Heisenberg's Quantum Mechanics - M. Razavy (World, 2011) BBS

HEISENBERGSQUANTUM

MECHANICS

7702 tp.indd 1 10/28/10 10:20 AM

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N E W J E R S E Y L O N D O N S I N G A P O R E B E I J I N G S H A N G H A I H O N G K O N G TA I P E I C H E N N A I

World Scientific

HEISENBERGSQUANTUM

MECHANICS

Mohsen RazavyUniversity of Alberta, Canada

7702 tp.indd 2 10/28/10 10:20 AM

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HEISENBERGS QUANTUM MECHANICS

ZhangFang - Heisenberg's Quan Mechanics.pmd 10/19/2010, 11:16 AM1

Dedicated to my great teachers

A.H. Zarrinkoob, M. Bazargan and J.S. Levinger

Preface

There is an abundance of excellent texts and lecture notes on quantum theoryand applied quantum mechanics available to the students and researchers. Themotivation for writing this book is to present matrix mechanics as it was firstdiscovered by Heisenberg, Born and Jordan, and by Pauli and bring it up todate by adding the contributions by a number of prominent physicists in the in-tervening years. The idea of writing a book on matrix mechanics is not new. In1965 H.S. Green wrote a monograph with the title Matrix Mechanics (Nord-hoff, Netherlands) where from the works of the pioneers in the field he collectedand presented a self-contained theory with applications to simple systems.

In most text books on quantum theory, a chapter or two are devoted tothe Heisenbergs matrix approach, but due to the simplicity of the Schrodingerwave mechanics or the elegance of the Feynman path integral technique, thesetwo methods have often been used to study quantum mechanics of systems withfinite degrees of freedom.

The present book surveys matrix and operator formulations of quantummechanics and attempts to answer the following basic questions: (a) whyand where the Heisenberg form of quantum mechanics is more useful than otherformulations and (b) how the formalism can be applied to specific problems?To seek answer to these questions I studied what I could find in the originalliterature and collected those that I thought are novel and interesting. My firstinclination was to expand on Greens book and write only about the matrixmechanics. But this plan would have severely limited the scope and coverage ofthe book. Therefore I decided to include and use the wave equation approachwhere it was deemed necessary. Even in these cases I tried to choose the ap-proach which, in my judgement, seemed to be closer to the concepts of matrixmechanics. For instance in discussing quantum scattering theory I followed thedeterminantal approach and the LSZ reduction formalism.

In Chapter 1 a brief survey of analytical dynamics of point particles ispresented which is essential for the formulation of quantum mechanics, and anunderstanding of the classical-quantum mechanical correspondence. In this partof the book particular attention is given to the question of symmetry and con-servation laws.

vii

viii Heisenbergs Quantum Mechanics

In Chapter 2 a short historical review of the discovery of matrix mechanicsis given and the original Heisenbergs and Borns ideas leading to the formu-lation of quantum theory and the discovery of the fundamental commutationrelations are discussed.

Chapter 3 is concerned with the mathematics of quantum mechanics,namely linear vector spaces, operators, eigenvalues and eigenfunctions. Herean entire section is devoted to the ways of constructing Hermitian operators,together with a discussion of the inconsistencies of various rules of associationof classical functions and quantal operators.

In Chapter 4 the postulates of quantum mechanics and their implica-tions are studied. A detailed review of the uncertainty principle for position-momentum, time-energy and angular momentum-angle and some applicationsof this principle is given. This is followed by an outline of the correspondenceprinciple. The question of determination of the state of the system from themeasurement of probabilities in coordinate and momentum space is also in-cluded in this chapter.

In Chapter 5 connections between the equation of motion, the Hamiltonianoperator and the commutation relations are examined, and Wigners argumentabout the nonuniqueness of the canonical commutation relations is discussed.In this chapter quantum first integrals of motion are derived and it is shownthat unlike their classical counterparts, these, with the exception of the energyoperator, are not useful for the quantal description of the motion.

In Chapter 6 the symmetries and conservation laws for quantum mechan-ical systems are considered. Also topics related to the Galilean invariance, masssuperselection rule and the time invariance are studied. In addition a brief dis-cussion of classical and quantum integrability and degeneracy is presented.

Chapter 7 deals with the application of Heisenbergs equations of motionin determining bound state energies of one-dimensional systems. Here Kleinsmethod and its generalization are considered. In addition the motion of a par-ticle between fixed walls is studied in detail.

Chapter 8 is concerned with the factorization method for exactly solvablepotentials and this is followed by a brief discussion of the supersymmetry andof shape invariance.

The two-body problem is the subject of discussion in Chapter 9, where theproperties of the orbital and spin angular momentum operators and determina-tion of their eigenfunctions are presented. Then the solution to the problem ofhydrogen atom is found following the original formulation of Pauli using RungeLenz vector.

In Chapter 10 methods of integrating Heisenbergs equations of motionare presented. Among them the discrete-time formulation pioneered by Benderand collaborators, the iterative solution for polynomial potentials advanced byZnojil and also the direct numerical method of integration of the equations ofmotion are mentioned.

The perturbation theory is studied in Chapter 11 and in Chapter 12 othermethods of approximation, mostly derived from Heisenbergs equations of mo-

Preface ix

tion are considered. These include the semi-classical approximation and varia-tional method.

Chapter 13 is concerned with the problem of quantization of equations ofmotion with higher derivatives, this part follows closely the work of Pais andUhlenbeck.

Potential scattering is the next topic which is considered in Chapter 14.Here the Schrodinger equation is used to define concepts such as cross sectionand the scattering amplitude, but then the deteminantal method of Schwingeris followed to develop the connection between the potential and the scatteringamplitude. After this, the time-dependent scattering theory, the scattering ma-trix and the LippmannSchwinger equation are studied. Other topics reviewedin this chapter are the impact parameter representation of the scattering am-plitude, the Born approximation and transition probabilities.

In Chapter 15 another feature of the wave nature of matter which is quan-tum diffraction is considered.

The motion of a charged particle in electromagnetic field is taken up inChapter 16 with a discussion of the AharonovBohm effect and the Berry phase.

Quantum many-body problem is reviewed in Chapter 17. Here systemswith many-fermion and with many-boson are reviewed and a brief review of thetheory of superfluidity is given.

Chapter 18 is about the quantum theory of free electromagnetic field witha discussion of coherent state of radiation and of Casimir force.

Chapter 19, contains the theory of interaction of radiation with matter.Finally in the last chapter, Chapter 20, a brief discussion of Bells inequal-

ities and its relation to the conceptual foundation of quantum theory is given.In preparing this book, no serious attempt has been made to cite all of

the important original sources and various attempts in the formulation and ap-plications of the Heisenberg quantum mechanics.

I am grateful to my wife for her patience and understanding while I waswriting this book, and to my daughter, Maryam, for her help in preparing themanuscript.

Edmonton, Canada, 2010

Contents

Preface vii

1 A Brief Survey of Analytical Dynamics 1

1.1 The Lagrangian and the Hamilton Principle . . . . . . . . . . . . 1

1.2 Noethers Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3 The Hamiltonian Formulation . . . . . . . . . . . . . . . . . . . . 8

1.4 Canonical Transformation . . . . . . . . . . . . . . . . . . . . . . 12

1.5 Action-Angle Variables . . . . . . . . . . . . . . . . . . . . . . . . 14

1.6 Poisson Brackets . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.7 Time Development of Dynamical Variables and PoissonBrackets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.8 Infinitesimal Canonical Transformation . . . . . . . . . . . . . . . 21

1.9 Action Principle with Variable End Points . . . . . . . . . . . . . 23

1.10 Symmetry and Degeneracy in Classical Dynamics . . . . . . . . . 27

1.11 Closed Orbits and Accidental Degeneracy . . . . . . . . . . . . . 30

1.12 Time-Dependent Exact Invariants . . . . . . . . . . . . . . . . . 32

2 Discovery of Matrix Mechanics 39

xi

xii Heisenbergs Quantum Mechanics

2.1 Equivalence of Wave and Matrix Mechanics . . . . . . . . . . . . 44

3 Mathematical Preliminaries 49

3.1 Vectors and Vector Spaces . . . . . . . . . . . . . . . . . . . . . . 49

3.2 Special Types of Operators . . . . . . . . . . . . . . . . . . . . . 55

3.3 Vector Calculus for the Operators . . . . . . . . . . . . . . . . . . 58

3.4 Construction of Hermitian and Self-Adjoint Operators . . . . . . 59

3.5 Symmetrization Rule . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.6 Weyls Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.7 Diracs Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.8 Von Neumanns Rules . . . . . . . . . . . . . . . . . . . . . . . . 67

3.9 Self-Adjoint Operators . . . . . . . . . . . . . . . . . . . . . . . . 67

3.10 Momentum Operator in a Curvilinear Coordinates . . . . . . . . 73

3.11 Summation Over Normal Modes . . . . . . . . . . . . . . . . . . 79

4 Postulates of Quantum Theory 83

4.1 The Uncertainty Principle . . . . . . . . . . . . . . . . . . . . . . 91

4.2 Application of the Uncertainty Principle for CalculatingBound State Energies . . . . . . . . . . . . . . . . . . . . . . . . 96

4.3 Time-Energy Uncertainty Relation . . . . . . . . . . . . . . . . . 98

4.4 Uncertainty Relations for Angular Momentum-AngleVariables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

4.5 Local Heisenberg Inequalities . . . . . . . . . . . . . . . . . . . . 106

4.6 The Correspondence Principle . . . . . . . . . . . . . . . . . . . . 112

4.7 Determination of the State of a System . . . . . . . . . . . . . . 116

Contents xiii

5 Equations of Motion, Hamiltonian Operator and theCommutation Relations 125

5.1 Schwingers Action Principle and Heisenbergs equationsof Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

5.2 Nonuniqueness of the Commutation Relations . . . . . . . . . . . 128

5.3 First Integrals of Motion . . . . . . . . . . . . . . . . . . . . . . . 132

6 Symmetries and Conservation Laws 139

6.1 Galilean Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . 140

6.2 Wave Equation and the Galilean Transformation . . . . . . . . . 141

6.3 Decay Problem in Nonrelativistic Quantum Mechanics andMass Superselection Rule . . . . . . . . . . . . . . . . . . . . . . 143

