, t f x 0, t 0 x f e ihδ/ e ihδ/... x 0 and then we could insert the identity everywhere along the path. Kx f, t f, x 0, t 0 x f e ihδ/ dx x x e ihδ/ dx 1 x 1 x 1 e ihδ/ x 0 dx Kx f, t f, x, t dx 1 Kx 1, t 1, x 0, t 0 The amplitude is the sum of all N-legged paths. dx N x N x N... dx N Kx, t, x N, t N... A paths A paths paths dx 1 dx...dx, A path K xn,x K x,x N... 1
2 Let s consider the j th term. Kx j+1, x j x j+1 e ihδ/ x j Since δ is small we can expand the exponential and we have Kx j+1, x j x j+1 1 ihδ/ + Oδ x j Then we can insert the identity dp j p j p j and Kx j+1, x j dp j x j+1 1 ih δ p j p j x j dp j x j+1 p j p j x j i x j+1 H δ p j p j x j dp j x j+1 p j p j x j i δ p j m + V x j+1 x j+1 p j p j x j dpj e ix j+1 x j p j / i δ π p j m + V x j+1e ix j+1 x j p j / dpj π eix j+1 x j p j / exp i δ p j m + V x j+1 dpj π eix j+1 x j p j / exp i δ H Now we write x j+1 x j /δ ẋ j and we have dpj Kx j+1, x j π eiδẋ jp j / e i δ H There are N such factors in the amplitude so dp j A path π exp i δ ẋ j p j Hx j, p j That s the amplitude for one path. Now integrate over all paths Kx N, x 0 j1 dx j dp j π exp i δ ẋ j p j Hx j, p j
3 As N the sum becomes an integral over all time and we write Kx N, x 0 Dxt Dpt exp i δ T dtẋp Hx, p This is the phase space path integral. If the Hamiltonian has the standard form H p + V x then we can integrate each of the terms in the sum m Kx N, x 0 We use j1 dx j exp i δ e αx βx dx 0 V x j π α eβ /4α dp j π exp i δ ẋ j p j p j m where the above holds for pure imaginary α if it is regarded as a limit, namely if α a + ib, a > 0 it is the limit as a 0. This is what it looks like dp π exp i δ p m ẋp m e i δ πi δ mẋ / Putting it all together K j1 dx j exp i δ N/ πiδ j1 V x j dx j exp i δ ẋ j m πiδ exp i δ mẋ j V x j The sum is an approximation of the action of a path passing through the points x 0, x 1, x,... K Dxte is[xt] is the configuration space path integral. 1.1 Free particle path integral The configuration space path integral for a free particle is N/ [ iδ ẋ j K dx j exp πiδ j1 3 ]
4 K πiδ N/ K πiδ K j1 N/ πiδ N/ j1 j1 dx j exp dx j exp [ iδ m [ im δ ] xj+1 x j δ x j+1 x j ] [ im dx j exp xn x + x x N x 1 x 0 ] δ where x 0 and x N are initial and final points. The integrals are Gaussian and can be evaluated exactly but since they are coupled it ain t pretty. Let s see if we can figure it out. First let s define y i δ 1 x i. Then K K πiδ N/ j1 1 δ dyj exp [ i y N y + y y N y 1 y 0 ] m N/ / δ dy j exp [ i y N y + y y N y 1 y 0 ] πiδ m j1 Let s do the y 1 integration first. dy 1 exp i y y 1 + y 1 y 0 dy 1 exp iy + y 0 + y 1 y 1 y + y 0 dy 1 exp i y 1 y + y 0 / exp iy + y0 exp i y + y 0 dz exp 1 i z exp i y y 0 iπ exp i y y 0 Next we do the y integration. iπ y 3 y + 1 y y 0 iπ dy exp 1 i dy exp 1 3y i y y 3 + y 0 exp 1 i y y 0 4
5 It looks like it goes to iπ dz exp 1 z y 3 + y 0 /3 exp 1 3 i i y y 0 iπ iπ 3 exp 1 y3 + y 0 /3 exp 1 i i y y 0 iπ exp 1 3 3i y 3 y 0 iπ Putting it all together we have N exp 1Ni y N y 0 N/ / δ iπ K lim N πiδ m N 1 exp 1Ni y n y 0 The answer is N/ / 1 πiδ K lim e imx x /Nδ N πiδ N m m 1/ K lim e imx x /Nδ N πinδ Since Nδ T we have 1/ K e imx x /T πit which is of course the same as we calculated directly. Now on further investigation we see that 1 K exp i m xn x 0 m 1 T exp i T L cl dt πit T πit 0 Cute huh. The coordinate space path integral for the free particle, the sum of the action through every possible point in space, reduces to simply the classical action. The propagator reduces to two factors, one being the phase exp i S cl 5
6 1. Harmonic oscillator path integral The coordinate space path integral for the harmonic oscillator is Now let s write K N/ πiδ j dx j exp iδ ẋ j 1 mω x j xt x cl t + yt, dx dy, ẋ ẋ cl + ẏ Then ẋ 1 mω x j ẋ cl 1 mω x cl + L ẋ ẋ cl ẏ + L x x cl y + ẏ 1 mω y Let s look at the middle term and convert the sum to an integral L dt x y + d L dt ẋ y d L dt ẋ y L ẋ y t N t0 + dt The first term is zero because y 0 yt N 0. So N/ i K lim exp N πiδ S cl j1 dy j exp iδ L x d dt ẏ L y 0 ẋ 1 mω y j The PI over y is independent of the endpoints. It is zero at each end. It will depend only on the total time T and if K exp i S mω cly T, Y T πi 1/ xt A cosωt + B sin ωt, x N A cos ωt + B, x 0 A Then B x N x 0 cos ωt / 1 S cl mẋ 1 mx dt 6
