Averaging II: Adiabatic Invariance for Integrable Systems (argued via the Averaging Principle)


 Brett Underwood
 1 years ago
 Views:
Transcription
1 Averaging II: Adiabatic Invariance for Integrable Systems (argued via the Averaging Principle In classical mechanics an adiabatic invariant is defined as follows[1]. Consider the Hamiltonian system with slowly varying external parameter λ: H = H(p, q; λ, λ = ɛt (1 ṗ = H q q = H p Definition: The quantity A(p, q; λ is an adiabatic invariant of the system above if for every κ > there is an ɛ > such that if < ɛ < ɛ and < t < 1/ɛ, then A(p(t, q(t; ɛt A(p(, q(; < κ. The principle of adiabatic action invariance is that the values of the actions I for an integrable system are indeed adiabatic invariants. (Note that most systems with N > 1 are not integrable, but one has to start somewhere. huh? Here s what we are saying. Consider a hamiltonian system with N position degrees of freedom (N total degrees of freedom that is a function of some fixed external parameter λ. We require that the system be integrable (have at least N independent constants of motion for any fixed value of λ in some region, say [, 1]. Since the system is integrable, its motion (trajectory is specified by I. That is, we can construct the actions I i = p dq from the i (hollow torus that the trajectory fills out. The torus, (or equivalently, the quantity I, can be considered to be a function of the starting point of the motion (p, q in phase space. It is also a function of λ. Now consider the system where λ = ɛt. I(p(, q(; is the action associated with the torus for a constant λ specified by the initial, instantaneous values of p, q, and λ. Likewise I(p(t, q(t; ɛt is the action associated with the torus (for constant λ specified by the values of p, q, and λ at the instant t. A quantity is an adiabatic invariant if we can make its maximum devation from its initial value arbitrarily small during a process in which we change an external parameter a lot (say from to 1 simply by varying the external parameter slowly enough. (Mentally translate these words into the 1
2 definition above. Our principle is that the I i are adiabatic invariants (see Arnold s quote regarding such principles. It is straightforward to get this principle of adiabatic action invariance from the averaging principle. Following Arnold: For a fixed λ we can canonically transform the system (1 into the actionangle variable system via a generating function that depends on λ. words, there is a generating function 1 F = F (I, q; λ such that via p = F q, In other φ = F I, H new = H old ( we get the new Hamiltonian system where I and φ are the actionangle variables (meaning that I i = p dq and i ( I H = φ I =, φ = ( H ω(i, λ (3 I φ Now let λ = ɛt. Now the canonical transformation generated by (the same function F is timedependent so that the last part of ( is changed to H new H old = ( F/ t I,q = ɛ F/ λ. Thus, we have the system of equations I = ɛf(i, φ; λ (4 φ = ω(i, λ + ɛg(i, φ; λ (5 λ = ɛ (6 where g = F I λ, f = F φ λ We apply the averaging principle. The averaged system has the form (7 J i = ɛ f i i = 1,,..., N J N+1 = ɛ (8 But f i = (π N π π ( F dφ 1... dφ N φ i λ = (do the φ i integral (9 Therefore, in the averaged system the J i components corresponding to the actions I i do not change. 1 Here we use a type generating function. If we considered the actions I to be a position (Q instead of a momentum, we could use a type 1 generating function. In this case ( would be altered.