6.4 Time-Reversal Invariance . . . . . . . . . . . . . . . . . . . . . . 146

6.5 Parity of a State . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

6.6 Permutation Symmetry . . . . . . . . . . . . . . . . . . . . . . . 150

6.7 Lattice Translation . . . . . . . . . . . . . . . . . . . . . . . . . . 153

6.8 Classical and Quantum Integrability . . . . . . . . . . . . . . . . 156

6.9 Classical and Quantum Mechanical Degeneracies . . . . . . . . . 157

7 Bound State Energies for One-Dimensional Problems 163

7.1 Kleins Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

7.2 The Anharmonic Oscillator . . . . . . . . . . . . . . . . . . . . . 166

7.3 The Double-Well Potential . . . . . . . . . . . . . . . . . . . . . . 171

7.4 Chasmans Method . . . . . . . . . . . . . . . . . . . . . . . . . . 173

7.5 Heisenbergs Equations of Motion for Impulsive Forces . . . . . . 175

7.6 Motion of a Wave Packet . . . . . . . . . . . . . . . . . . . . . . 178

xiv Heisenbergs Quantum Mechanics

7.7 Heisenbergs and Newtons Equations of Motion . . . . . . . . . . 181

8 Exactly Solvable Potentials, Supersymmetry and ShapeInvariance 191

8.1 Energy Spectrum of the Two-Dimensional Harmonic Oscillator . 192

8.2 Exactly Solvable Potentials Obtained from HeisenbergsEquation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

8.3 Creation and Annihilation Operators . . . . . . . . . . . . . . . . 198

8.4 Determination of the Eigenvalues by Factorization Method . . . 201

8.5 A General Method for Factorization . . . . . . . . . . . . . . . . 206

8.6 Supersymmetry and Superpotential . . . . . . . . . . . . . . . . . 212

8.7 Shape Invariant Potentials . . . . . . . . . . . . . . . . . . . . . . 216

8.8 Solvable Examples of Periodic Potentials . . . . . . . . . . . . . . 221

9 The Two-Body Problem 227

9.1 The Angular Momentum Operator . . . . . . . . . . . . . . . . . 230

9.2 Determination of the Angular Momentum Eigenvalues . . . . . . 232

9.3 Matrix Elements of Scalars and Vectors and the Selection Rules . 236

9.4 Spin Angular Momentum . . . . . . . . . . . . . . . . . . . . . . 239

9.5 Angular Momentum Eigenvalues Determined from theEigenvalues of Two Uncoupled Oscillators . . . . . . . . . . . . . 241

9.6 Rotations in Coordinate Space and in Spin Space . . . . . . . . . 243

9.7 Motion of a Particle Inside a Sphere . . . . . . . . . . . . . . . . 245

9.8 The Hydrogen Atom . . . . . . . . . . . . . . . . . . . . . . . . . 247

9.9 Calculation of the Energy Eigenvalues Using the RungeLenzVector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

Contents xv

9.10 Classical Limit of Hydrogen Atom . . . . . . . . . . . . . . . . . 256

9.11 Self-Adjoint Ladder Operator . . . . . . . . . . . . . . . . . . . . 260

9.12 Self-Adjoint Ladder Operator for Angular Momentum . . . . . . 261

9.13 Generalized Spin Operators . . . . . . . . . . . . . . . . . . . . . 262

9.14 The Ladder Operator . . . . . . . . . . . . . . . . . . . . . . . . 263

10 Methods of Integration of Heisenbergs Equations of Motion 269

10.1 Discrete-Time Formulation of the Heisenbergs Equationsof Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269

10.2 Quantum Tunneling Using Discrete-Time Formulation . . . . . . 273

10.3 Determination of Eigenvalues from Finite-Difference Equations . 276

10.4 Systems with Several Degrees of Freedom . . . . . . . . . . . . . 278

10.5 Weyl-Ordered Polynomials and BenderDunne Algebra . . . . . 282

10.6 Integration of the Operator Differential Equations . . . . . . . . 287

10.7 Iterative Solution for Polynomial Potentials . . . . . . . . . . . . 291

10.8 Another Numerical Method for the Integration of theEquations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . 295

10.9 Motion of a Wave Packet . . . . . . . . . . . . . . . . . . . . . . 299

11 Perturbation Theory 309

11.1 Perturbation Theory Applied to the Problem of aQuartic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . 313

11.2 Degenerate Perturbation Theory . . . . . . . . . . . . . . . . . . 321

11.3 Almost Degenerate Perturbation Theory . . . . . . . . . . . . . . 323

11.4 van der Waals Interaction . . . . . . . . . . . . . . . . . . . . . . 325

11.5 Time-Dependent Perturbation Theory . . . . . . . . . . . . . . . 327

xvi Heisenbergs Quantum Mechanics

11.6 The Adiabatic Approximation . . . . . . . . . . . . . . . . . . . . 329

11.7 Transition Probability to the First Order . . . . . . . . . . . . . 333

12 Other Methods of Approximation 337

12.1 WKB Approximation for Bound States . . . . . . . . . . . . . . . 337

12.2 Approximate Determination of the Eigenvalues forNonpolynomial Potentials . . . . . . . . . . . . . . . . . . . . . . 340

12.3 Generalization of the Semiclassical Approximation toSystems with N Degrees of Freedom . . . . . . . . . . . . . . . . 343

12.4 A Variational Method Based on Heisenbergs Equationof Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350

12.5 RaleighRitz Variational Principle . . . . . . . . . . . . . . . . . 354

12.6 Tight-Binding Approximation . . . . . . . . . . . . . . . . . . . . 355

12.7 Heisenbergs Correspondence Principle . . . . . . . . . . . . . . . 356

12.8 Bohr and Heisenberg Correspondence and the Frequenciesand Intensities of the Emitted Radiation . . . . . . . . . . . . . . 361

13 Quantization of the Classical Equations of Motion withHigher Derivatives 371

13.1 Equations of Motion of Finite Order . . . . . . . . . . . . . . . . 371

13.2 Equation of Motion of Infinite Order . . . . . . . . . . . . . . . . 374

13.3 Classical Expression for the Energy . . . . . . . . . . . . . . . . . 376

13.4 Energy Eigenvalues when the Equation of Motion is ofInfinite Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378

14 Potential Scattering 381

14.1 Determinantal Method in Potential Scattering . . . . . . . . . . 389

14.2 Two Solvable Problems . . . . . . . . . . . . . . . . . . . . . . . 395

Contents xvii

14.3 Time-Dependent Scattering Theory . . . . . . . . . . . . . . . . 399

14.4 The Scattering Matrix . . . . . . . . . . . . . . . . . . . . . . . . 402

14.5 The LippmannSchwinger Equation . . . . . . . . . . . . . . . . 404

14.6 Analytical Properties of the Radial Wave Function . . . . . . . . 415

14.7 The Jost Function . . . . . . . . . . . . . . . . . . . . . . . . . . 418

14.8 Zeros of the Jost Function and Bound Sates . . . . . . . . . . . 421

14.9 Dispersion Relation . . . . . . . . . . . . . . . . . . . . . . . . . 423

14.10 Central Local Potentials having Identical Phase Shifts andBound States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424

14.11 The Levinson Theorem . . . . . . . . . . . . . . . . . . . . . . . 426

14.12 Number of Bound States for a Given Partial Wave . . . . . . . . 427

14.13 Analyticity of the S-Matrix and the Principle of Casuality . . . 429

14.14 Resonance Scattering . . . . . . . . . . . . . . . . . . . . . . . . 431

14.15 The Born Series . . . . . . . . . . . . . . . . . . . . . . . . . . . 433

14.16 Impact Parameter Representation of the Scattering Amplitude . 437

14.17 Determination of the Impact Parameter Phase Shift from theDifferential Cross Section . . . . . . . . . . . . . . . . . . . . . . 442

14.18 Elastic Scattering of Identical Particles . . . . . . . . . . . . . . 444

14.19 Transition Probability . . . . . . . . . . . . . . . . . . . . . . . . 447

14.20 Transition Probabilities for Forced Harmonic Oscillator . . . . . 448

15 Quantum Diffraction 459

15.1 Diffraction in Time . . . . . . . . . . . . . . . . . . . . . . . . . . 460

15.2 High Energy Scattering from an Absorptive Target . . . . . . . . 463

xviii Heisenbergs Quantum Mechanics

16 Motion of a Charged Particle in Electromagnetic Fieldand Topological Quantum Effects for Neutral Particles 467

16.1 The AharonovBohm Effect . . . . . . . . . . . . . . . . . . . . 471

16.2 Time-Dependent Interaction . . . . . . . . . . . . . . . . . . . . 480

16.3 Harmonic Oscillator with Time-Dependent Frequency . . . . . . 481

16.4 Heisenbergs Equations for Harmonic Oscillator withTime-Dependent Frequency . . . . . . . . . . . . . . . . . . . . . 483

16.5 Neutron Interferometry . . . . . . . . . . . . . . . . . . . . . . . 489

16.6 Gravity-Induced Quantum Interference . . . . . . . . . . . . . . 491

16.7 Quantum Beats in Waveguides with Time-DependentBoundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493

16.8 Spin Magnetic Moment . . . . . . . . . . . . . . . . . . . . . . . 500

16.9 SternGerlach Experiment . . . . . . . . . . . . . . . . . . . . . 503

16.10 Precession of Spin Magnetic Moment in a ConstantMagnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506

16.11 Spin Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . 508

16.12 A Simple Model of Atomic Clock . . . . . . . . . . . . . . . . . . 511

16.13 Berrys Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514

17 Quantum Many-Body Problem 525

17.1 Ground State of Two-Electron Atom . . . . . . . . . . . . . . . 526

17.2 Hartree and HartreeFock Approximations . . . . . . . . . . . . 529

17.3 Second Quantization . . . . . . . . . . . . . . . . . . . . . . . . . 537

17.4 Second-Quantized Formulation of the Many-Boson Problem . . . 538

17.5 Many-Fermion Problem . . . . . . . . . . . . . . . . . . . . . . . 542

17.6 Pair Correlations Between Fermions . . . . . . . . . . . . . . . . 547

Contents xix

17.7 Uncertainty Relations for a Many-Fermion System . . . . . . . . 551

17.8 Pair Correlation Function for Noninteracting Bosons . . . . . . . 554

17.9 Bogoliubov Transformation for a Many-Boson System . . . . . . 557

17.10 Scattering of Two Quasi-Particles . . . . . . . . . . . . . . . . . 565

17.11 Bogoliubov Transformation for Fermions Interacting throughPairing Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571

17.12 Damped Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . 578

18 Quantum Theory of Free Electromagnetic Field 589

18.1 Coherent State of the Radiation Field . . . . . . . . . . . . . . . 592

18.2 Casimir Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599

18.3 Casimir Force Between Parallel Conductors . . . . . . . . . . . . 601

18.4 Casimir Force in a Cavity with Conducting Walls . . . . . . . . . 603

19 Interaction of Radiation with Matter 607

19.1 Theory of Natural Line Width . . . . . . . . . . . . . . . . . . . 611

19.2 The Lamb Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617

19.3 Heisenbergs Equations for Interaction of an Atom withRadiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 621

20 Bells Inequality 631

20.1 EPR Experiment with Particles . . . . . . . . . . . . . . . . . . . 631

20.2 Classical and Quantum Mechanical Operational Conceptsof Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 640

20.3 Collapse of the Wave Function . . . . . . . . . . . . . . . . . . . 643

20.4 Quantum versus Classical Correlations . . . . . . . . . . . . . . . 645

Index 653

Chapter 1

A Brief Survey ofAnalytical Dynamics

1.1 The Lagrangian and the Hamilton Principle

We can formulate the laws of motion of a mechanical system with N degrees offreedom in terms of Hamiltons principle. This principle states that for everymotion there is a well-defined function of the N coordinates qi and N velocitiesqi which is called the Lagrangian, L, such that the integral

S = t2t1

L (qi, qi, t) dt, (1.1)

takes the least possible value (or extremum) when the system occupies positionsqi(t1) and qi(t2) at the times t1 and t2 [1],[2].