7 1 1 dt ωa sin ωt + ωb cos ωt mω A cos ωt + B sin ωt dt mω A sin ωt + B cos ωt AB sin ωt cos ωt A cos ωt + B sin ωt + AB sin ωt cos ωt 1 dt mω B A cos ωt AB sin ωt mω 4 B A sin ωt + AB cos ωt T 0 mω 4 B A + ABcos ωt 1 mω 4 x N x 0 cos ωt x 0 sin ωt sin ωt cos ωt 1 +x 0 x n x 0 cos ωt mω 4 x N + x 0 cos ωt x N x 0 cos ωt +x 0 x n x 0 cos ωt cos ωt 1 mω 4 x N + x 0 cos ωt x N x 0 cos ωt cos ωt 1 +x 0 x n x 0 cos ωt cos ωt sin ωt cos ωt mω 4 x N cos ωt + x 1 0 cos ωt 4x N x 0 mω x N + x 0 cos ωt x N x Principle of Least Action Consider the configuration space path integral K Dxte is[xt]/. It says that a particle going from initial to final position and time takes all possible paths. The classical path is included but it gets no special mention. Every path has precisely unit magnitude. The contributions from the classical path and the totally wild path are the same. It turns out that the amplitudes interfere with each 7
8 other in a very special way. Consider two neighboring paths xt and x t and let x t xt + ηt, with ηt small. Then we can write the action S[x ] S[x + η] S[x] + dtηt δs[x] δxt + Oη The contribution of the two paths to the PI is A e is[x]/ 1 + exp i dtηt δs[x] δxt The phase difference between the two paths is 1 dtηt δs[x]. Smaller larger phase δxt difference. Even paths that are very close together will have large phase difference for small and on average they will interfere destructively. This is true except for one exceptional path, that which extremizes the action, namely the classical path x c t. For this path S[x c + η] S[x c ] + Oη. The classical path and a close neighbor will have actions which differ by much less than two randomly chosen but equally close paths. If the problem is classical action, paths near the classical path will on average interfere constructively small phase difference whereas for random paths the interference will be on average destructive. Classically, the particles motion is governed by the principle that the action is stationary. 8
Feynman Path Integrals in Quantum Mechanics
Feynman Path Integrals in Quantum Mechanics Christian Egli October, 2004 Abstract This text is written as a report to the seminar course in theoretical physics at KTH, Stockholm. The idea of this work
More information221A Lecture Notes Path Integral
A Lecture Notes Path Integral Feynman s Path Integral Formulation Feynman s formulation of quantum mechanics using the so-called path integral is arguably the most elegant. It can be stated in a single
More information8.323 Relativistic Quantum Field Theory I
MIT OpenCourseWare http://ocw.mit.edu 8.33 Relativistic Quantum Field Theory I Spring 008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. MASSACHUSETTS
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 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 informationChapter 2: Complex numbers
Chapter 2: Complex numbers Complex numbers are commonplace in physics and engineering. In particular, complex numbers enable us to simplify equations and/or more easily find solutions to equations. We
More informationPath Integrals and Quantum Mechanics
Path Integrals and Quantum Mechanics Martin Sandström Department Of Physics Umeȧ University Supervisor: Jens Zamanian October 1, 015 Abstract In this thesis we are investigating a different formalism of
More informationQuantum Field Theory. Victor Gurarie. Fall 2015 Lecture 9: Path Integrals in Quantum Mechanics
Quantum Field Theory Victor Gurarie Fall 015 Lecture 9: Path Integrals in Quantum Mechanics 1 Path integral in quantum mechanics 1.1 Green s functions of the Schrödinger equation Suppose a wave function
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 informationPath Intergal. 1 Introduction. 2 Derivation From Schrödinger Equation. Shoichi Midorikawa
Path Intergal Shoichi Midorikawa 1 Introduction The Feynman path integral1 is one of the formalism to solve the Schrödinger equation. However this approach is not peculiar to quantum mechanics, and M.