3 I. ADIABATIC INVARIANCE IN A MORE GENERAL CONTEXT The word adiabatic appears in at least two contexts in physics. The most common context perhaps is in thermodynamics, and the second is related to sufficiently slow changes in dynamical systems, which is how we have used the word here. The two uses of adiabatic have a somewhat imprecise link regarding processes in which entropy does not change. In thermodynamics, an adiabatic process is a process in which there is no flow of heat into or out of the system considered. One can, however, do work on a system during an adiabatic process, by say, changing the volume. So the energy can certainly change. If the adiabatic process is also reversible (carried out near equilibrium then the process is isoentropic, the entropy does not change. An example of an adiabatic, nonreversible process is the free expansion of a gas, during which the entropy of the gas (and the universe increases. If we restrict ourselves to equilibrium thermodynamics however, adiabaticity implies that entropy is constant. In classical and quantum mechanics, adiabaticity has an analogous meaning, referring to a process which is in some way ordered or reversible. In quantum mechanics the adiabatic theorem states that a system that is in an energy eigenstate at time will evolve so that it always in an energy eigenstate (though the energy (eigenvalue and the eigenstate are functions of λ, given that the external parameter λ is varied with ɛ. If the system is varied too quickly then it ends up in a noticable superposition of the new eigenstates. The energy of the new state is uncertain. Since the result is a pure state however, the quantum entropy is still zero. Classically, if the process is too fast then I is not only nonconstant, but its evolution depends on the initial phases φ(. That is, the surface of the initial torus evolves not to a single new torus (at time t, but to an ensemble of them. If the initial phase was not known and only the I could be observed, this would be like an increase in entropy. Besides invariance of the action for integrable systems, there is also a principle of ergodic adiabatic invariance [, 3] for completely chaotic systems (systems which, for fixed λ, the only constant of the motion, for any trajectory, is the hamiltonian H. The motion initially defines a N 1 dimensional energy shell in the N dimensional phase space. If λ is varied infinitely slowly, this shell evolves to a new energy shell such that the phase space volume Ω inside the shell is constant (Ω is the adiabatic invariant. If not, it evolves to an ensemble of 3
4 shells (a N dimensional region and Ω isn t well defined. Again, same entropy (or at least uncertainty idea. II. HARMONIC OSCILLATOR EXAMPLE I = (π 1 p dq (1 H = p /m + (1/mω q = E (11 p = ± me m ω q (1 I = 1 qmax π me m ω q dq q min (it is customary to integrate in sense that I is positive = 1 qmax me 1 mω π E q dq = 1 π I = E ω q min umax=1 me 1 u 1 π/ π/ u min = 1 1 u du cos y dy, sin y = u du mω /(E, u = mω E ( cos y dy, (nd term (13 Therefore adiabatic invariance of actions predicts that as a parameter (m or ω of the HO is changed by a significant amount but very slowly, the ratio of the oscillator s energy to its frequency remains approximately constant. The direct check to see whether or not this is true for the HO involves essentially some of the same averaging and timescale related arguments as used in our proof of the averaging theorem. Here is a quick version with only one little leap of faith. Say we are changing ω from ω to ω uniformly over a long time τ = 1/ɛ ω(t = ω (1 + ɛt( sec 1 (14 4
5 Now Ė = H t = (1/mq (ω ω = ɛmω ω(t 1Hz (15 Imagine integrating Ė over the long time τ = 1/ɛ. Since Ė is always positive we are going to get a change in energy E that is O(1. So E is definitely not an adiabatic invariant. Now check the time derivative of the action I = E/ω (I ll suppress the extra Hz unit here. d E = dt( Ė ω ω E ω ω = ɛmq ω + ɛ E ω ω (16 = ɛ ω ω (mω q E = ɛ ω (U E ω U T = ɛω (17 ω Now lets integrate d(e/ω/dt over the long time τ to get the change in I I = ɛω τ U T ω dt (18 Is I of O(1 or O(ɛ? If it is O(1, like E then I is not an adiabatic invariant. If it is O(ɛ, then we can make I arbitrarily small by making the rate of change, ɛ, arbitrily small (although the total change in the external parameter is large. Look at the integrand in (18. The 1/ω changes slowly with time. But (U T rapidly oscillates between +E and E, two oscillations per period of the HO motion. So we do not expect the quantity t (U T /ω dt to be assymptotically proportional to t and thus we expect τ (U T /ω dt to not be O(1/ɛ but to be of some smaller order such as O(1 or O(ɛ. Then by (18 we see that I is indeed O(ɛ or smaller, and it is not O(1. Thus I = E/ω is indeed an adiabatic invariant as the principle predicted. [1] V. I. Arnold, Mathematical Methods of Classical Mechanics, nd Ed., SpringerVerlag, New York, [] E. Ott, Phys. Rev. Lett. 4, 168 (1979. [3] C. Jarzynski, Phys. Rev. Lett. 71, 839 (