The set of N independent quantities {qi} which completely defines theposition of the system of N degrees of freedom are called generalized coordinatesand their time derivatives are called generalized velocities.

The requirement that S be a minimum (or extremum) implies that L mustsatisfy the EulerLagrange equation

L

qi ddt

(L

qi

)= 0, i = 1, N. (1.2)

The mathematical form of these equations remain invariant under a point trans-formation. Let us consider a non-singular transformation of the coordinates fromthe set of N {qi} s to another set of N {Qi} s given by the equations

Qi = Qi (q1, , qN ) , i = 1, N, (1.3)

1

2 Heisenbergs Quantum Mechanics

and its inverse transform given by the N equations

qj = qj (Q1, , QN ) , j = 1, N. (1.4)Now let F (q1, , qN , q1, , qN ) be a twice differentiable function of 2N vari-ables q1, , qN , q1, , qN . We note that this function can be written as afunction of Qj s and Qj s if we replace qi s and qi s by Qj s and Qj s using Eq.(1.4). Now by direct differentiation we find that(

qi ddt

qi

)F(qi(Qj), qi(Qj , Qj)

)=

Nj=1

(Qjqi

)(

Qj ddt

Qj

)F(qi(Qj), qi(Qj , Qj)

), i = 1, N.

(1.5)

Thus if L(Q1, QN ) has a vanishing EulerLagrange derivative i.e.[

Qj ddt

(

Qj

)]L = 0, (1.6)

then Eq. (1.5) implies that

L

qi ddt

(L

qi

)= 0, i = 1, N. (1.7)

This result shows that we can express the motion of the system either in termsof the generalized coordinates qi and generalized velocities qi or in terms of Qjand Qj .

For simple conservative systems for which potential functions of the typeV (q1, , qN , t) can be found, the Lagrangian L has a simple form:

L = T (q1, , qn; q1, , qN ) V (q1, , qN , t), (1.8)where T is the kinetic energy of the particles in the system under considerationand V is their potential energy. However given the force law acting on thei-th particle of the system as Fi(q1, , qN ; q1, , qN ), in general, a uniqueLagrangian cannot be found. For instance we observe that the EulerLagrangederivative of any total time derivative of a function F of qi, qi i.e. ddtF(qi, qi, t)is identically zero;[

qi ddt

(

qi

)]dF(qi, qi, t)

dt 0, i = 1, N. (1.9)

Therefore we can always add a total time derivative dFdt to the Lagrangian with-

out affecting the resulting equations of motion.

Lagrangian Formulation 3

The inverse problem of classical mechanics is that of determination ofthe Lagrangian (or Hamiltonian) when the force law Fj(qi, qi, t) is known. Thenecessary and sufficient conditions for the existence of the Lagrangian has beenstudied in detail by Helmholtz [3][6]. In general, for a given set of Fj s, Lsatisfies a linear partial differential equation. To obtain this equation we startwith the EulerLagrange equation (1.2), find the total time derivative of Lqiand then replace qi by Fimi . In this way we obtain

2L

qit+j

(Fjmj

)(2L

qiqj

)+j

(2L

qiqj

)qiqj L

qi= 0, i = 1, N.

(1.10)This set of equations yield the Lagrangian function. But as was stated earlier Lis not unique even for conservative systems. The advantage of the Lagrangianformulation is that it contains information about the symmetries of the motionwhich, in general, cannot be obtained from the equations of motion alone.

For instance let us consider the Lagrangian for the motion of a free particle.In a reference frame in which space is homogeneous and isotropic and time ishomogeneous, i.e. an inertial frame, a free particle which is at rest at a giveninstant of time, always remains at rest. Because of the homogeneity of spaceand time, the Lagrangian L cannot depend either on the position of the particler nor on time t. Thus it can only be a function of velocity r. Now if the velocityof the particle is r relative to a frame S, then in another frame S which ismoving with a small velocity v with respect to S the velocity is r, and theLagrangian is

L(r2)

= L[(r + v)2

] L [r2]+ 2r v L

r2, (1.11)

where we have ignored higher order terms in v. Since the equation of motionshould have the same form in every frame, therefore the difference betweenL(r2)

and L(r2)

must be a total time derivative (Galilean invariance). For aconstant v this implies that L

r2 must be a constant and we choose this constantto be m2 . Thus we arrive at a unique Lagrangian for the motion of a free particle.

L =12mr2. (1.12)

As a second example let us consider a system consisting of two particleseach of mass m interacting with each other with a potential V (|r1 r2|), wherer1 and r2 denote the positions of the two particles. The standard Lagrangianaccording to Eq. (1.8) is

L1 =12m(r21 + r

22

) V (|r1 r2|), (1.13)and this generates the equations of motion

md2rid t2

= iV (|r1 r2|), i = 1, 2. (1.14)


A Lagrangian equivalent to L1, is given by [8]

L2 = m (r1 r2) V (|r1 r2|), (1.15)and this L2 also yields the equations of motion (1.14). However the symmetriesof the two Lagrangians L1 and L2 are different. The Lagrangian L1 is invariantunder the rotation of the six-dimensional space r1 and r2, whereas L2 is not.

The requirement of the invariance under the full Galilean group whichincludes the conservation of energy, the angular momentum and the motion ofthe center of mass, restricts the possible forms of the Lagrangian (apart froma total time derivative) but still leaves certain arbitrariness. Here we want toinvestigate this point and see whether by imposition of the Galilean invariancewe can determine a unique form for the Lagrangian or not.

Consider a system of N pairwise interacting particles with the equationsof motion

mjd2rjd t2

= jV, j = 1, 2, , N, (1.16)where V depends on the relative coordinates of the particles rj rk and hence

j

jV = 0. (1.17)

This means that the forces are acting only between the particles of the system.Thus from (1.16) we have the law of conservation of the total linear momentum;

d

dt

j

mjvj = 0, (1.18)

where vj = ddtrj . The conservation law (1.18) also follows from the Lagrangian

L =j

12mjv2j V. (1.19)

Now under the Galilean transformation vj vj + v, and L will change to Lwhere L L is a total time derivative. Therefore the resulting equations ofmotion, (1.16), will remain unchanged. Since

jmjvj is constant, Eq. (1.18),

we can add any function ofjmjvj to the Lagrangian without affecting the

equations of motion. If we denote this new Lagrangian which is found by theaddition of the constant term F

(jmjvj

)to L by L[F ], then we observe that

if in L[F ] we replace vi by vi + v, then L[F ] L[F ] will not be a total timederivative unless F is of the form

F =1

2

i

(mivi)2, (1.20)

where is a constant with the dimension of mass. From this result it fol-lows that the general form of L[F ], can be rejected on the ground that it is

Lagrangian Formulation 5

not invariant under the full Galilean group [9],[10]. With the addition of theterm 12

i (mivi)

2, the resulting Lagrangian is now Galilean invariant but isdependent on the parameter is [9],[10];

L() =j

12mjv2j +

12

i

(mivi)2 V. (1.21)

What is interesting about this Lagrangian is that for any two values of theparameter , say and , L()L() is not a total time derivative, and inthis sense the two Lagrangians are inequivalent.

The equations of motion derived from (1.21) are given by

d

dt

mkvk + mk

j

mjvj

= kV, k = 1, 2, , N. (1.22)Noting that ddt

jmjvj is zero, Eqs. (1.22) are the same as the equations of

motion (1.16). However the relation between the canonical momentum pk andthe velocity vk is now more complicated:

pk =L

vk= mkvk +

mk

j

mjvj , (1.23)

or solving for vk in terms of pk we have

vk =pkmk

j pj

+jmj

. (1.24)

Velocity-Dependent Forces If the velocity-dependent forces are suchthat the system can be described by a Lagrangian of the form

L =12mv2 V (r,v, t), (1.25)

then the generalized force Fi can be written as

Fi = Vxi

+d

dt

(V

vi

), (1.26)

a result which follows from the EulerLagrange equation. Now according toHelmholtz for the existence of the Lagrangian such a generalized force can beat most a linear function of acceleration, and it must satisfy the Helmholtzidentities [3]:

Fivj

=Fjvi

, (1.27)

Fivj

+Fjvi

=d

dt

(Fivj

+Fjvi

), (1.28)


andFixj Fjxi

=12d

dt

(Fivj Fjvi

), (1.29)

If we assume as in (1.25) that the generalized force is independent of accelerationthen from (1.28) it follows that

2Fivjvk

= 0. (1.30)

By integrating this equation once we get

Fivj

=e

c

k

ijkBk(r, t), (1.31)

where c and e are constants, and B(r, t) is a vector function. The symbol ijkdenotes the totally antisymmetric tensor (LeviCivita symbol) defined by

123 = 231 = 312 = 1, (1.32)

321 = 213 = 132 = 1. (1.33)

Now we integrate (1.31) a second time and we find that the generalized force isvelocity-dependent and of Lorentz form

F(r,v, t) = eE(r, t) +e

cv B(r, t), (1.34)

where E(r, t) is a vector function of r and t. The two vector functions E(r, t) andB(r, t) are not independent of each other. From the three Helmholtz conditions,Eq. (1.27) is trivially satisfied in this case since F is independent of v. But inorder to satisfy (1.29) we find that E and B cannot be chosen arbitrarily. Toget the connection between these two vectors we substitute (1.34) in (1.29) andwe find (

E + 1c

Bt

)k

= vk( B). (1.35)

Since the velocities and coordinates are independent, each side of (1.35) mustbe equal to zero, and thus we find two Maxwell type equations [7]

E = 1c

Bt, (1.36)

and

.B = 0. (1.37)

Noethers Theorem 7

1.2 Noethers Theorem

The symmetries and the conservation laws associated with a given Lagrangiancan be found by applying Noethers theorem to the Lagrangian of a system ofparticles. Let us consider the change in the Lagrangian under an infinitesimaltransformation of the generalized coordinates

qi qi + qi, i = 1, , N, (1.38)

which to the order of qi is

L(qi + qi, qi + qi) L(qi, qi)=

j

qjL(qi, qi)qj

+j

dqjdt

L(qi, qi)qj

=j

[L(qi, qi)

qj ddt

L(qi, qi)qj

]qj +

d

dt

j

qjL(qi, qi)qj

. (1.39)

If the Lagrangian remains unchanged under this transformation, i.e.