More informationPath integrals in quantum mechanics
Path integrals in quantum mechanics Phys V3500/G8099 handout #1 References: there s a nice discussion of this material in the first chapter of L.S. Schulman, Techniques and applications of path integration.
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 informationCreation and Destruction Operators and Coherent States
Creation and Destruction Operators and Coherent States WKB Method for Ground State Wave Function state harmonic oscillator wave function, We first rewrite the ground < x 0 >= ( π h )1/4 exp( x2 a 2 h )
More informationSolutions for homework 5
1 Section 4.3 Solutions for homework 5 17. The following equation has repeated, real, characteristic roots. Find the general solution. y 4y + 4y = 0. The characteristic equation is λ 4λ + 4 = 0 which has
More informationEach problem is worth 34 points. 1. Harmonic Oscillator Consider the Hamiltonian for a simple harmonic oscillator. 2ml 2 0. d 2
Physics 443 Prelim # with solutions March 7, 8 Each problem is worth 34 points.. Harmonic Oscillator Consider the Hamiltonian for a simple harmonic oscillator H p m + mω x (a Use dimensional analysis to
More informationMath 1302, Week 8: Oscillations
Math 302, Week 8: Oscillations T y eq Y y = y eq + Y mg Figure : Simple harmonic motion. At equilibrium the string is of total length y eq. During the motion we let Y be the extension beyond equilibrium,
More informationMonte Carlo simulations of harmonic and anharmonic oscillators in discrete Euclidean time
Monte Carlo simulations of harmonic and anharmonic oscillators in discrete Euclidean time DESY Summer Student Programme, 214 Ronnie Rodgers University of Oxford, United Kingdom Laura Raes University of
More informationOne-Sided Laplace Transform and Differential Equations
One-Sided Laplace Transform and Differential Equations As in the dcrete-time case, the one-sided transform allows us to take initial conditions into account. Preliminaries The one-sided Laplace transform
More informationAssignment 8. [η j, η k ] = J jk
Assignment 8 Goldstein 9.8 Prove directly that the transformation is canonical and find a generating function. Q 1 = q 1, P 1 = p 1 p Q = p, P = q 1 q We can establish that the transformation is canonical
More informationPeriodic functions: simple harmonic oscillator
Periodic functions: simple harmonic oscillator Recall the simple harmonic oscillator (e.g. mass-spring system) d 2 y dt 2 + ω2 0y = 0 Solution can be written in various ways: y(t) = Ae iω 0t y(t) = A cos
More informationTwo dimensional oscillator and central forces
Two dimensional oscillator and central forces September 4, 04 Hooke s law in two dimensions Consider a radial Hooke s law force in -dimensions, F = kr where the force is along the radial unit vector and
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 informationSeminar 6: COUPLED HARMONIC OSCILLATORS
Seminar 6: COUPLED HARMONIC OSCILLATORS 1. Lagrangian Equations of Motion Let consider a system consisting of two harmonic oscillators that are coupled together. As a model, we will use two particles attached
More informationVibrational Motion. Chapter 5. P. J. Grandinetti. Sep. 13, Chem P. J. Grandinetti (Chem. 4300) Vibrational Motion Sep.