Assignment 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 informationAnatoli Polkovnikov Boston University
Anatoli Polkovnikov Boston University L. D Alessio BU M. Bukov BU C. De Grandi Yale V. Gritsev Amsterdam M. Kolodrubetz Berkeley C.W. Liu BU P. Mehta BU M. Tomka BU D. Sels BU A. Sandvik BU T. Souza BU
More informationarxiv: v2 [hepth] 7 Apr 2015
Statistical Mechanics Derived From Quantum Mechanics arxiv:1501.05402v2 [hepth] 7 Apr 2015 YuLei Feng 1 and YiXin Chen 1 1 Zhejiang Institute of Modern Physics, Zhejiang University, Hangzhou 310027,
More informationIV. Classical Statistical Mechanics
IV. Classical Statistical Mechanics IV.A General Definitions Statistical Mechanics is a probabilistic approach to equilibrium macroscopic properties of large numbers of degrees of freedom. As discussed
More informationQuantum Mechanical Foundations of Causal Entropic Forces
Quantum Mechanical Foundations of Causal Entropic Forces Swapnil Shah North Carolina State University, USA snshah4@ncsu.edu Abstract. The theory of Causal Entropic Forces was introduced to explain the
More informationPhysics 443, Solutions to PS 2
. Griffiths.. Physics 443, Solutions to PS The raising and lowering operators are a ± mω ( iˆp + mωˆx) where ˆp and ˆx are momentum and position operators. Then ˆx mω (a + + a ) mω ˆp i (a + a ) The expectation
More information= 0. = q i., q i = E
Summary of the Above Newton s second law: d 2 r dt 2 = Φ( r) Complicated vector arithmetic & coordinate system dependence Lagrangian Formalism: L q i d dt ( L q i ) = 0 n secondorder differential equations
More informationChapter 2 Thermodynamics
Chapter 2 Thermodynamics 2.1 Introduction The First Law of Thermodynamics is a statement of the existence of a property called Energy which is a state function that is independent of the path and, in the
More informationCoherent states, beam splitters and photons
Coherent states, beam splitters and photons S.J. van Enk 1. Each mode of the electromagnetic (radiation) field with frequency ω is described mathematically by a 1D harmonic oscillator with frequency ω.
More informationGeneral Formula for the Efficiency of QuantumMechanical Analog of the Carnot Engine
Entropy 2013, 15, 14081415; doi:103390/e15041408 Article OPEN ACCESS entropy ISSN 10994300 wwwmdpicom/journal/entropy General Formula for the Efficiency of QuantumMechanical Analog of the Carnot Engine
More informationElectric 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 informationMultiple scale methods
Multiple scale methods G. Pedersen MEK 3100/4100, Spring 2006 March 13, 2006 1 Background Many physical problems involve more than one temporal or spatial scale. One important example is the boundary layer
More informationStatistical Mechanics
Statistical Mechanics Contents Chapter 1. Ergodicity and the Microcanonical Ensemble 1 1. From Hamiltonian Mechanics to Statistical Mechanics 1 2. Two Theorems From Dynamical Systems Theory 6 3. The Microcanonical
More informationGeneral formula for the efficiency of quantummechanical analog of the Carnot engine
General formula for the efficiency of quantummechanical analog of the Carnot engine Sumiyoshi Abe Department of Physical Engineering, Mie University, Mie 5148507, Japan Abstract: An analog of the Carnot
More informationSingle Particle Motion
Single Particle Motion C ontents Uniform E and B E =  guiding centers Definition of guiding center E gravitation Non Uniform B 'grad B' drift, B B Curvature drift Grad B drift, B B invariance of µ. Magnetic
More information4. Supplementary Notes on Time and Space Evolution of a Neutrino Beam
Lecture Notes for Quantum Physics II & III 8.05 & 8.059 Academic Year 1996/1997 4. Supplementary Notes on Time and Space Evolution of a Neutrino Beam c D. Stelitano 1996 As an example of a twostate system
More informationJoint Entrance Examination for Postgraduate Courses in Physics EUF
Joint Entrance Examination for Postgraduate Courses in Physics EUF Second Semester 013 Part 1 3 April 013 Instructions: DO NOT WRITE YOUR NAME ON THE TEST. It should be identified only by your candidate
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 informationCurves in the configuration space Q or in the velocity phase space Ω satisfying the EulerLagrange (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 informationHomework 4. Goldstein 9.7. Part (a) Theoretical Dynamics October 01, 2010 (1) P i = F 1. Q i. p i = F 1 (3) q i (5) P i (6)