L(qi + qi, qi + qi) L(qi, qi) = 0, (1.40)

then the quantity

N =j

qjL(qi, qi)

qi, (1.41)

which is called the Noether charge is conserved. Here we assumed that L satisfiesthe EulerLagrange equation, and therefore each term in the square bracket in(1.39) is zero.

When L (qi, qi, t) does not depend on a particular coordinate, say qk, thenclearly the momentum conjugate to this coordinate, pk = Lq , is conserved. Thisparticular coordinate is called ignorable or cyclic coordinate.

As an example let us consider the infinitesimal transformation{ r1 + r2 +

, (1.42)

that is an infinitesimal space translation with an arbitrary small vector. Usingthe Lagrangian (1.13) we find that

N = m (r1 + r2) , (1.43)

is the conserved quantity. Since is an arbitrary small vector, therefore the totalmomentum P = m (r1 + r2) remains constant. Now both of the Lagrangians L1and L2, Eqs. (1.13) and (1.15) have the same Noether charge given by (1.43),but as we mentioned earlier they have different properties under space rotation.


1.3 The Hamiltonian Formulation

In classical mechanics the Hamiltonian is defined as a function of the canonicalcoordinates and momenta which satisfies Hamiltons principle

t2t1

{Ni=1

qipi H(pi, qi)}dt = 0, (1.44)

i.e. the integral in (1.44) between any two arbitrary times t1 and t2 for theactual path must be minimum (or extremum) [2].

This requirement gives us 2N Hamiltons canonical equations [1],[2]

qi =H(pi, qi)

pi, i = 1, 2, N, (1.45)

and

pi = H(pi, qi)qi

i = 1, 2, N. (1.46)

These first order differential equations for qi(t) and pi(t) determine theposition and momentum of the i-th degree of freedom as a function of time ifwe assume that the initial conditions

qi(t = 0) = q0(0), and pi(t = 0) = p0(0), (1.47)

are known.By eliminating pi s between the two sets of equations (1.45) and (1.46)

we find the Newton equations of motion

miqi = Fi

(q1, q2 qN , H

p1 H

pN

), i = 1, 2, N. (1.48)

If we only require that the Hamiltonian generate the correct equations ofmotion in coordinate space, viz Eq. (1.48), then by differentiating (1.45) withrespect to time and substituting for qi, pi and qi using Eqs. (1.45), (1.46) and(1.48) we find

qi =Nk=1

(H

pk

2H

piqk Hqk

2H

pipk

)=

1mi

Fi

(q1, qN , H

pi H

pN

), i = 1, 2, N. (1.49)

The solutions of this set of nonlinear partial differential equations yield thedesired Hamiltonians (H is not unique as will be shown later) and each ofthese Hamiltonians generates the motion in coordinate space (1.48). In thisformulation, the canonical momenta pi s are dummy variables and are related

Hamiltonian Formulation 9

to the coordinates and generalized velocities via Eqs. (1.45).q-equivalent Hamiltonians The complete solution of (1.49) even

for a simple one-dimensional motion with the force law F (q, q) is not known.However if we assume that F is derivable from a velocity-independent potentialfunction, i.e.

F (q) = (V (q)q

), (1.50)

then we can solve Eq. (1.49) in the following way:We write Eq. (1.49) as [11]

1m

V

q=2H

qp

H

p

2H

p2H

q, (1.51)

or rearranging the terms we have

q

{m

2

(H

p

)2+ V (q)

}= m

2H

p2H

q. (1.52)

We note that the quantity inside the curly bracket in (1.52) is the HamiltonianH0 which is also the total energy of the particle. Thus if G(H0) is an arbitraryfunction of H0, we have

G

p=

dG

dH0mq

q

p=

dG

dH0m2H

p2H

p. (1.53)

By substituting for m2Hp2 from Eqs. (1.52) in (1.53) it follows that

G

p

H

q=H

p

(dG

dH0

H0q

)=H

p

G

q, (1.54)

and therefore

H = G

[m

2

(H

p

)2+ V (q)

]. (1.55)

The canonical momentum, p, in this case is not equal to the mechanicalmomentum unless G(H0) = H0. To find the relation between p and mq, wewrite the Lagrangian L as

L = qpH(q, p), (1.56)and then use the definition of p to find it in terms of q

p =L

q= p+ q

p

q dGdH0

mq. (1.57)

Thus we have1m

p

q=

dG

dH0, (1.58)


or

p =

dG

dH0mdq + g(q), (1.59)

where g(q) is an arbitrary function of q. For systems with two or more degreesof freedom or when Fi is not derivable from a potential this method does notwork (see also [12],[13].)

We can also try to determine the Hamiltonian H(pi, qi) such that thecanonical equations (1.45) and (1.46) yield the motion of the system in phasespace. In this case, pi s are not dummy variables but are directly related to thegeneralized velocities, i.e. instead of (1.46) we find the solutions of (1.45) and

pi = H(pi, qi)qi

= Fi

(q1, q2 qN , H

p1 H

pN

)i = 1, 2, N. (1.60)

Even this set of equations of motion will not give us a unique Hamiltonian.Thus we can have a set of q-equivalent (or coordinate equivalent) or a smallerset of pq-equivalent for a given set of forces Fi. For the classical description ofmotion any member of either set is acceptable.Let us consider two simple examples:

(1) - Assuming that V (q) is positive for all values of q and taking G(H0) =

H120 , by solving (1.55) we get [11]

H(p, q) = A (V (q))12 cosh

(2mp

). (1.61)

(2) - If we choose H to be the inverse positive square root of energy,

G(H0) = H 120 we find [11]

H(p, q) =

[1 ( 2

m

)p2(V (q))2

V (q)

] 12

. (1.62)

pq-equivalent Hamiltonians Now let us examine pq-equivalent Hamil-tonians. For a one-dimensional case the equations of motion in coordinate spaceis

md2q

d t2+V (q)q

= 0, (1.63)

and the equations for the mechanical momentum mq = p can be written as

md2p

d t2+2V (q)q2

p = 0. (1.64)

The solution of these equations subject to the initial conditions give us q(t) andp(t).

Hamiltonian Formulation 11

The case of the harmonic oscillator V (q) = 12m2q2 is particularly simple

since the two equations uncouple, and we have

mdq

dtd

(dq

dt

)= m2qdq, (1.65)

and

mdp

dtd

(dp

dt

)= m2pdp. (1.66)

These equations can be integrated to yield

12

[(H

p

)2+ 2q2

]= C(H), (1.67)

12

[(H

q

)2+ 2p2

]= D(H), (1.68)

where C and D are arbitrary functions of H. Rather than trying to find themost general solution of (1.67) and (1.68), let us consider a class of solutionswhich we can find in the following way [14]. Let

H = (H0) = (p2

2m+

12m2q2

), (1.69)

then by differentiating H with respect to p and q and substituting in C(H) andD(H) we find that if

=H0

, (1.70)

thenC(H) =

1m

(

)2H1(H0), (1.71)

andD(H) = m2H1(H0), (1.72)

where is an energy-dependent frequency. Depending on our choice of wehave an infinite set of pq-equivalent Hamiltonians. For instance we can choosethe Hamiltonian to be

H = (H0) =Hj0j1

, (1.73)

where is a constant with the dimension of energy.Galilean Invariant Hamiltonians The same type of ambiguity which

we found for the Lagrangian L(), Eq. (1.21), also appears in the classicalHamiltonian formulation. Thus for a system of interacting particles when L()and pk are given by (1.21) and (1.23), the corresponding Hamiltonian obtainedfrom the definition H() =

k pk vk L() is

H() =k

p2k2mk

(

j pj)2

2+ 2jmj

+ V. (1.74)


1.4 Canonical Transformation

A set of transformationspj pj(Pi, Qi)

qj qj(Pi, Qi)j = 1, 2, N, (1.75)

is called canonical if the form of Hamiltons equations of motion, Eqs. (1.45)and (1.46), are preserved. Thus the canonical transformation is, in a way, anextension of the point transformation of N space coordinates qi qi(Qj) tothe transformation of 2N coordinates of (pi(Pj , Qj), qi(Pj , Qj)) space, called thephase space and is given by (1.75). Under the point transformation the EulerLagrange equations are left unchanged, whereas under the canonical transfor-mation the form of the canonical equations (1.45) and (1.46) remain the same.Thus if H(pi, qi) is transformed to K(Pj , Qj), then Eqs. (1.45) and (1.46) arereplaced by

Qi =K(Pj , Qj)

Pi, Pi = K(Pj , Qj)

Qi, i = 1, 2, N, (1.76)

respectively.We can derive the connection between H and K by noting that according

to the Hamilton principle Eq. (1.44)

Ldt =

j

pjdqj H(pi, qi, t)dt = 0, (1.77)

and when the same principle is applied to K(Pj , Qj) we find

Ldt =

j

PjdQj K(Pi, Qi, t)dt = 0. (1.78)

As we have seen earlier the difference between L and L

must be a totaltime derivative dF1dt . Thus

i

pidqi Hdt =i

PidQi Kdt+ dF1, (1.79)

where F1 is called the generating function for the canonical transformation.Rewriting (1.79) as

dF1(qi, Qi, t) =i

(pidqi PidQi) + (K H)dt, (1.80)

