heisenbergs quantum mechanics

HEISENBERG’S

QUANTUM
MECHANICS

7702 tp.indd 1

10/28/10 10:20 AM


This page intentionally left blank


HEISENBERG’S

QUANTUM
MECHANICS

Mohsen Razavy
University of Alberta, Canada

World Scientific
NEW JERSEY

7702 tp.indd 2

•

LONDON

•

SINGAPORE

•

BEIJING

•

SHANGHAI

•

HONG KONG

•

TA I P E I

•

CHENNAI

10/28/10 10:20 AM


Published by
World Scientific Publishing Co. Pte. Ltd.
5 Toh Tuck Link, Singapore 596224

USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.

HEISENBERG’S QUANTUM MECHANICS
Copyright © 2011 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
electronic or mechanical, including photocopying, recording or any information storage and retrieval
system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright
Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to
photocopy is not required from the publisher.

ISBN-13
ISBN-10
ISBN-13
ISBN-10

978-981-4304-10-8
981-4304-10-7
978-981-4304-11-5 (pbk)
981-4304-11-5 (pbk)

Printed in Singapore.

ZhangFang - Heisenberg's Quan Mechanics.pmd
1

10/19/2010, 11:16 AM


Dedicated to my great teachers
A.H. Zarrinkoob, M. Bazargan and J.S. Levinger


This page intentionally left blank


Preface
There is an abundance of excellent texts and lecture notes on quantum theory
and applied quantum mechanics available to the students and researchers. The
motivation for writing this book is to present matrix mechanics as it was first
discovered by Heisenberg, Born and Jordan, and by Pauli and bring it up to
date by adding the contributions by a number of prominent physicists in the intervening years. The idea of writing a book on matrix mechanics is not new. In
1965 H.S. Green wrote a monograph with the title “Matrix Mechanics” (Nordhoff, Netherlands) where from the works of the pioneers in the field he collected
and presented a self-contained theory with applications to simple systems.
In most text books on quantum theory, a chapter or two are devoted to
the Heisenberg’s matrix approach, but due to the simplicity of the Schr¨
odinger
wave mechanics or the elegance of the Feynman path integral technique, these
two methods have often been used to study quantum mechanics of systems with
finite degrees of freedom.
The present book surveys matrix and operator formulations of quantum
mechanics and attempts to answer the following basic questions: (a) — why
and where the Heisenberg form of quantum mechanics is more useful than other
formulations 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 original

literature and collected those that I thought are novel and interesting. My first
inclination was to expand on Green’s book and write only about the matrix
mechanics. But this plan would have severely limited the scope and coverage of
the book. Therefore I decided to include and use the wave equation approach
where it was deemed necessary. Even in these cases I tried to choose the approach which, in my judgement, seemed to be closer to the concepts of matrix
mechanics. For instance in discussing quantum scattering theory I followed the
determinantal approach and the LSZ reduction formalism.
In Chapter 1 a brief survey of analytical dynamics of point particles is
presented which is essential for the formulation of quantum mechanics, and an
understanding of the classical-quantum mechanical correspondence. In this part
of the book particular attention is given to the question of symmetry and conservation laws.
vii


viii

Heisenberg’s Quantum Mechanics

In Chapter 2 a short historical review of the discovery of matrix mechanics
is given and the original Heisenberg’s and Born’s ideas leading to the formulation of quantum theory and the discovery of the fundamental commutation
relations are discussed.
Chapter 3 is concerned with the mathematics of quantum mechanics,
namely linear vector spaces, operators, eigenvalues and eigenfunctions. Here
an entire section is devoted to the ways of constructing Hermitian operators,
together with a discussion of the inconsistencies of various rules of association
of classical functions and quantal operators.
In Chapter 4 the postulates of quantum mechanics and their implications are studied. A detailed review of the uncertainty principle for positionmomentum, time-energy and angular momentum-angle and some applications
of this principle is given. This is followed by an outline of the correspondence
principle. The question of determination of the state of the system from the
measurement of probabilities in coordinate and momentum space is also included in this chapter.