Theoretical Dynamics October 01, 2010 Instructor: Dr. Thomas Cohen Homework 4 Submitted by: Vivek Saxena Goldstein 9.7 Part (a) F 1 (q, Q, t) F 2 (q, P, t) P i F 1 Q i (1) F 2 (q, P, t) F 1 (q, Q, t) +
More information8.333: Statistical Mechanics I Problem Set # 5 Due: 11/22/13 Interacting particles & Quantum ensembles
8.333: Statistical Mechanics I Problem Set # 5 Due: 11/22/13 Interacting particles & Quantum ensembles 1. Surfactant condensation: N surfactant molecules are added to the surface of water over an area
More information2.3 Damping, phases and all that
2.3. DAMPING, PHASES AND ALL THAT 107 2.3 Damping, phases and all that If we imagine taking our idealized mass on a spring and dunking it in water or, more dramatically, in molasses), then there will be
More informationThe Wave Function. Chapter The Harmonic Wave Function
Chapter 3 The Wave Function On the basis of the assumption that the de Broglie relations give the frequency and wavelength of some kind of wave to be associated with a particle, plus the assumption that
More informationEnergy Barriers and Rates  Transition State Theory for Physicists
Energy Barriers and Rates  Transition State Theory for Physicists Daniel C. Elton October 12, 2013 Useful relations 1 cal = 4.184 J 1 kcal mole 1 = 0.0434 ev per particle 1 kj mole 1 = 0.0104 ev per particle
More informationDissipation and the Relaxation to Equilibrium
1 Dissipation and the Relaxation to Equilibrium Denis J. Evans, 1 Debra J. Searles 2 and Stephen R. Williams 1 1 Research School of Chemistry, Australian National University, Canberra, ACT 0200, Australia
More informationChapter 9: Statistical Mechanics
Chapter 9: Statistical Mechanics Chapter 9: Statistical Mechanics...111 9.1 Introduction...111 9.2 Statistical Mechanics...113 9.2.1 The Hamiltonian...113 9.2.2 Phase Space...114 9.2.3 Trajectories and
More informationLectures on Periodic Orbits
Lectures on Periodic Orbits 11 February 2009 Most of the contents of these notes can found in any typical text on dynamical systems, most notably Strogatz [1994], Perko [2001] and Verhulst [1996]. Complete
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 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 waveparticle duality of matter requires we describe entities through some waveform
More informationElectron in a Box. A wave packet in a square well (an electron in a box) changing with time.
Electron in a Box A wave packet in a square well (an electron in a box) changing with time. Last Time: Light Wave model: Interference pattern is in terms of wave intensity Photon model: Interference in
More informationChE 503 A. Z. Panagiotopoulos 1
ChE 503 A. Z. Panagiotopoulos 1 STATISTICAL MECHANICAL ENSEMLES 1 MICROSCOPIC AND MACROSCOPIC ARIALES The central question in Statistical Mechanics can be phrased as follows: If particles (atoms, molecules,
More informationPhysics 121, April 3, Equilibrium and Simple Harmonic Motion. Physics 121. April 3, Physics 121. April 3, Course Information
Physics 121, April 3, 2008. Equilibrium and Simple Harmonic Motion. Physics 121. April 3, 2008. Course Information Topics to be discussed today: Requirements for Equilibrium (a brief review) Stress and
More informationThe LightFront Vacuum
The LightFront Vacuum Marc Herrmann and W. N. Polyzou Department of Physics and Astronomy, The University of Iowa, Iowa City, IA 52242, USA (Dated: February 2, 205) Background: The vacuum in the lightfront
More informationSummary: angular momentum derivation
Summary: angular momentum derivation L = r p L x = yp z zp y, etc. [x, p y ] = 0, etc. () () (3) Angular momentum commutation relations [L x, L y ] = i hl z (4) [L i, L j ] = i hɛ ijk L k (5) LeviCivita
More informationPhysics 115C Homework 3
Physics 115C Homework 3 Problem 1 In this problem, it will be convenient to introduce the Einstein summation convention. Note that we can write S = i S i i where the sum is over i = x,y,z. In the Einstein
More informationStatistical methods in atomistic computer simulations
Statistical methods in atomistic computer simulations Prof. Michele Ceriotti, michele.ceriotti@epfl.ch This course gives an overview of simulation techniques that are useful for the computational modeling
More informationChapter 5: Thermal Properties of Crystal Lattices
Chapter 5: Thermal Properties of Crystal Lattices Debye January 30, 07 Contents Formalism. The Virial Theorem............................. The Phonon Density of States...................... 5 Models of
More information1 Heisenberg Representation
1 Heisenberg Representation What we have been ealing with so far is calle the Schröinger representation. In this representation, operators are constants an all the time epenence is carrie by the states.