Canonical Transformation 13

we havepi =

F1qi

, (1.81)

Pi = F1Qi

, (1.82)

andK = H +

F1t

. (1.83)

A different but more useful case is when the generating function dependson old coordinates qi and the new momenta Pi. By writing (1.80) as

dF2 = d(F1(qi, Qi, t) +

i

PiQi

)=i

pidqi + (K H)dt, (1.84)

we obtainpi =

F2qi

, Qi =F2Pi

, K = H +F2t

. (1.85)

An interesting example of the generating function of this type, F2, is forthe motion of the particle in a time-dependent harmonic oscillator potentialwhere the Hamiltonian is

H(p, q, t) =1

2m[p2 + 2(t)q2

]. (1.86)

For this problem the generating function F2 is

F2 = m2(t)(t)

q2 q(t)

(2P q

2

2(t)

) 12

P sin1[

q

(t)

2P

]+(n+

12

)piP, (1.87)

where n is an integer and where

pi2 sin1

[q

(t)

2P

] pi

2. (1.88)

In Eq. (1.87) the signs are taken according to whether(pm (t)

(t)q)

ispositive or negative. The function (t) appearing in Eqs. (1.87) and (1.88) is asolution of the nonlinear differential equation

m2(t) + 2(t)(t) 13(t)

= 0, (1.89)

with the boundary conditions (0) = 1 and (0) = 0. In this problem F2depends explicitly on time, and therefore the new Hamiltonian K(P,Q, t) isgiven by

K(P,Q, t) = H(p, q, t) +F2t

=1m

(P

2(t)

). (1.90)


We note that

Q =F2q

= tan1 p2(t)qm(t)(t), (1.91)

and

P =12

[q2

2(t)+ ((t)pmq(t))2

]. (1.92)

The variable Q does not appear in K(P,Q, t), therefore Q is a cyclic coordinate,and the momentum conjugate to it, P , is a constant of motion [15].

Let us also note that for the special generating function F3 =i gi(qj , t)Pi,

the canonical transformation reduces to the point transformation (1.3).

1.5 Action-Angle Variables

A very important property of the canonical transformation is that it leaves thevolume in phase space unchanged. We want to show this important propertyof the transformation for a system with one degree of freedom. To prove thisresult which is referred to as Liouville theorem, let us consider the canonicaltransformation Q = Q(p, q), P = P (p, q). By taking the time derivative of Qwe have

Q =Q

pp+

Q

qq = Q

p

H

q+Q

q

H

p. (1.93)

Since the transformation is canonical and is independent of time, therefore thenew Hamiltonian is just the old Hamiltonian written in terms of Q and P ,H(p, q) H(P,Q). The form of the Hamilton canonical equations are thesame whether expressed in terms of (p, q) or (P, Q), Eq. (1.76). Thus

Q =H

P=H

q

q

P+H

p

p

P. (1.94)

By comparing (1.93) and (1.94) we find that

q

P= Q

p, and

p

P=Q

q. (1.95)

Similarly we calculate P by differentiating P (p, q) with respect to time andcomparing the result with P = HQ . In this way we get another pair ofequations:

P

p=

q

Q, and

P

q= p

Q. (1.96)

Now let us consider the transformed phase space areaPdQ =

Jdqdp, (1.97)

Action-Angle Variables 15

where J is the Jacobian of the transformation

J =D(P,Q)D(p, q)

=[ Pp

Pq

Qp

Qq

]. (1.98)

This Jacobian J can also be written as

J =D(P,Q)D(P,q)

D(p,q)D(P,q)

=

(Qq

)(pP

) = 1, (1.99)where in getting the last step we have used (1.95). Therefore Eq. (1.97) withJ = 1 shows that under canonical transformation the area (or volume) in phasespace is preserved.

Definition of the Action-Angle Variables We define the action-angle variables which we denote by {, I} in the following way: We first calcu-late the area under the curve p(q, E), where E is the energy of the particle

A(E) =p(q, E)dq = 2

q2q1

[2m(E V (q))] 12 dq. (1.100)

Then we find the area under the curve I(E);

A(E) =Id =

2pi0

Id = 2piI. (1.101)

Since the transformation {p, q} to {, I} is canonical therefore A(E) = A(E)and from Eqs. (1.100) and (1.101) we have

I =1

2pi

p(E, q)dq. (1.102)

Equation (1.102) defines the action variable. The conjugate of action variable is which varies between pi and pi. This relation can be generalized to a systemwith N degrees of freedom where for the k-th degree of freedom we have

Ik =1

2pi

pk(E, q1, , qN )dqk. (1.103)

This action is conjugate to the angle variable k, where k changes betweenpi < k pi. Thus all angle variables are periodic with period 2pi.

For a system of two degrees of freedom, the phase space is four-dimensionaland the toroid is a two-dimensional surface lying in a three-dimensional energyshell. We can generalize this concept to a system of N degrees of freedom inthe following way:

For the bounded motion of an integrable system we can transform thephase space coordinates (p, q) = (p1, p2, pN ; q1, q2, qN ) to (I, ) =(I1, I2, IN ; 1, 2, N ), where is an N -dimensional angle variable,


pi k pi. By this transformation the Hamiltonian becomes a function of I,H = H(I), i.e. all k s become cyclic coordinates. Thus we have

k =H

Ik=E(I)Ik

= k(I), (1.104)

andIk = H

k= 0. (1.105)

Since I is a constant vector in N -dimensions, Eq. (1.104) can be integrated toyield

k = kt+ k. (1.106)

In this relation k is a characteristic angular frequency of the motion and k isa phase shift. For a completely separable system k depends on Ik alone. Sincethe motion is periodic in k, therefore k(q, I) s are all multi-valued functions ofthe coordinates q. Now any single valued function A(p, q), when expressed interms of I and is a periodic function of the angle variables with each variablehaving a period 2pi. Hence we can expand the function A(I, ) of the dynamicalvariables, Ii, i in terms of the Fourier series

A(I, ) =

j1=

jN=

Aj1j2jN exp [i (j11 + j22 + + jnN )] .

(1.107)By substituting from (1.106) in (1.107) we have

A(I, ) =

j1=

jN=

Aj1j2jN exp[it

(j1E

I1+ j2

E

I2+ + jn E

IN

)].

(1.108)We note that each term in this sum is a periodic function of time with thefrequency

Nl=1

jlE

Il, (1.109)

but these frequencies are not generally commensurable, therefore the sum is nota periodic function of time. In particular we can choose A to be either pk or qkwhich shows that pk and qk are also non-periodic functions of time.

For an integrable system of N degrees of freedom, the motion of a phasepoint (p1, p2 pN ; q1, q2 qN ) in 2N -dimensional phase space will be con-fined to an invariant toroid of N dimensions. That is this invariant toroid occupythe whole phase space of bounded integrable motions.

If the system of N degrees of freedom is not integrable, subject to cer-tain conditions we can still express p and q in terms of k s. The parametricrepresentation of p and q as functions of N -dimensional angle variable

P = PT () = (pT (), qT ()) , (1.110)

Action-Angle Variables 17

Figure 1.1: The phase space point (1, 2, I1, I2) for a system with two degrees of freedom isperiodic in each 1 and 2 and this point moves in a region which is a toroid. The two curvesC1 and C2 define the action integrals I1 and I2.

defines an N -dimensional toroid, T , in the 2N -dimensional phase space [16][18]. The action in this case is

Ik =Ck

p dq, (1.111)

where p and q are the N -dimensional vectors and Ck s are N independent closedcurves on toroid [17]. In the case of a conservative system with two degrees offreedom such a toroid is shown in Fig. 1.1.

For the bounded motion of a system the action-angle variables makes thedescription of the motion very simple [16][18]. Let us consider a motion of onedegree of freedom with the Hamiltonian

H(p, q) =p2

2m+ V (q) = E, (1.112)

where E, the total energy, is a constant. If we solve (1.112) for p(q, E) we findthe two-valued function

p(q, E) =

2m(E V (q)). (1.113)

Since this canonical variable is multivalued, we try to find another set of con-jugate variables (, I) in such a way that H becomes independent of , and Iand becomes a constant of motion. Suppose that we have found these variables,then we have the canonical equations of motion

dI

dt= 0 = H(I)

, (1.114)


andd

dt=H(I)I

= constant. (1.115)

Equation (1.115) can be integrated easily with the result that is a linearfunction of time

= (I)t+ . (1.116)

1.6 Poisson Brackets

The Hamilton canonical equations (1.45) and (1.46) can be written in a conciseform if we introduce the Poisson bracket of two dynamical variables u(pi, qi)and v(pi, qi) by

{u(pi, qi), v(pi, qi)}pi,qi =Ni=1

(u

qi

v

pi upi

v

qi

). (1.117)

From this definition it readily follows that

{qi, pj} = ij , {qi, qj} = {pi, pj} = 0. (1.118)Also by direct differentiation we find

{u, v}pi,qi =Nj=1

{u, v}Pj ,Qj{Qj , Pj}pi,qi . (1.119)

If the transformation

Qj Qj(pi, qi, t), and Pj Pj(pi, qi, t), (1.120)is canonical then

i

(pidqi PidQi) , (1.121)

is a complete differential and we have

{Qj , Pj}pi,qi = ij . (1.122)Thus (1.119) reduces to

{u, v}pi,qi = {u, v}Pi,Qi . (1.123)This result shows that the Poisson bracket {u, v}pi,qi remains unchanged bya canonical transformation of one set of canonical variables to another. Tosimplify the notation we suppress the subscripts pi, qi and denote the Poissonbracket by {u, v}.

Poisson Brackets 19

Some of the other properties of Poisson bracket are as follows:(1) - Skew symmetry

{u, v} = {v, u}. (1.124)(2) - Linearity

{u+ v,w} = {u,w}+ {v, w}, (1.125)where and are constants.

(3) - Leibniz property

{uv,w} = u{v, w}+ v{u,w}, (1.126)where u, v and w are functions of pi and qi.

(4) - Jacobi identity

{{u, v}, w}+ {{v, w}, u}+ {{w, u}, v} = 0. (1.127)The first three results follow directly from the definition of the Poisson bracketand the proof of the last relation is straightforward but lengthy (see Ref. [1]).