Vibrational Motion Chapter 5 P. J. Grandinetti Chem. 4300 Sep. 13, 2017 P. J. Grandinetti (Chem. 4300) Vibrational Motion Sep. 13, 2017 1 / 20 Simple Harmonic Oscillator Simplest model for harmonic oscillator
More informationQuantum Mechanics C (130C) Winter 2014 Assignment 7
University of California at San Diego Department of Physics Prof. John McGreevy Quantum Mechanics C (130C) Winter 014 Assignment 7 Posted March 3, 014 Due 11am Thursday March 13, 014 This is the last problem
More informationFeynman s path integral approach to quantum physics and its relativistic generalization
Feynman s path integral approach to quantum physics and its relativistic generalization Jürgen Struckmeier j.struckmeier@gsi.de, www.gsi.de/ struck Vortrag im Rahmen des Winterseminars Aktuelle Probleme
More informationStatistical Mechanics Solution Set #1 Instructor: Rigoberto Hernandez MoSE 2100L, , (Dated: September 4, 2014)
CHEM 6481 TT 9:3-1:55 AM Fall 214 Statistical Mechanics Solution Set #1 Instructor: Rigoberto Hernandez MoSE 21L, 894-594, hernandez@gatech.edu (Dated: September 4, 214 1. Answered according to individual
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 informationIntroduction to Instantons. T. Daniel Brennan. Quantum Mechanics. Quantum Field Theory. Effects of Instanton- Matter Interactions.
February 18, 2015 1 2 3 Instantons in Path Integral Formulation of mechanics is based around the propagator: x f e iht / x i In path integral formulation of quantum mechanics we relate the propagator to
More informationStarting from Heat Equation
Department of Applied Mathematics National Chiao Tung University Hsin-Chu 30010, TAIWAN 20th August 2009 Analytical Theory of Heat The differential equations of the propagation of heat express the most
More informationProblem 1: Lagrangians and Conserved Quantities. Consider the following action for a particle of mass m moving in one dimension
105A Practice Final Solutions March 13, 01 William Kelly Problem 1: Lagrangians and Conserved Quantities Consider the following action for a particle of mass m moving in one dimension S = dtl = mc dt 1
More informationUniversity of California, Berkeley Department of Mechanical Engineering ME 104, Fall Midterm Exam 1 Solutions
University of California, Berkeley Department of Mechanical Engineering ME 104, Fall 2013 Midterm Exam 1 Solutions 1. (20 points) (a) For a particle undergoing a rectilinear motion, the position, velocity,
More informationLecture 7. Please note. Additional tutorial. Please note that there is no lecture on Tuesday, 15 November 2011.
Lecture 7 3 Ordinary differential equations (ODEs) (continued) 6 Linear equations of second order 7 Systems of differential equations Please note Please note that there is no lecture on Tuesday, 15 November
More informationAlternative Method for Determining the Feynman Propagator of a Non-Relativistic Quantum Mechanical Problem
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 3 (007, 110, 1 pages Alternative Method for Determining the Feynman Propagator of a Non-Relativistic Quantum Mechanical Problem Marcos
More informationA comment on Path integral for the quantum harmonic oscillator
A comment on Path integral for the quantum harmonic oscillator Kiyoto Hira (Dated: May 8, 203) An elementary derivation of quantum harmonic oscillator propagator using the path integral have been reconsidered
More informationOscillations. Simple Harmonic Motion of a Mass on a Spring The equation of motion for a mass m is attached to a spring of constant k is
Dr. Alain Brizard College Physics I (PY 10) Oscillations Textbook Reference: Chapter 14 sections 1-8. Simple Harmonic Motion of a Mass on a Spring The equation of motion for a mass m is attached to a spring