More informationLecture 12. The harmonic oscillator
Lecture 12 The harmonic oscillator 107 108 LECTURE 12. THE HARMONIC OSCILLATOR 12.1 Introduction In this chapter, we are going to find explicitly the eigenfunctions and eigenvalues for the timeindependent
More informationWhat is Classical Molecular Dynamics?
What is Classical Molecular Dynamics? Simulation of explicit particles (atoms, ions,... ) Particles interact via relatively simple analytical potential functions Newton s equations of motion are integrated
More informationQuantum Physics 2: Homework #6
Quantum Physics : Homework #6 [Total 10 points] Due: 014.1.1(Mon) 1:30pm Exercises: 014.11.5(Tue)/11.6(Wed) 6:30 pm; 56105 Questions for problems: 민홍기 hmin@snu.ac.kr Questions for grading: 모도영 modori518@snu.ac.kr
More informationA Short Essay on Variational Calculus
A Short Essay on Variational Calculus Keonwook Kang, Chris Weinberger and Wei Cai Department of Mechanical Engineering, Stanford University Stanford, CA 943054040 May 3, 2006 Contents 1 Definition of
More informationAA242B: MECHANICAL VIBRATIONS
AA242B: MECHANICAL VIBRATIONS 1 / 50 AA242B: MECHANICAL VIBRATIONS Undamped Vibrations of ndof Systems These slides are based on the recommended textbook: M. Géradin and D. Rixen, Mechanical Vibrations:
More informationarxiv: v1 [math.ds] 19 Dec 2012
arxiv:1212.4559v1 [math.ds] 19 Dec 2012 KAM theorems and open problems for infinite dimensional Hamiltonian with short range Xiaoping YUAN December 20, 2012 Abstract. Introduce several KAM theorems for
More informationTopics in Representation Theory: Cultural Background
Topics in Representation Theory: Cultural Background This semester we will be covering various topics in representation theory, see the separate syllabus for a detailed list of topics, including some that
More informationPHY4604 Introduction to Quantum Mechanics Fall 2004 Final Exam SOLUTIONS December 17, 2004, 7:30 a.m. 9:30 a.m.
PHY464 Introduction to Quantum Mechanics Fall 4 Final Eam SOLUTIONS December 7, 4, 7:3 a.m. 9:3 a.m. No other materials allowed. If you can t do one part of a problem, solve subsequent parts in terms
More information5.74 Introductory Quantum Mechanics II
MIT OpenCourseWare http://ocw.mit.edu 5.74 Introductory Quantum Mechanics II Spring 9 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Andrei Tokmakoff,
More information16.1. PROBLEM SET I 197
6.. PROBLEM SET I 97 Answers: Problem set I. a In one dimension, the current operator is specified by ĵ = m ψ ˆpψ + ψˆpψ. Applied to the left hand side of the system outside the region of the potential,
More informationThe goal of equilibrium statistical mechanics is to calculate the diagonal elements of ˆρ eq so we can evaluate average observables < A >= Tr{Â ˆρ eq
Chapter. The microcanonical ensemble The goal of equilibrium statistical mechanics is to calculate the diagonal elements of ˆρ eq so we can evaluate average observables < A >= Tr{Â ˆρ eq } = A that give
More information1 Multiplicity of the ideal gas
Reading assignment. Schroeder, section.6. 1 Multiplicity of the ideal gas Our evaluation of the numbers of microstates corresponding to each macrostate of the twostate paramagnet and the Einstein model
More informationarxiv: v2 [condmat.statmech] 16 Mar 2012
arxiv:119.658v2 condmat.statmech] 16 Mar 212 Fluctuation theorems in presence of information gain and feedback Sourabh Lahiri 1, Shubhashis Rana 2 and A. M. Jayannavar 3 Institute of Physics, Bhubaneswar
More information8.044 Lecture Notes Chapter 4: Statistical Mechanics, via the counting of microstates of an isolated system (Microcanonical Ensemble)
8.044 Lecture Notes Chapter 4: Statistical Mechanics, via the counting of microstates of an isolated system (Microcanonical Ensemble) Lecturer: McGreevy 4.1 Basic principles and definitions.........................