Now let be some function of the pi s and qi s and time, then its totaltime derivative is given by

ddt

=t

+j

(qj

qj +pj

pj

). (1.128)

Substituting for qj and pj from canonical equations (1.45) and (1.46) we obtain

ddt

=t

+ {H,}. (1.129)

If does not explicitly depend on time and remains a constant of motion, then

{H,} = 0, (1.130)and is called an integral of motion.

Written in terms of the Poisson brackets the equation of motion (1.49)has a simple form

1mi

Fi = {H, {H, qi}} . (1.131)Poisson Brackets for Galilean Invariant Hamiltonian Again it

is interesting to examine the Poisson bracket found from the Lagrangian (1.21)and its Hamiltonian counterpart (1.74). According to Eq. (1.23) the relationbetween pk and vk involves the velocity of the other particles. For simplicity letus consider the case of two interacting particles in one dimension, where from(1.24) we have

v1 =p1m1 p1 + p2+m1 +m2

, and v2 =p2m2 p1 + p2+m1 +m2

. (1.132)


From these relations we can calculate the Poisson brackets

{x1, m1v1} =(

1 +m2

)(

+m1 +m2

), (1.133)

{x1, m2v2} =(m2

)(

+m1 +m2

), (1.134)

and similar relations for {x2, m2v2} and {x2, m1v1}. Thus for an arbitraryvalue of all the coordinates xi fail to have a vanishing Poisson bracket with allthe canonical momenta mixi. Therefore, in quantum theory, for the Hamilto-nian H(), none of these quantities can be sharply measured. As (1.133) showsthe Poisson bracket of the coordinate x1 and the mechanical momentum m1v1is zero if we choose = m2. This result indicates that if the Lagrangian suchas (1.21) is used as a classical basis for the quantum mechanical formulationthen it is possible to violate the uncertainty principle x(m1v1) [10].

1.7 Time Development of Dynamical Variablesand Poisson Brackets

Let us consider a dynamical quantity u(pi, qi) and let us assume that its initialvalue u(pi(0), qi(0)) is known. We want to find the time evolution of this quan-tity. Suppressing the canonical coordinates and momenta we write u(t) as aTaylor series

u(t) = u(0) +t1!

(du

dt

)t=0

+(t)2

2!

(d2u

d t2

)t=0

+ . (1.135)

We express the derivatives of u in terms of the Poisson brackets(du

dt

)t=0

= {u,H}t=0 ,(d2u

d t2

)t=0

= {{u,H} , H}t=0 , . (1.136)

Substituting these derivatives in (1.135) we obtain

u(t) = u(0) +t1!{u,H}t=0 +

(t)2

2!{{u,H} , H}t=0 + . (1.137)

This last relation shows that u(t) can be expressed in terms of the initialvalues pi(0) and qi(0). For instance if we want to solve the one-dimensionalproblem of a harmonic oscillator with the Hamiltonian

H =1

2mp2 +

12m2q2, (1.138)

Infinitesimal Canonical Transformation 21

and determine the coordinate q(t) in terms of q(0) and p(0) we first find thePoisson brackets

{q,H}t=0 =p(0)m

, (1.139)

{{q,H} ,H}t=0 = 2q(0), (1.140)

{{{q,H} ,H} ,H} = 2

mp(0), (1.141)

and so on. Then from the expansion

q(t) = q(0) +t1!{q,H}t=0 +

(t)2

2!{{q,H} ,H}t=0 + , (1.142)

we get

q(t) = q(0)[1

2(t)2

2!+4(t)4

4!

]+

p(0)m

[t

2(t)3

3!+

]= q(0) cos(t) +

p(0)m

sin(t), (1.143)

a solution which is valid for all values of (t). We will see later that the quantumversion of (1.137) can be used to solve the Heisenberg equation of motion.

1.8 Infinitesimal Canonical Transformation

If the generator of the canonical transformation F2(pi, qi, t) is of the form

F2(Pi, qi, t) =i

Piqi + G(Pi, qi, t), (1.144)

where is a very small positive number, then according to (1.85) we have

Qi =F2Pi

= qi + G

Pi qi + G

pi, (1.145)

andpi =

F2qi

= Pi + G

qi. (1.146)

i.e. the canonical coordinates and momenta are changed by an amount propor-tional to . In this case G is called the generator of the infinitesimal transfor-mation. We can also write (1.145) and (1.146) as

qi = Qi qi = Gpi

, (1.147)


and

pi = Pi pi = Gqi

. (1.148)

The infinitesimal transformation generated by G takes a simple form if wewrite it in terms of the Poisson bracket. Let us consider an arbitrary dynamicalvariable (pi, qi, t) and determine how it will be changed by an infinitesimalcanonical transformation. We note that after such a transformation (pi, qi, t)changes to

(pi + pi, qi + qi, t) = (pi, qi, t) +Ni=1

(qi

qi +pi

pi

)

= (pi, qi, t) + Ni=1

(qi

G

pi pi

G

qi

)= (pi, qi, t) + {, G},

(1.149)

where we have used Eqs. (1.147) and (1.148). Denoting the difference (pi +pi, qi + qi, t) (pi, qi, t) by (pi, qi, t), this last relation can be written as

(pi, qi, t) = {, G}. (1.150)

In particular if we choose (pi, qi, t) to be the Hamiltonian of the system, H,then

H = {H,G} = (dG

dt G

t

). (1.151)

Here Eq. (1.129) has been used to write H in terms of the time derivativesof G. This relation shows that if G does not depend on time explicitly, and ifH is invariant under the infinitesimal transformation, then dGdt = 0 and G is aconstant of motion (i.e. a conserved quantity).

Among the constants of motion for a system of interacting particles thefollowing constants are of particular interest:

(a) - The infinitesimal generator for space translation of two interactingparticles, when the potential between them is V (|r1 r2|) can be written as

G = (p1 + p2) . (1.152)

(b) - The generator for the rotation about the axis a by an angle is

G = a (r p). (1.153)

(c) - For the Galilean transformation of a system of n particles we have{ ri ri + vtpi pi +miv

, i = 1, n, (1.154)

Action Principle 23

and the generator for transformation by infinitesimal velocity v is given by

v N = v ni=1

(pitmiri) . (1.155)

In the case of the Galilean transformation given by Eqs. (1.154) and(1.155), the vector N for a system of n interacting particles isolated from therest of universe is a constant of motion

dNdt

=ni=1

(pi miri) +ni

dpidt

= 0. (1.156)

Let us note that while under this transformation the equations of motion remaininvariant, the Hamiltonian H will change. For example for a single particle,H = p

2

2m , changes to H where

H =1

2m(p +mv)2 . (1.157)

Combining these transformations we obtain the most general form of in-finitesimal transformation of space-time displacements for a system of n parti-cles. Such a transformation is given by

G = P + J + v N tH, (1.158)

where P =ni=1 pi is the total momentum of the system, J is its total angular

momentum, H is the total Hamiltonian, and N which is called the boost, de-scribes the Galilean transformation.

1.9 Action Principle with Variable End Points

Earlier we discussed the standard approach for obtaining the equations of motionfrom the action principle. However it is possible to perform a more generalvariation, where in addition to qj(t) s we vary the time and the end points t1and t2. The transformation of qj(t) and t are as follows:

qj(t) qj(t) = qj(t) + qj , t t = t+ t. (1.159)

These variations generate a variation S of the action

S = t2+t2t1+t1

L (qj + qj , qj + qj , t+ t) dt t2t1

L (qj , qj , t) dt. (1.160)


Using the usual technique of the calculus of variation we find S to be

S = t2t1

Nj=1

(L

qj ddt

L

qj

)qjdt+

Nj=1

pj qj Htt2t1

, (1.161)

where in (1.161) H and qj are defined by

H =Nj=1

pj qj L, (1.162)

and

qj = qj(t+ t) qj(t) = qj(t) qj(t) + qjt= qj + qjt. (1.163)

The first term on the right hand-side of (1.161) is zero since L satisfies theEulerLagrange equation (1.2), therefore

S = (G(t2)G(t1)), (1.164)where

G(t) =Nj=1

pj qj Ht. (1.165)

This quantity G(t) is the generator of a canonical transformation, i.e. for anydynamical variable (pk, qk) the Poisson bracket Eq. (1.150) is satisfied by and G(t).Let us consider the following two cases:

(1) - Ifqj qj + qj , pj pj , and t t, (1.166)

then

G(t) =Nj=1

pj qj , (1.167)

and{qj , G(t)} = qj , and {pj , G(t)} = 0, (1.168)

as is expected.(2) - If qj = pj = 0 and t t+ t then

{qj , G(t)} = {qj , Ht} = Hpj

t = qjt, (1.169)

and{pj , G(t)} = {pj , Ht} = H

qjt = pjt. (1.170)

Action Principle 25

Since qj = pj = 0,

qj = qj + qjt = 0, pj = pj + pjt = 0, (1.171)

we haveqj = {qj , G(t)} , (1.172)

andpj = {pj , G(t)} . (1.173)

These results show that G(t) is indeed the generator of infinitesimal transfor-mation [19],[20].

Quantum Version of the Action Principle for Cases Where{qj , qk} Depends on qk s It is important to note that this formulationof the Schwinger action principle when applied to quantum mechanical prob-lems yields the correct result in the cases where the commutator

[qj , qk] , (1.174)

is a c-number, i.e.(ihm jk

). As we have seen before, for a general curvilinear co-

ordinate, the Hamiltonian is a complicated quadratic function of pj , and thus inquantum theory the corresponding commutator, (see Eq. (1.174)), will dependon qk and is a q-number. For these and also the problems related to dissipativeforces depending on velocity, the action integral corresponding to the classicalHamiltons principle should be modified.

The classical Hamiltonian function which we defined by the variation ofthe action integral, Eq. (1.44), is now replaced by the quantum action integral

S (qi, qi, pii, t1, t2) = t2t1

{k

12[pik, q

k]+

+D (qi, pii)H(qi, pii, t)} dt,(1.175)

where [ , ]+ denotes the anticommutator[pik, q

k]+

= pik qk + qkpik, (1.176)

and all of the operators are assumed to be Hermitian. In Eq. (1.175) theHermitian operator D is introduced such that the q-number variation (DH)of (D H) differs from

12

[H

qk, qk

]+

+12

[H

pik, pik

]+

+H

tt, (1.177)

by a total time derivative of an operator G.This operator, D (qi, pii), is uniquely determined by the variation H of

the Hamiltonian. Thus if t is not varied, then the variation t of D is such thatthe relation

tD dGdt

= tH 12k

[H

qk, qk

]+

12

k

[H

pik, pik

]+

, (1.178)


holds. Furthermore just as in classical dynamics we assume that and ddtcommute with each other;

d

dtqk =

(dqk

dt

),

d

dt(t) =

(dt

dt

)= 0. (1.179)

Under these conditions the variation of S in the action integral (1.175) willdepend on the end points t1 and t2;

S = J(t2) J(t1). (1.180)Now let us consider the variation of S [21],[22].