More informationVariational Principles
Part IB Variational Principles Year 218 217 216 215 214 213 212 211 21 218 Paper 1, Section I 4B 46 Variational Principles Find, using a Lagrange multiplier, the four stationary points in R 3 of the function
More informationPoincaré Map, Floquet Theory, and Stability of Periodic Orbits
Poincaré Map, Floquet Theory, and Stability of Periodic Orbits CDS140A Lecturer: W.S. Koon Fall, 2006 1 Poincaré Maps Definition (Poincaré Map): Consider ẋ = f(x) with periodic solution x(t). Construct
More information1 Imaginary Time Path Integral
1 Imaginary Time Path Integral For the so-called imaginary time path integral, the object of interest is exp( τh h There are two reasons for using imaginary time path integrals. One is that the application
More informationChemistry 532 Problem Set 7 Spring 2012 Solutions
Chemistry 53 Problem Set 7 Spring 01 Solutions 1. The study of the time-independent Schrödinger equation for a one-dimensional particle subject to the potential function leads to the differential equation
More informationWave Phenomena Physics 15c
Wave Phenomena Physics 15c Lecture Harmonic Oscillators (H&L Sections 1.4 1.6, Chapter 3) Administravia! Problem Set #1! Due on Thursday next week! Lab schedule has been set! See Course Web " Laboratory
More informationQuantum Physics III (8.06) Spring 2016 Assignment 3
Quantum Physics III (8.6) Spring 6 Assignment 3 Readings Griffiths Chapter 9 on time-dependent perturbation theory Shankar Chapter 8 Cohen-Tannoudji, Chapter XIII. Problem Set 3. Semi-classical approximation
More informationMathematical Economics: Lecture 2
Mathematical Economics: Lecture 2 Yu Ren WISE, Xiamen University September 25, 2012 Outline 1 Number Line The number line, origin (Figure 2.1 Page 11) Number Line Interval (a, b) = {x R 1 : a < x < b}
More informationVibrations and Waves Physics Year 1. Handout 1: Course Details
Vibrations and Waves Jan-Feb 2011 Handout 1: Course Details Office Hours Vibrations and Waves Physics Year 1 Handout 1: Course Details Dr Carl Paterson (Blackett 621, carl.paterson@imperial.ac.uk Office
More informationPHY 396 K. Problem set #5. Due October 9, 2008.
PHY 396 K. Problem set #5. Due October 9, 2008.. First, an exercise in bosonic commutation relations [â α, â β = 0, [â α, â β = 0, [â α, â β = δ αβ. ( (a Calculate the commutators [â αâ β, â γ, [â αâ β,
More informationGaussians Distributions, Simple Harmonic Motion & Uncertainty Analysis Review. Lecture # 5 Physics 2BL Summer 2015
Gaussians Distributions, Simple Harmonic Motion & Uncertainty Analysis Review Lecture # 5 Physics 2BL Summer 2015 Outline Significant figures Gaussian distribution and probabilities Experiment 2 review
More informationPH.D. PRELIMINARY EXAMINATION MATHEMATICS
UNIVERSITY OF CALIFORNIA, BERKELEY Dept. of Civil and Environmental Engineering FALL SEMESTER 2014 Structural Engineering, Mechanics and Materials NAME PH.D. PRELIMINARY EXAMINATION MATHEMATICS Problem
More informationWave Phenomena Physics 15c. Lecture 2 Damped Oscillators Driven Oscillators
Wave Phenomena Physics 15c Lecture Damped Oscillators Driven Oscillators What We Did Last Time Analyzed a simple harmonic oscillator The equation of motion: The general solution: Studied the solution m
More informationIntegration in the Complex Plane (Zill & Wright Chapter 18)
Integration in the omplex Plane Zill & Wright hapter 18) 116-4-: omplex Variables Fall 11 ontents 1 ontour Integrals 1.1 Definition and Properties............................. 1. Evaluation.....................................