In Chapter 5 connections between the equation of motion, the Hamiltonian
operator and the commutation relations are examined, and Wigner’s argument
about the nonuniqueness of the canonical commutation relations is discussed.
In this chapter quantum first integrals of motion are derived and it is shown
that unlike their classical counterparts, these, with the exception of the energy
operator, are not useful for the quantal description of the motion.
In Chapter 6 the symmetries and conservation laws for quantum mechanical systems are considered. Also topics related to the Galilean invariance, mass
superselection rule and the time invariance are studied. In addition a brief discussion of classical and quantum integrability and degeneracy is presented.
Chapter 7 deals with the application of Heisenberg’s equations of motion
in determining bound state energies of one-dimensional systems. Here Klein’s
method and its generalization are considered. In addition the motion of a particle between fixed walls is studied in detail.
Chapter 8 is concerned with the factorization method for exactly solvable
potentials and this is followed by a brief discussion of the supersymmetry and
of shape invariance.
The two-body problem is the subject of discussion in Chapter 9, where the
properties of the orbital and spin angular momentum operators and determination of their eigenfunctions are presented. Then the solution to the problem of
hydrogen atom is found following the original formulation of Pauli using Runge–
Lenz vector.
In Chapter 10 methods of integrating Heisenberg’s equations of motion
are presented. Among them the discrete-time formulation pioneered by Bender
and collaborators, the iterative solution for polynomial potentials advanced by
Znojil and also the direct numerical method of integration of the equations of
motion are mentioned.
The perturbation theory is studied in Chapter 11 and in Chapter 12 other
methods of approximation, mostly derived from Heisenberg’s equations of mo-


Preface

ix

tion are considered. These include the semi-classical approximation and variational method.
Chapter 13 is concerned with the problem of quantization of equations of
motion with higher derivatives, this part follows closely the work of Pais and
Uhlenbeck.
Potential scattering is the next topic which is considered in Chapter 14.
Here the Schr¨
odinger equation is used to define concepts such as cross section
and the scattering amplitude, but then the deteminantal method of Schwinger
is followed to develop the connection between the potential and the scattering
amplitude. After this, the time-dependent scattering theory, the scattering matrix and the Lippmann–Schwinger equation are studied. Other topics reviewed
in this chapter are the impact parameter representation of the scattering amplitude, the Born approximation and transition probabilities.
In Chapter 15 another feature of the wave nature of matter which is quantum diffraction is considered.
The motion of a charged particle in electromagnetic field is taken up in
Chapter 16 with a discussion of the Aharonov–Bohm effect and the Berry phase.
Quantum many-body problem is reviewed in Chapter 17. Here systems
with many-fermion and with many-boson are reviewed and a brief review of the
theory of superfluidity is given.
Chapter 18 is about the quantum theory of free electromagnetic field with
a 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 Bell’s inequalities 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 applications of the Heisenberg quantum mechanics.
I am grateful to my wife for her patience and understanding while I was
writing this book, and to my daughter, Maryam, for her help in preparing the
manuscript.

Edmonton, Canada, 2010

This page intentionally left blank


Contents
Preface

vii

1 A Brief Survey of Analytical Dynamics

1

1.1

The Lagrangian and the Hamilton Principle . . . . . . . . . . . .

1

1.2

Noether’s 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 Poisson
Brackets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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
xi

. . . . . . . . . . . . . . . . .

39


xii

Heisenberg’s Quantum Mechanics
2.1

Equivalence of Wave and Matrix Mechanics . . . . . . . . . . . .

3 Mathematical Preliminaries

44

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

Weyl’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

3.7

Dirac’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

64

3.8

Von Neumann’s 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 Calculating
Bound State Energies . . . . . . . . . . . . . . . . . . . . . . . .

96

4.3

Time-Energy Uncertainty Relation . . . . . . . . . . . . . . . . .