More informationGraduate Preliminary Examination, Thursday, January 6, Part I 2
Graduate Preliminary Examination, Thursday, January 6, 2011  Part I 2 Section A. Mechanics 1. ( Lasso) Picture from: The Lasso: a rational guide... c 1995 Carey D. Bunks A lasso is a rope of linear mass
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 wellspecified, the problem posed in nonrelativistic quantum mechanics is to
More informationLecture Notes for PHY 405 Classical Mechanics
Lecture Notes for PHY 405 Classical Mechanics From Thorton & Marion s Classical Mechanics Prepared by Dr. Joseph M. Hahn Saint Mary s University Department of Astronomy & Physics September 1, 2005 Chapter
More informationThe Particle in a Box
Page 324 Lecture 17: Relation of Particle in a Box Eigenstates to Position and Momentum Eigenstates General Considerations on Bound States and Quantization Continuity Equation for Probability Date Given:
More informationMinimum Fuel Optimal Control Example For A Scalar System
Minimum Fuel Optimal Control Example For A Scalar System A. Problem Statement This example illustrates the minimum fuel optimal control problem for a particular firstorder (scalar) system. The derivation
More informationarxiv:grqc/ v2 30 Oct 2005
International Journal of Modern Physics D c World Scientific Publishing Company arxiv:grqc/0505111v2 30 Oct 2005 ENTROPY AND AREA IN LOOP QUANTUM GRAVITY JOHN SWAIN Department of Physics, Northeastern
More informationSupplement on Lagrangian, Hamiltonian Mechanics
Supplement on Lagrangian, Hamiltonian Mechanics Robert B. Griffiths Version of 28 October 2008 Reference: TM = Thornton and Marion, Classical Dynamics, 5th edition Courant = R. Courant, Differential and
More informationFoundations of Chemical Kinetics. Lecture 19: Unimolecular reactions in the gas phase: RRKM theory
Foundations of Chemical Kinetics Lecture 19: Unimolecular reactions in the gas phase: RRKM theory Marc R. Roussel Department of Chemistry and Biochemistry Canonical and microcanonical ensembles Canonical
More informationBrownian Motion and Langevin Equations
1 Brownian Motion and Langevin Equations 1.1 Langevin Equation and the Fluctuation Dissipation Theorem The theory of Brownian motion is perhaps the simplest approximate way to treat the dynamics of nonequilibrium
More informationPhysics 70007, Fall 2009 Answers to Final Exam
Physics 70007, Fall 009 Answers to Final Exam December 17, 009 1. Quantum mechanical pictures a Demonstrate that if the commutation relation [A, B] ic is valid in any of the three Schrodinger, Heisenberg,
More informationOnedimensional harmonic oscillator. motivation. equation, energy levels. eigenfunctions, Hermite polynomials. classical analogy
Onedimensional harmonic oscillator motivation equation, energy levels eigenfunctions, Hermite polynomials classical analogy Onedimensional harmonic oscillator 05/0 Harmonic oscillator = potential
More informationAn introduction to Birkhoff normal form
An introduction to Birkhoff normal form Dario Bambusi Dipartimento di Matematica, Universitá di Milano via Saldini 50, 0133 Milano (Italy) 19.11.14 1 Introduction The aim of this note is to present an
More informationThe 3 dimensional Schrödinger Equation
Chapter 6 The 3 dimensional Schrödinger Equation 6.1 Angular Momentum To study how angular momentum is represented in quantum mechanics we start by reviewing the classical vector of orbital angular momentum
More informationTwo and ThreeDimensional Systems
0 Two and ThreeDimensional Systems Separation of variables; degeneracy theorem; group of invariance of the twodimensional isotropic oscillator. 0. Consider the Hamiltonian of a twodimensional anisotropic
More informationPhysics 342 Lecture 27. Spin. Lecture 27. Physics 342 Quantum Mechanics I
Physics 342 Lecture 27 Spin Lecture 27 Physics 342 Quantum Mechanics I Monday, April 5th, 2010 There is an intrinsic characteristic of point particles that has an analogue in but no direct derivation from
More informationChapter 38 Quantum Mechanics
Chapter 38 Quantum Mechanics Units of Chapter 38 381 Quantum Mechanics A New Theory 372 The Wave Function and Its Interpretation; the DoubleSlit Experiment 383 The Heisenberg Uncertainty Principle
More informationconventions and notation
Ph95a lecture notes, //0 The Bloch Equations A quick review of spin conventions and notation The quantum state of a spin particle is represented by a vector in a twodimensional complex Hilbert space
More informationChapter 1. Principles of Motion in Invariantive Mechanics
Chapter 1 Principles of Motion in Invariantive Mechanics 1.1. The EulerLagrange 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 informationActionAngle Variables and KAMTheory in General Relativity
ActionAngle Variables and KAMTheory in General Relativity Daniela Kunst, Volker Perlick, Claus Lämmerzahl Center of Space Technology and Microgravity University of Bremen, Germany Workshop in Oldenburg
More informationQuantum Mechanics in Three Dimensions
Physics 342 Lecture 21 Quantum Mechanics in Three Dimensions Lecture 21 Physics 342 Quantum Mechanics I Monday, March 22nd, 21 We are used to the temporal separation that gives, for example, the timeindependent
More informationNotes on wavefunctions IV: the Schrödinger equation in a potential and energy eigenstates.