More informationEvaluation of integrals
Evaluation of certain contour integrals: Type I Type I: Integrals of the form 2π F (cos θ, sin θ) dθ If we take z = e iθ, then cos θ = 1 (z + 1 ), sin θ = 1 (z 1 dz ) and dθ = 2 z 2i z iz. Substituting
More informationThe Path Integral Formulation of Quantum Mechanics
Based on Quantum Mechanics and Path Integrals by Richard P. Feynman and Albert R. Hibbs, and Feynman s Thesis The Path Integral Formulation Vebjørn Gilberg University of Oslo July 14, 2017 Contents 1 Introduction
More information5 Applying the Fokker-Planck equation
5 Applying the Fokker-Planck equation We begin with one-dimensional examples, keeping g = constant. Recall: the FPE for the Langevin equation with η(t 1 )η(t ) = κδ(t 1 t ) is = f(x) + g(x)η(t) t = x [f(x)p
More informationPath Integrals in Quantum Mechanics
Path Integrals in Quantum Mechanics Michael Fowler 10/4/07 Huygen s Picture of Wave Propagation If a point source of light is switched on, the wavefront is an expanding sphere centered at the source. Huygens
More informationWaves and the Schroedinger Equation
Waves and the Schroedinger Equation 5 april 010 1 The Wave Equation We have seen from previous discussions that the wave-particle duality of matter requires we describe entities through some wave-form
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 informationSymmetries 2 - Rotations in Space
Symmetries 2 - Rotations in Space This symmetry is about the isotropy of space, i.e. space is the same in all orientations. Thus, if we continuously rotated an entire system in space, we expect the system
More informationGeneral Physics I. Lecture 14: Sinusoidal Waves. Prof. WAN, Xin ( 万歆 )
General Physics I Lecture 14: Sinusoidal Waves Prof. WAN, Xin ( 万歆 ) xinwan@zju.edu.cn http://zimp.zju.edu.cn/~xinwan/ Motivation When analyzing a linear medium that is, one in which the restoring force
More informationPARABOLIC POTENTIAL WELL
APPENDIX E PARABOLIC POTENTIAL WELL An example of an extremely important class of one-dimensional bound state in quantum mechanics is the simple harmonic oscillator whose potential can be written as V(x)=
More informationAnalytical Mechanics ( AM )
Analytical Mechanics ( AM ) Olaf Scholten KVI, kamer v8; tel nr 6-55; email: scholten@kvinl Web page: http://wwwkvinl/ scholten Book: Classical Dynamics of Particles and Systems, Stephen T Thornton & Jerry
More informationLecture 7. More dimensions
Lecture 7 More dimensions 67 68 LECTURE 7. MORE DIMENSIONS 7.1 Introduction In this lecture we generalize the concepts introduced so far to systems that evolve in more than one spatial dimension. While
More informationInfinite Series. 1 Introduction. 2 General discussion on convergence
Infinite Series 1 Introduction I will only cover a few topics in this lecture, choosing to discuss those which I have used over the years. The text covers substantially more material and is available for
More informationMath 4263 Homework Set 1
Homework Set 1 1. Solve the following PDE/BVP 2. Solve the following PDE/BVP 2u t + 3u x = 0 u (x, 0) = sin (x) u x + e x u y = 0 u (0, y) = y 2 3. (a) Find the curves γ : t (x (t), y (t)) such that that
More informationDegree Master of Science in Mathematical Modelling and Scientific Computing Mathematical Methods I Thursday, 12th January 2012, 9:30 a.m.- 11:30 a.m.
Degree Master of Science in Mathematical Modelling and Scientific Computing Mathematical Methods I Thursday, 12th January 2012, 9:30 a.m.- 11:30 a.m. Candidates should submit answers to a maximum of four
More informationNormal Modes, Wave Motion and the Wave Equation Hilary Term 2011 Lecturer: F Hautmann
Normal Modes, Wave Motion and the Wave Equation Hilary Term 2011 Lecturer: F Hautmann Part A: Normal modes ( 4 lectures) Part B: Waves ( 8 lectures) printed lecture notes slides will be posted on lecture
More informationThursday, August 4, 2011
Chapter 16 Thursday, August 4, 2011 16.1 Springs in Motion: Hooke s Law and the Second-Order ODE We have seen alrealdy that differential equations are powerful tools for understanding mechanics and electro-magnetism.