98

4.4

Uncertainty Relations for Angular Momentum-Angle
Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

4.5

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

4.6

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

4.7

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


Contents
5 Equations of Motion, Hamiltonian Operator and the
Commutation Relations

xiii

125

5.1

Schwinger’s Action Principle and Heisenberg’s equations
of 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 and

Mass 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

Klein’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

7.2

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

7.3

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

7.4

Chasman’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . 173

7.5

Heisenberg’s Equations of Motion for Impulsive Forces . . . . . . 175

7.6

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


xiv

Heisenberg’s Quantum Mechanics
7.7

Heisenberg’s and Newton’s Equations of Motion . . . . . . . . . . 181

8 Exactly Solvable Potentials, Supersymmetry and Shape
Invariance

191

8.1

Energy Spectrum of the Two-Dimensional Harmonic Oscillator . 192

8.2

Exactly Solvable Potentials Obtained from Heisenberg’s
Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

8.3

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

8.4

Determination of the Eigenvalues by Factorization Method

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

. . . 201

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 the
Eigenvalues 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 Runge–Lenz
Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Heisenberg’s Equations of Motion 269
10.1 Discrete-Time Formulation of the Heisenberg’s Equations
of 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 Bender–Dunne 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 the
Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . 295
10.9 Motion of a Wave Packet . . . . . . . . . . . . . . . . . . . . . . 299

11 Perturbation Theory

309

11.1 Perturbation Theory Applied to the Problem of a
Quartic 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

Heisenberg’s 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 for
Nonpolynomial Potentials . . . . . . . . . . . . . . . . . . . . . . 340
12.3 Generalization of the Semiclassical Approximation to
Systems with N Degrees of Freedom . . . . . . . . . . . . . . . . 343
12.4 A Variational Method Based on Heisenberg’s Equation
of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350
12.5 Raleigh–Ritz Variational Principle . . . . . . . . . . . . . . . . . 354
12.6 Tight-Binding Approximation . . . . . . . . . . . . . . . . . . . . 355
12.7 Heisenberg’s Correspondence Principle . . . . . . . . . . . . . . . 356
12.8 Bohr and Heisenberg Correspondence and the Frequencies
and Intensities of the Emitted Radiation . . . . . . . . . . . . . . 361

13 Quantization of the Classical Equations of Motion with
Higher 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 of
Infinite 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 Lippmann–Schwinger 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 and
Bound 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 the
Differential 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

Heisenberg’s Quantum Mechanics

16 Motion of a Charged Particle in Electromagnetic Field
and Topological Quantum Effects for Neutral Particles

467

16.1 The Aharonov–Bohm Effect . . . . . . . . . . . . . . . . . . . . 471
16.2 Time-Dependent Interaction . . . . . . . . . . . . . . . . . . . . 480
16.3 Harmonic Oscillator with Time-Dependent Frequency . . . . . . 481
16.4 Heisenberg’s Equations for Harmonic Oscillator with
Time-Dependent Frequency . . . . . . . . . . . . . . . . . . . . . 483
16.5 Neutron Interferometry . . . . . . . . . . . . . . . . . . . . . . . 489
16.6 Gravity-Induced Quantum Interference . . . . . . . . . . . . . . 491
16.7 Quantum Beats in Waveguides with Time-Dependent
Boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493
16.8 Spin Magnetic Moment . . . . . . . . . . . . . . . . . . . . . . . 500
16.9 Stern–Gerlach Experiment . . . . . . . . . . . . . . . . . . . . . 503
16.10 Precession of Spin Magnetic Moment in a Constant
Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506
16.11 Spin Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . 508
16.12 A Simple Model of Atomic Clock . . . . . . . . . . . . . . . . . . 511
16.13 Berry’s Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514

17 Quantum Many-Body Problem

525

17.1 Ground State of Two-Electron Atom . . . . . . . . . . . . . . . 526
17.2 Hartree and Hartree–Fock 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 through
Pairing 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 Heisenberg’s Equations for Interaction of an Atom with
Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 621
20 Bell’s Inequality