Notes on wavefunctions IV: the Schrödinger equation in a potential and energy eigenstates. We have now seen that the wavefunction for a free electron changes with time according to the Schrödinger Equation
More informationPAPER 310 COSMOLOGY. Attempt no more than THREE questions. There are FOUR questions in total. The questions carry equal weight.
MATHEMATICAL TRIPOS Part III Wednesday, 1 June, 2016 9:00 am to 12:00 pm PAPER 310 COSMOLOGY Attempt no more than THREE questions. There are FOUR questions in total. The questions carry equal weight. STATIONERY
More informationCanonical Quantization C6, HT 2016
Canonical Quantization C6, HT 016 Uli Haisch a a Rudolf Peierls Centre for Theoretical Physics University of Oxford OX1 3PN Oxford, United Kingdom Please send corrections to u.haisch1@physics.ox.ac.uk.
More informationQuantumMechanical Carnot Engine
QuantumMechanical Carnot Engine Carl M. Bender 1, Dorje C. Brody, and Bernhard K. Meister 3 1 Department of Physics, Washington University, St. Louis MO 63130, USA Blackett Laboratory, Imperial College,
More informationDiffraction effects in entanglement of two distant atoms
Journal of Physics: Conference Series Diffraction effects in entanglement of two distant atoms To cite this article: Z Ficek and S Natali 007 J. Phys.: Conf. Ser. 84 0007 View the article online for updates
More informationPhys 7221 Homework # 8
Phys 71 Homework # 8 Gabriela González November 15, 6 Derivation 56: Torque free symmetric top In a torque free, symmetric top, with I x = I y = I, the angular velocity vector ω in body coordinates with
More informationChapter 11. Hamiltonian formulation Legendre transformations
Chapter 11 Hamiltonian formulation In reformulating mechanics in the language of a variational principle and Lagrangians in Chapter 4, we learned about a powerful new technology that helps us unravel dynamics
More informationGEOMETRIC QUANTIZATION
GEOMETRIC QUANTIZATION 1. The basic idea The setting of the Hamiltonian version of classical (Newtonian) mechanics is the phase space (position and momentum), which is a symplectic manifold. The typical
More informationAssignments VIII and IX, PHYS 301 (Classical Mechanics) Spring 2014 Due 3/21/14 at start of class
Assignments VIII and IX, PHYS 301 (Classical Mechanics) Spring 2014 Due 3/21/14 at start of class Homeworks VIII and IX both center on Lagrangian mechanics and involve many of the same skills. Therefore,
More informationLecture 8. The Second Law of Thermodynamics; Energy Exchange
Lecture 8 The Second Law of Thermodynamics; Energy Exchange The second law of thermodynamics Statistics of energy exchange General definition of temperature Why heat flows from hot to cold Reading for
More informationTwolevel systems coupled to oscillators
Twolevel systems coupled to oscillators RLE Group Energy Production and Conversion Group Project Staff Peter L. Hagelstein and Irfan Chaudhary Introduction Basic physical mechanisms that are complicated
More informationControl of chaos in Hamiltonian systems
Control of chaos in Hamiltonian systems G. Ciraolo, C. Chandre, R. Lima, M. Vittot Centre de Physique Théorique CNRS, Marseille M. Pettini Osservatorio Astrofisico di Arcetri, Università di Firenze Ph.