More informationReview of Classical Mechanics
Review of Classical Mechanics VBS/MRC Review of Classical Mechanics 0 Some Questions Why does a tennis racket wobble when flipped along a certain axis? What do hear when you pluck a Veena string? How do
More informationWave Phenomena Physics 15c
Wave Phenomena Physics 15c Lecture Harmonic Oscillators (H&L Sections 1.4 1.6, Chapter 3) Administravia! Problem Set #1! Due on Thursday next week! Lab schedule has been set! See Course Web " Laboratory
More informationSTABILITY. Phase portraits and local stability
MAS271 Methods for differential equations Dr. R. Jain STABILITY Phase portraits and local stability We are interested in system of ordinary differential equations of the form ẋ = f(x, y), ẏ = g(x, y),
More informationThe Path Integral: Basics and Tricks (largely from Zee)
The Path Integral: Basics and Tricks (largely from Zee) Yichen Shi Michaelmas 03 Path-Integral Derivation x f, t f x i, t i x f e H(t f t i) x i. If we chop the path into N intervals of length ɛ, then
More informationt L(q, q)dt (11.1) ,..., S ) + S q 1
Chapter 11 WKB and the path integral In this chapter we discuss two reformulations of the Schrödinger equations that can be used to study the transition from quantum mechanics to classical mechanics. They
More informationApplied Nuclear Physics (Fall 2006) Lecture 2 (9/11/06) Schrödinger Wave Equation
22.101 Applied Nuclear Physics (Fall 2006) Lecture 2 (9/11/06) Schrödinger Wave Equation References -- R. M. Eisberg, Fundamentals of Modern Physics (Wiley & Sons, New York, 1961). R. L. Liboff, Introductory
More informationMTH3101 Spring 2017 HW Assignment 4: Sec. 26: #6,7; Sec. 33: #5,7; Sec. 38: #8; Sec. 40: #2 The due date for this assignment is 2/23/17.
MTH0 Spring 07 HW Assignment : Sec. 6: #6,7; Sec. : #5,7; Sec. 8: #8; Sec. 0: # The due date for this assignment is //7. Sec. 6: #6. Use results in Sec. to verify that the function g z = ln r + iθ r >
More informationTutorial 6 (week 6) Solutions
THE UNIVERSITY OF SYDNEY PURE MATHEMATICS Linear Mathematics 9 Tutorial 6 week 6 s Suppose that A and P are defined as follows: A and P Define a sequence of numbers { u n n } by u, u, u and for all n,
More informationSolutions of Spring 2008 Final Exam
Solutions of Spring 008 Final Exam 1. (a) The isocline for slope 0 is the pair of straight lines y = ±x. The direction field along these lines is flat. The isocline for slope is the hyperbola on the left
More informationFourier Sin and Cos Series and Least Squares Convergence
Fourier and east Squares Convergence James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University May 7, 28 Outline et s look at the original Fourier sin
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 informationCorrections to Quantum Theory for Mathematicians
Corrections to Quantum Theory for Mathematicians B C H January 2018 Thanks to ussell Davidson, Bruce Driver, John Eggers, Todd Kemp, Benjamin ewis, Jeff Margrave, Alexander Mukhin, Josh asmussen, Peter
More information4. Complex Oscillations
4. Complex Oscillations The most common use of complex numbers in physics is for analyzing oscillations and waves. We will illustrate this with a simple but crucially important model, the damped harmonic
More informationQUALIFYING EXAMINATION, Part 1. Solutions. Problem 1: Mathematical Methods. x 2. 1 (1 + 2x 2 /3!) ] x 2 1 2x 2 /3
QUALIFYING EXAMINATION, Part 1 Solutions Problem 1: Mathematical Methods (a) Keeping only the lowest power of x needed, we find 1 x 1 sin x = 1 x 1 (x x 3 /6...) = 1 ) 1 (1 x 1 x /3 = 1 [ 1 (1 + x /3!)
More informationX(t)e 2πi nt t dt + 1 T
HOMEWORK 31 I) Use the Fourier-Euler formulae to show that, if X(t) is T -periodic function which admits a Fourier series decomposition X(t) = n= c n exp (πi n ) T t, then (1) if X(t) is even c n are all
More informationB Ordinary Differential Equations Review
B Ordinary Differential Equations Review The profound study of nature is the most fertile source of mathematical discoveries. - Joseph Fourier (1768-1830) B.1 First Order Differential Equations Before
More informationMAS 315 Waves 1 of 7 Answers to Example Sheet 3. NB Questions 1 and 2 are relevant to resonance - see S2 Q7
MAS 35 Waves o 7 Answers to Example Sheet 3 NB Questions and are relevant to resonance - see S Q7. The CF is A cosnt + B sin nt (i ω n: Try PI y = C cosωt. OK provided C( ω + n = C = GS is y = A cosnt
More informationQuantum Mechanics. p " The Uncertainty Principle places fundamental limits on our measurements :
Student Selected Module 2005/2006 (SSM-0032) 17 th November 2005 Quantum Mechanics Outline : Review of Previous Lecture. Single Particle Wavefunctions. Time-Independent Schrödinger equation. Particle in
More informationSolution of Additional Exercises for Chapter 4
1 1. (1) Try V (x) = 1 (x 1 + x ). Solution of Additional Exercises for Chapter 4 V (x) = x 1 ( x 1 + x ) x = x 1 x + x 1 x In the neighborhood of the origin, the term (x 1 + x ) dominates. Hence, the
More information/ \ ( )-----/\/\/\/ \ / In Lecture 3 we offered this as an example of a first order LTI system.