631

20.1 EPR Experiment with Particles . . . . . . . . . . . . . . . . . . . 631
20.2 Classical and Quantum Mechanical Operational Concepts
of Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 640
20.3 Collapse of the Wave Function . . . . . . . . . . . . . . . . . . . 643
20.4 Quantum versus Classical Correlations . . . . . . . . . . . . . . . 645
Index

653


This page intentionally left blank


Chapter 1

A Brief Survey of
Analytical Dynamics
1.1

The Lagrangian and the Hamilton Principle

We can formulate the laws of motion of a mechanical system with N degrees of
freedom in terms of Hamilton’s principle. This principle states that for every
motion there is a well-defined function of the N coordinates qi and N velocities
q˙i which is called the Lagrangian, L, such that the integral
t2

S=

L (qi , q˙i , t) dt,

(1.1)

t1

takes the least possible value (or extremum) when the system occupies positions
qi (t1 ) and qi (t2 ) at the times t1 and t2 [1],[2].
The set of N independent quantities {qi } which completely defines the
position of the system of N degrees of freedom are called generalized coordinates
and their time derivatives are called generalized velocities.
The requirement that S be a minimum (or extremum) implies that L must
satisfy the Euler–Lagrange equation
d
∂L
−
∂qi
dt

∂L
∂ q˙i

= 0,

i = 1, · · · N.

(1.2)

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

Qi = Qi (q1 , · · · , qN ) ,
1

i = 1, · · · N,

(1.3)


2

Heisenberg’s 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 , q˙1 , · · · , qN
˙ ) be a twice differentiable function of 2N variables q1 , · · · , qN , q˙1 , · · · , qN
˙ . We note that this function can be written as a
function of Qj s and Q˙ j s if we replace qi s and q˙i s by Qj s and Q˙ j s using Eq.
(1.4). Now by direct differentiation we find that
d ∂
∂
−
∂qi
dt ∂ q˙i
N

=
j=1

∂Qj
∂qi

F qi (Qj ), q˙i (Qj , Q˙ j )
∂
d ∂
−
∂Qj
dt ∂ Q˙ j

F qi (Qj ), q˙i (Qj , Q˙ j ) ,

i = 1, · · · N.
(1.5)

Thus if L(Q1 , · · · Q˙ N ) has a vanishing Euler–Lagrange derivative i.e.
d
∂
−
∂Qj
dt

∂
∂ Q˙ j

(1.6)

L = 0,

then Eq. (1.5) implies that
d
∂L
−
∂qi
dt

∂L
∂ q˙i

= 0,

i = 1, · · · N.

(1.7)

This result shows that we can express the motion of the system either in terms
of the generalized coordinates qi and generalized velocities q˙i or in terms of Qj
and Q˙ j .
For simple conservative systems for which potential functions of the type
V (q1 , · · · , qN , t) can be found, the Lagrangian L has a simple form:
L = T (q1 , · · · , qn ; q˙1 , · · · , q˙N ) − V (q1 , · · · , qN , t),

(1.8)

where T is the kinetic energy of the particles in the system under consideration
and V is their potential energy. However given the force law acting on the

i-th particle of the system as Fi (q1 , · · · , qN ; q˙1 , · · · , q˙N ), in general, a unique
Lagrangian cannot be found. For instance we observe that the Euler–Lagrange
d
F(qi , q˙i , t)
derivative of any total time derivative of a function F of qi , q˙i i.e. dt
is identically zero;
d
∂
−
∂qi
dt

∂
∂ q˙i

dF(qi , q˙i , t)
≡ 0,
dt

Therefore we can always add a total time derivative
out affecting the resulting equations of motion.

i = 1, · · · N.
dF
dt

(1.9)

to the Lagrangian with-

Lagrangian Formulation

3

The inverse problem of classical mechanics is that of determination of
the Lagrangian (or Hamiltonian) when the force law Fj (qi , q˙i , t) is known. The
necessary and sufficient conditions for the existence of the Lagrangian has been
studied in detail by Helmholtz [3]–[6]. In general, for a given set of Fj s, L
satisfies a linear partial differential equation. To obtain this equation we start
with the Euler–Lagrange equation (1.2), find the total time derivative of ∂∂L
q˙i
Fi
and then replace q¨i by m
.
In
this
way
we
obtain
i
∂2L
+
∂ q˙i ∂t

j

∂2L
∂ q˙i ∂ q˙j

Fj
mj

+
j

∂2L
∂ q˙i ∂qj

q˙i q˙j −

∂L
= 0,
∂qi

i = 1, · · · N.