S = t2t1

d

{12

k

[pik, q

k]+

+ (D H)t}

+ t2t1

12

k

[dqk H

pikdt, pik

]+

t2t1

12

k

[dpik +

H

qkdt, qk

]+

+ t2t1

(dH H

tdt

)t. (1.181)

By assuming that the variations qk, pik and t are all independent, from(1.181) it follows that

d

dtqk =

H

pik, (1.182)

d

dtpik = H

qk, (1.183)

andd

dtH = H

t, (1.184)

J =12

k

[pik, q

k]+

+ (D H)t. (1.185)

Thus we have obtained Heisenbergs equations of motion from the action integralwithout assuming that

[qj , qk

]is a c-number. We note that here qk and pij are

canonically conjugate variables, i.e.[qj , pik

]= ihjk, (1.186)

while the other commutators of qj , qk and pij , pik vanish.Equations (1.182)(1.184) are Heisenbergs equations of motion, however

unlike the c-number variation discussed earlier, (1.184) cannot be derived from(1.182) and (1.183). Moreover the canonical commutation relation that we foundby the Schwinger variational principle cannot be obtained directly from theseequations.

Examples Let us consider two forms of Hamiltonian for the motion of

Symmetry and Degeneracy 27

a particle given in the curvilinear coordinates (see Eq. (3.216)). The first oneis

H(qj , pij

)=

1

2m

i,j

1g

(piig gijpij

)+ V

(qj), (1.187)

where g and gij are functions of qj s. The second form is that of the Hamiltonian

H(qj , pij

)=i,j

1

2m

(pii g

ijpij)

+ V(qj), (1.188)

which has been used to describe the velocity-dependent forces between twonucleons [12]. For these Hamiltonians the velocity operator is defined by

qk =H

pik=

1

2

[pij , g

jk]+. (1.189)

Using (1.189) we can determine the commutator of qj and qk operator which isdefined by [

qj , qk]

= ihgjk, (1.190)

and is not a c-number.

1.10 Symmetry and Degeneracy inClassical Dynamics

When G(qi, pi) is a first integral of motion which does not depend explicitly ontime then from (1.151) it follows that {H,G} = 0. Now suppose that we havefound a number of independent first integrals of motion G1(qi, pi), , Gn(qi, pi)then each of these Gi s can be used to generate a group of infinitesimal canonicaltransformation (1.149) and (1.150), i.e.

qk = {qk, Gi}, pi = {pk, Gi}. (1.191)These continuous transformations will make a group provided that the firstintegrals, Gk, satisfy the conditions

{Gk, Gm} =i

CikmGi, (1.192)

where Cikm are constants which may also depend on the total energy. In whatfollows we consider three specific examples with different symmetry groups.

Isotropic Two-Dimensional Harmonic Oscillator As a first ex-ample consider the simple case of an isotropic harmonic oscillator given by theHamiltonian

H =1

2m

(p21 + p

22

)+

1

2m2

(q21 + q

22

). (1.193)


There are three first integrals of motion G1, G2 and G3;

G1 =12

(q1p2 p1q2) , (1.194)

G2 =14

[1m

(p21 p22

)+m

(q21 q22

)], (1.195)

and

G3 =12

(1m

p1p2 +mq1q2

). (1.196)

The first equation G1 is the angular momentum and the second one is the energydifference between the two oscillators which is known as correlation [23], [24].

The set of functions G1, G2 and G3 is closed under the Poisson bracketoperation, i.e.

{Gi, Gj} = Gk, (1.197)where the indices form a cyclic permutation of 1, 2 and 3. In the case of theisotropic harmonic oscillator (1.193) we have a conserved second rank tensorwith the components

Tij =pipj2m

+12m2xix. (1.198)

By differentiating Tij with respect to t and then substituting for pi and pi fromthe equations of motion we find that this derivative is zero and Tij is a constant.Here a second rank tensor rather than a vector is conserved since there are twoaxes of symmetry, the semi-major as well as the semi-minor axes. On the otherhand in the Kepler problem which we will consider next, the conserved quantityis a vector and there is one axis of symmetry.

Two-Dimensional Kepler Problem For the Kepler problem in twodimensions with the Hamiltonian

H =1

2m(p21 + p

22

) Ze2(q21 + q

22)

12, (1.199)

we have the following first integrals

L3 = q1p2 q2p1, (1.200)

R1 = L3p2mZe2

+q1

(q21 + q22)

12, (1.201)

andR2 =

L3p1mZe2

+q2

(q21 + q22)

12. (1.202)

The vector R with components R1 and R2 is the two-dimensional form of theRungeLenz vector (see the next section) [2]. By calculating the Poisson brack-ets for L3, R1 and R2 we find

{L3, R1} = R2, (1.203)

Symmetry and Degeneracy 29

{R2, L3} = R1, (1.204)and

{R1, R2} = 1mZ2e4

(2H)L3. (1.205)We can transform these to the standard form of (1.192) by choosing

G1 =mZe22|E| R1, (1.206)

G2 =mZe22|E| R2, (1.207)

andG3 = L3, (1.208)

noting that E which is the total energy of the particle is negative.The RungeLenz vector R, with the components given by (1.201) and

(1.202) can be written as a vector equation

R =1m

(p L) Ze2

, (1.209)

where is the position vector with components q1 and q2. This vector is directedfrom the focus of the orbit to its perihelion (or the point of closest approach).By multiplying (1.209) by we find

R = R cos = 1m (p L) Ze2. (1.210)

Using the vector identity

(p L) = L ( p) = L2, (1.211)from Eq. (1.210) we obtain the equation of orbit

() =L2

Ze2 +R cos=

d

1 + cos, (1.212)

where d = L2

Ze2 is the semi-latus rectum of the ellipse and =R

mZe2 is its eccen-tricity. Note that R vanishes for a circle ( = 0) since there is no unique majoraxis.

Two-Dimensional Anisotropic Harmonic Oscillator The prob-lem of a special two-dimensional anisotropic oscillator provides an examplewhere the symmetries and the group structure of the classical motion is notcarried over to the quantized motion.

Consider the Hamiltonian of this system for the general case which is givenby

H =(p21

2m1+

12m1

21q

21

)+(p22

2m2+

12m2

22q

22

). (1.213)


If the ratio of the two frequencies (1/2) is rational then the classical orbitcloses on itself. Let us assume that there are integers m and n such that

n1 = m2 = , (1.214)

For this case we introduce two constants 1 = m11 and 2 = m22 and twovariables bi and bi

bi =pi iiqi

2i, bi =

pi + iiqi2i

, i = 1, 2, (1.215)

and write the two first integrals as

F1 =12

[bn1 (bm2 ) + (bn1 )

bm2 ] , (1.216)

andF2 =

i2

[bn1 (bm2 ) (bn1 )bm2 ] . (1.217)

The three Gi s which form the group (1.192) are given by [25]

G1 =12

F1 (b1b1) 12 (n1) (b2b2)

12 (m1) , (1.218)

G2 =12

F2 (b1b1) 12 (n1) (b2b2)

12 (m1) , (1.219)

andG3 =

12

(b2b2 b1b1) . (1.220)We note that here G1 and G2 are not always algebraic and for some m and nvalues they are multivalued functions. When any one of the Gi s is not alge-braic, then the existence of its quantum counterpart is in doubt [25].

1.11 Closed Orbits and Accidental Degeneracy

For the three-dimensional Kepler problem in addition to the angular momentumvector, there is the RungeLenz vector, R, which is conserved [24]. This vector,R, is in the direction of the semi-major axis of the elliptic orbit, and since it isa constant vector the orbit is closed and there is no precession of the axis. Thevector R, the angular momentum vector L and the associated elliptic orbit of themotion are shown in Fig. 1.2. More about this vector will be mentioned whenwe discuss Paulis method of obtaining the spectra of the hydrogen atom by thematrix method in Sec. 9.8. First, let us consider the types of attractive centralpotentials that lead to closed orbits. According to Bertrands theorem the

Closed Orbits and Accidental Degeneracy 31

Figure 1.2: The RungeLenz vector, R and the angular momentum vector L for an ellipticorbit of the classical problem with k

rpotential (Kepler problem).

only central forces giving closed orbits for all bound particles are the Coulombpotential (or Kepler problem) and the three-dimensional harmonic oscillator[26][28].

For noncentral forces we can have accidental degeneracy for a wide classof potentials [29],[30]. For instance consider the two-dimensional motion of aparticle in the xy-plane when the Hamiltonian is given by

H =1

2m(p2x + p

2y

)+

12m2x2

[(x)

(1 + 1)2+

(x)(1 1)2

]+

12m2y2

[(y)

(1 + 2)2+

(y)(1 2)2

], (1.221)

where (x) is the step function

(x) ={ 0 for x < 0

1 for x > 0, (1.222)

and 1 and 2 satisfy the inequalities

1 < 1 < 1, 1 < 2 < 1. (1.223)We can write (1.221) in terms of the action-angle variables defined by [2]

Ix =1

2pi

[2m( V1(x))]

12 dx, (1.224)

andIy =

12pi

[2m(E V2(y))]

12 dy, (1.225)


1 0.5 0.5

1

y

x

1

2

2

1

Figure 1.3: The closed orbit found from the Hamiltonian (1.221) for m = 1, = 1, 1 = 0.4and 2 = 0.2. Apart from the energy there are no apparent symmetries or conserved quantitiesfor this motion.

where E is the total energy and is the separation constant. The potentialsV1(x) and V2(x) are given by the third and fourth terms in (1.221) respectively.By carrying out the integrations in (1.224) and (1.225) and replacing E by Hwe find a simple expression for the Hamiltonian

H = (Ix + Iy). (1.226)

This clearly shows the degeneracy of motion since the two periods given by

x =1

2piH

Ix=

2pi, y =

12pi

H

Jx=

2pi, (1.227)

are equal. A closed orbit for the noncentral force derived from the Hamiltonian(1.221) is displayed in Fig. 1.3.The example that we have just considered is for a potential which is separable inCartesian coordinates, but one can construct similar cases in other coordinates[29].