18.03 Class 17, March 12, 2010 Linearity and time invariance [1] RLC [2] Superposition III [3] Time invariance [4] Review of solution methods [1] We've spent a lot of time with mx" + bx' + cx = q(t). There
More informationLecture 6: Differential Equations Describing Vibrations
Lecture 6: Differential Equations Describing Vibrations In Chapter 3 of the Benson textbook, we will look at how various types of musical instruments produce sound, focusing on issues like how the construction
More informationChemistry 432 Problem Set 1 Spring 2018 Solutions
Chemistry 43 Problem Set 1 Spring 018 Solutions 1. A ball of mass m is tossed into the air at time t = 0 with an initial velocity v 0. The ball experiences a constant acceleration g from the gravitational
More informationMath 211. Substitute Lecture. November 20, 2000
1 Math 211 Substitute Lecture November 20, 2000 2 Solutions to y + py + qy =0. Look for exponential solutions y(t) =e λt. Characteristic equation: λ 2 + pλ + q =0. Characteristic polynomial: λ 2 + pλ +
More information2 Resolvents and Green s Functions
Course Notes Solving the Schrödinger Equation: Resolvents 057 F. Porter Revision 09 F. Porter Introduction Once a system is well-specified, the problem posed in non-relativistic quantum mechanics is to
More informationForced Response - Particular Solution x p (t)
Governing Equation 1.003J/1.053J Dynamics and Control I, Spring 007 Proessor Peacoc 5/7/007 Lecture 1 Vibrations: Second Order Systems - Forced Response Governing Equation Figure 1: Cart attached to spring
More information221A Lecture Notes Steepest Descent Method
Gamma Function A Lecture Notes Steepest Descent Method The best way to introduce the steepest descent method is to see an example. The Stirling s formula for the behavior of the factorial n! for large
More informationPHY 396 L. Solutions for homework set #21.
PHY 396 L. Solutions for homework set #21. Problem 1a): The difference between a circle and a straight line is that on a circle the path of a particle going from point x 0 to point x does not need to be
More informationThe Harmonic Oscillator
The Harmonic Oscillator Math 4: Ordinary Differential Equations Chris Meyer May 3, 008 Introduction The harmonic oscillator is a common model used in physics because of the wide range of problems it can
More informationDerivation of the harmonic oscillator propagator using the Feynman path integral and
Home Search Collections Journals About Contact us My IOPscience Derivation of the harmonic oscillator propagator using the Feynman path integral and recursive relations This content has been downloaded
More information1 Simple Harmonic Oscillator
Physics 1a Waves Lecture 3 Caltech, 10/09/18 1 Simple Harmonic Oscillator 1.4 General properties of Simple Harmonic Oscillator 1.4.4 Superposition of two independent SHO Suppose we have two SHOs described
More informationEven-Numbered Homework Solutions
-6 Even-Numbered Homework Solutions Suppose that the matric B has λ = + 5i as an eigenvalue with eigenvector Y 0 = solution to dy = BY Using Euler s formula, we can write the complex-valued solution Y
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 informationPhysics 351 Monday, January 22, 2018
Physics 351 Monday, January 22, 2018 Phys 351 Work on this while you wait for your classmates to arrive: Show that the moment of inertia of a uniform solid sphere rotating about a diameter is I = 2 5 MR2.
More informationVibrations and Waves MP205, Assignment 4 Solutions
Vibrations and Waves MP205, Assignment Solutions 1. Verify that x = Ae αt cos ωt is a possible solution of the equation and find α and ω in terms of γ and ω 0. [20] dt 2 + γ dx dt + ω2 0x = 0, Given x
More informationMAT389 Fall 2016, Problem Set 11
MAT389 Fall 216, Problem Set 11 Improper integrals 11.1 In each of the following cases, establish the convergence of the given integral and calculate its value. i) x 2 x 2 + 1) 2 ii) x x 2 + 1)x 2 + 2x
More information