(1.10)
This set of equations yield the Lagrangian function. But as was stated earlier L
is not unique even for conservative systems. The advantage of the Lagrangian
formulation is that it contains information about the symmetries of the motion
which, 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 is
homogeneous, i.e. an inertial frame, a free particle which is at rest at a given
instant of time, always remains at rest. Because of the homogeneity of space
and time, the Lagrangian L cannot depend either on the position of the particle
r nor on time t. Thus it can only be a function of velocity r˙ . Now if the velocity
of the particle is r˙ relative to a frame S, then in another frame S which is
moving with a small velocity v with respect to S the velocity is r˙ , and the

Lagrangian is
2
L r˙ 2 = L (˙r + v) ≈ L r˙ 2 + 2˙r · v

∂L
,
∂ r˙ 2

(1.11)

where we have ignored higher order terms in v. Since the equation of motion
should have the same form in every frame, therefore the difference between
L r˙ 2 and L r˙ 2 must be a total time derivative (Galilean invariance). For a
constant v this implies that ∂∂L
r˙ 2 must be a constant and we choose this constant
.
Thus
we
arrive
at
a
unique
Lagrangian for the motion of a free particle.
to be m
2
L=

1 2
m˙r .
2

(1.12)

As a second example let us consider a system consisting of two particles
each of mass m interacting with each other with a potential V (|r1 − r2 |), where
r1 and r2 denote the positions of the two particles. The standard Lagrangian
according to Eq. (1.8) is
L1 =

1
m r˙ 21 + r˙ 22 − V (|r1 − r2 |),
2

(1.13)

and this generates the equations of motion
m

d2 ri
= −∇i V (|r1 − r2 |),
d t2

i = 1, 2.

(1.14)


4

Heisenberg’s Quantum Mechanics

A Lagrangian equivalent to L1 , is given by [8]
L2 = m (˙r1 · r˙ 2 ) − V (|r1 − r2 |),

(1.15)

and this L2 also yields the equations of motion (1.14). However the symmetries
of the two Lagrangians L1 and L2 are different. The Lagrangian L1 is invariant
under the rotation of the six-dimensional space r1 and r2 , whereas L2 is not.
The requirement of the invariance under the full Galilean group which
includes the conservation of energy, the angular momentum and the motion of
the center of mass, restricts the possible forms of the Lagrangian (apart from
a total time derivative) but still leaves certain arbitrariness. Here we want to
investigate this point and see whether by imposition of the Galilean invariance
we can determine a unique form for the Lagrangian or not.
Consider a system of N pairwise interacting particles with the equations
of motion
d2 rj
(1.16)
mj 2 = −∇j V, j = 1, 2, · · · , N,
dt
where V depends on the relative coordinates of the particles rj − rk and hence
∇j V = 0.

(1.17)

j

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
where vj =

d
dt rj .

mj vj = 0,

(1.18)

j

The conservation law (1.18) also follows from the Lagrangian
L=
j

1
mj vj2 − V.
2

(1.19)

Now under the Galilean transformation vj → vj + v, and L will change to L
where L − L is a total time derivative. Therefore the resulting equations of
motion, (1.16), will remain unchanged. Since j mj vj is constant, Eq. (1.18),
we can add any function of j mj vj to the Lagrangian without affecting the
equations of motion. If we denote this new Lagrangian which is found by the
addition of the constant term F
j mj vj 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 time
derivative unless F is of the form
F =

1
2µ

2

(mi vi ) ,

(1.20)

i

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