More information9 The conservation theorems: Lecture 23
9 The conservation theorems: Lecture 23 9.1 Energy Conservation (a) For energy to be conserved we expect that the total energy density (energy per volume ) u tot to obey a conservation law t u tot + i
More information140a Final Exam, Fall 2006., κ T 1 V P. (? = P or V ), γ C P C V H = U + PV, F = U TS G = U + PV TS. T v. v 2 v 1. exp( 2πkT.
40a Final Exam, Fall 2006 Data: P 0 0 5 Pa, R = 8.34 0 3 J/kmol K = N A k, N A = 6.02 0 26 particles/kilomole, T C = T K 273.5. du = TdS PdV + i µ i dn i, U = TS PV + i µ i N i Defs: 2 β ( ) V V T ( )
More informationThe Quantum Heisenberg Ferromagnet
The Quantum Heisenberg Ferromagnet Soon after Schrödinger discovered the wave equation of quantum mechanics, Heisenberg and Dirac developed the first successful quantum theory of ferromagnetism W. Heisenberg,
More informationInternal Degrees of Freedom
Physics 301 16Oct2002 151 Internal Degrees of Freedom There are several corrections we might make to our treatment of the ideal gas If we go to high occupancies our treatment using the MaxwellBoltzmann
More informationAdiabatic quantum computation a tutorial for computer scientists
Adiabatic quantum computation a tutorial for computer scientists Itay Hen Dept. of Physics, UCSC Advanced Machine Learning class UCSC June 6 th 2012 Outline introduction I: what is a quantum computer?
More informationThe Notsosimple Pendulum: Balancing a Pencil on its Point Peter Lynch, UCD, Dublin, May 2014
The Notsosimple Pendulum: Balancing a Pencil on its Point Peter Lynch, UCD, Dublin, May 204 arxiv:406.25v [nlin.si] 4 Jun 204 ABSTRACT. Does quantum mechanics matter at everyday scales? We generally
More information9 The conservation theorems
9 The conservation theorems 9.1 Energy Conservation (a) For energy to be conserved we expect that the total energy density (energy per volume ) u tot to obey a conservation law t u tot + i S i tot = 0
More informationCaltech Ph106 Fall 2001
Caltech h106 Fall 2001 ath for physicists: differential forms Disclaimer: this is a first draft, so a few signs might be off. 1 Basic properties Differential forms come up in various parts of theoretical
More informationEnsembles and incomplete information
p. 1/32 Ensembles and incomplete information So far in this course, we have described quantum systems by states that are normalized vectors in a complex Hilbert space. This works so long as (a) the system
More informationClassical and Quantum Conjugate Dynamics The Interplay Between Conjugate Variables
Chapter 1 Classical and Quantum Conjugate Dynamics The Interplay Between Conjugate Variables Gabino TorresVega Additional information is available at the end of the chapter http://dx.doi.org/10.5772/53598
More informationPhysics 505 Homework No. 8 Solutions S Spinor rotations. Somewhat based on a problem in Schwabl.
Physics 505 Homework No 8 s S8 Spinor rotations Somewhat based on a problem in Schwabl a) Suppose n is a unit vector We are interested in n σ Show that n σ) = I where I is the identity matrix We will
More informationLandau s Fermi Liquid Theory
Thors Hans Hansson Stockholm University Outline 1 Fermi Liquids Why, What, and How? Why Fermi liquids? What is a Fermi liquids? Fermi Liquids How? 2 Landau s Phenomenological Approach The free Fermi gas
More informationLecture 4 Quantum mechanics in more than onedimension
Lecture 4 Quantum mechanics in more than onedimension Background Previously, we have addressed quantum mechanics of 1d systems and explored bound and unbound (scattering) states. Although general concepts
More informationarxiv:condmat/ v1 29 Dec 1996
Chaotic enhancement of hydrogen atoms excitation in magnetic and microwave fields Giuliano Benenti, Giulio Casati Università di Milano, sede di Como, Via Lucini 3, 22100 Como, Italy arxiv:condmat/9612238v1
More informationLikewise, any operator, including the most generic Hamiltonian, can be written in this basis as H11 H
Finite Dimensional systems/ilbert space Finite dimensional systems form an important subclass of degrees of freedom in the physical world To begin with, they describe angular momenta with fixed modulus
More informationI.I Stability Conditions
I.I Stability Conditions The conditions derived in section I.G are similar to the well known requirements for mechanical stability. A particle moving in an eternal potential U settles to a stable equilibrium
More information