As we will discuss later these classically degenerate systems may or maynot remain degenerate in quantum mechanics. In other words closed orbits inclassical motion do not necessarily imply accidental degeneracies in their quan-tum spectra.

1.12 Time-Dependent Exact Invariants

For some time-dependent potentials, Noethers theorem may not be an easyway to find conserved quantities. This is the case for example, when we want to

Exact Invariants 33

determine a first integral of motion which explicitly depends on time. Howeverwe find the conserved quantity by the application of a method devised by Lewisand Leach [31].

Let us consider the one-dimensional Hamiltonian for a particle of unitmass

H =12p2 + V (q, t), (1.228)

and denote the invariant first integral by I(p, q, t). Since I(p, q, t) is invariant,we have

dI

dt=I

t+ {I,H} = I

t+(I

q

H

p Ip

H

q

)= 0. (1.229)

For the general solution of (1.229) when H is given by (1.228) we assume thatI(p, q, t) can be expanded as a power series in p. However this procedure, unlessthe power series terminates, does not give us a simple expression for I(p, q, t).Therefore let us limit our investigation to the possibility of obtaining a functionI(p, q, t) which is at most quadratic in p;

I(p, q, t) = p2f2(q, t) + pf1(q, t) + f0(q, t). (1.230)

By substituting (1.228) and (1.230) in (1.229) and equating the coefficients ofp3, p2, p and p0 equal to zero we find

f2(q, t)q

= 0, (1.231)

f2(q, t)t

+f1(q, t)q

= 0, (1.232)

f0(q, t)q

+f1(q, t)

t 2f2(q, t)V (q, t)

q= 0, (1.233)

andf0(q, t)

t f1(q, t)V (q, t)

q= 0. (1.234)

Integrations of (1.231) and (1.232) are trivial:

f2(q, t) = 2a(t), f1(q, t) = b(t) 2a(t)q, (1.235)where a(t) and b(t) are arbitrary functions of time and where the dot denotedifferentiation with respect to time. If we substitute for f1(q, t) and f2(q, t) in(1.233) we get

f0(q, t)q

4a(t)V (q, t)q

+ b(t) 2a(t)q = 0. (1.236)

This equation can easily be integrated with the result that

V (q, t) =1

4a(t)

[f0(q, t) a(t)q2 + b(t)q

]+ g(t), (1.237)


where again g(t) is an arbitrary function of t. If we substitute (1.235) and(1.237) in (1.233) we find

(2a(t)q b(t))(f0(q, t)q

2a(t)q + b(t))

+ 4a(t)f0(q, t)

t= 0. (1.238)

This is a partial differential equation of first order for f0(q, t).Using the method of characteristics we solve (1.238) to get the general

solution of this partial differential equation [32]

f0(q, t) = G[

qa(t)

+14

t b(t)dta

32 (t)

]+

12a(t)

(a(t)q 1

2b(t))2

. (1.239)

In thisexpression G is an arbitrary function of its argument. It is simpler torewrite (1.239) in terms of 1(t), (t) and G(z) where these functions of timeare defined by:

1(t) = 2a(t), (1.240)

(t) = 1(t)8

t b(t)dta

32 (t)

, (1.241)

andG(z) = G(2z). (1.242)

Equation (1.237) shows that the general form of the potential consists of twoparts: the fixed part which is a quadratic function of q and an arbitrary part

V (q, t) = F (t)q + 12

2(t)q2 +1

21(t)U(q (t)1(t)

), (1.243)

where U is an arbitrary function of its argument.Now comparing (1.243) with (1.237) we find that the arbitrary functions

F (t),2(t), 1(t) and (t) are all related by the following relations:

1(t) + 2(t)1(t) K31(t)

= 0, (1.244)

(t) + 2(t)(t) = F (t). (1.245)

with K a constant. In terms of the functions (t) and 1(t) we can write theinvariant function I(p, q, t) as

I(p, q, t) =12

[1(t) (p (t)) 1(t)(q (t))]2

+12K

(q (t)1(t)

)2+ U

(q (t)1(t)

). (1.246)

For instance if the Hamiltonian is given by

H =12p2 +

12

2(t)q2, (1.247)

Bibliography 35

and if we also choose G to beG = 1

2x2, (1.248)

then 1(t) satisfies the differential equation

1(t) + 2(t)1(t) 131

= 0. (1.249)

A scale transformation given by

1(t) =10(t). (1.250)

where 0 is a constant, changes (1.249) to the form

(t) + 2(t)(t) 20

3= 0. (1.251)

This is the form that later we will use in the corresponding quantum mechanicalproblem.

Bibliography

[1] L.D. Landau and E.M. Lifshitz, Mechanics, (Pergamon Press, 1960), p.136.

[2] H. Goldstein, C. Poole and J. Safco, Classical Mechanics, Third Edition(Addison-Wesley, San Francisco, 2002).

[3] H. Helmholtz, Uber die physikaliche Bedeutung Prinzips der kleinstenWirkung, J. Reine und Angew. Math. 100, 137 (1887).

[4] R.M. Santilli, Foundations of Theoretical Mechanics, (Springer, New York,1987).

[5] P. Havas, The range of application of the Lagrange formalism, Nuovo Ci-mento. 5, 363 (1957).

[6] P. Havas, The connection between conservation laws and the invariancegroups: Folklore, fiction and fact, Acta Physica Austriaca, 38, 145 (1973).

[7] R.J. Hughes, On Feynmans proof of the Maxwell equations, Am. J. Phys.60, 301 (1992).

[8] Y. Takahashi, Lecture notes, University of Alberta, (unpublished) (1991).

[9] E.H. Kerner, How certain is the uncertainty principle?, Contemp. Phys. 36,329 (1995).


[10] E.H. Kerner, An essential ambiguity of quantum theory, Found. Phys. Lett.7, 241 (1994).

[11] F.J. Kennedy, Jr. and E.H. Kerner, Note on the inequivalence of classicaland quantum Hamiltonians, Am. J. Phys. 33, 463 (1965).

[12] D.G. Currie and E.J. Saletan, q-equivalent particle Hamiltonians, I: Theclassical one-dimensional case, J. Math. Phys. 7, 967 (1966).

[13] G. Gelman and E.J. Saletan, q-equivalent particle Hamiltonians, II: Thetwo-dimensional classical oscillator, Nuovo Cimento. 18B, 53 (1973).

[14] M. Razavy, Some counterexamples in quantum mechanics, Iranian J. Sci.Tech. 23, 329 (2001).

[15] H.R. Lewis, Jr., Classical and quantum systems with time-dependentharmonic-oscillator-type Hamiltonians, Phys. Rev. Lett. 18, 510 (1967).

[16] For a simple account of the action-angle variables see I. Percival and D.Richards, Introduction to Dynamics, (Cambridge University Press, Cam-bridge, 1982), Chapter 7.

[17] I.C. Percival, Variational principles for the invariant toroids of classicaldynamics, J. Phys. A. 7, 794 (1974).

[18] I.C. Percival and N. Pomphrey, Vibrational quantization of polyatomicmolecules, Mol. Phys. 31, 97 (1976).

[19] J. Schwinger, The theory of quantized fields. I, Phys. Rev. 82, 914 (1951).

[20] See P. Roman, Advanced Quantum Mechanics, (Addison-Wesley, Reading1965), Chapter 1.

[21] M.Z. Shaharir, The modified Hamilton-Schwinger action principle, J. Phys.A7, 553 (1974).

[22] H.A. Cohen and M.Z. Shaharir, The action principle in quantum mechanics,Int. J. Theo. Phys. 11, 289 (1974).

[23] V.A. Dulock and H.V. McIntosh, On the degeneracy of the two-Dimensionalharmonic oscillator, Am. J. Phys. 33, 109 (1965).

[24] H.V. McIntosh, Symmetry and Degeneracy in Group Theory and its appli-cations, edited by E.M. Loebl, (Academic Press, New York, 1971), Vol. II,p. 28.

[25] J.M. Jauch and E.L. Hill, On the problem of degeneracy in quantum me-chanics, Phys. Rev. 57, 641 (1940).

[26] L.S. Brown, Forces giving no orbit precession, Am. J. Phys. 46, 930 (1978).

Bibliography 37

[27] D.M. Fradkin, Existence of the dynamic symmetries O4 and SU3 for allclassical central potential problems, Prog. Theor. Phys. 37, 798 (1967).

[28] D.F. Greenberg, Accidental degeneracy, Am. J. Phys. 34, 1101 (1966).

[29] M. Razavy, Quantization of periodic motions with nonprecessing orbits andthe accidental degeneracy of the spectral lines, Hadronic J. 7, 493 (1984).

[30] J.F. Marko and M. Razavy, Dynamical symmetries of a class of actionequivalent Hamiltonians, Lett. Nuovo Cimento 40, 533 (1984).

[31] H.R. Lewis and P.G.L. Leach, A direct approach to finding exact invariantsfor one-dimensional time-dependent classical Hamiltonian, J. Math. Phys.23, 2371 (1982).

[32] See for instance, C.R. Chester, Techniques in Partial Differential Equa-tions, (McGraw-Hill, New York, 1971), Chapter 8.

Chapter 2

Discovery of MatrixMechanics

Historians of science unanimously agree that the new quantum mechanics wasborn on July 29, 1925 with the publication of a paper by Heisenberg under thetitle of Uber quantentheoretische Umdeutung kinematischer und mechanischerBeziehungen [1][10]. The novel ideas contained in this paper paved the wayfor a complete departure from the classical description of atomic physics, andadvanced a new formulation of the laws of micro-physics. First and foremostHeisenberg thought of replacing the classical dynamics of the Bohr atom by aformulation based exclusively on relations between quantities that are actuallyobservables. Thus he abandoned the idea of determining the coordinates andmomenta of the electron as functions of time, but at the same time he retainedthe mathematical form of the second law of motion. Now if Bohrs orbits are notobservables, the spectral lines and their intensities are. In the classical Larmorformula for radiation from an accelerating electron the energy radiated by theelectron, in cgs units is [11]

P =(

2e2

3c3

)x2, (2.1)

where e is the charge of the electron, c the speed of light and x is the ac-celeration. For a harmonically bound electron, the average power for the -thharmonic with amplitude x(n) is

P =4e2

3c3[ (n)]4 |x(n)|2. (2.2)

39


For such a bound electron, the position x(n, t), where the stationary state islabeled by n, can be written as a Fourier series

x(n, t) =

=ae

i(n)t. (2.3)

Heisenberg observed that according to

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