heisenbergs quantum mechanics

Nord-In most text books on quantum theory, a chapter or two are devoted tothe Heisenberg’s matrix approach, but due to the simplicity of the Schr¨odingerwave mechanics or the elegance of

QUANTUM MECHANICS

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

HEISENBERG’S QUANTUM MECHANICS

Mohsen Razavy

University of Alberta, Canada

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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.

Copyright © 2011 by World Scientific Publishing Co Pte Ltd.

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Printed in Singapore.

HEISENBERG’S QUANTUM MECHANICS

Dedicated to my great teachers

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

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 In

1965 H.S Green wrote a monograph with the title “Matrix Mechanics” hoff, Netherlands) where from the works of the pioneers in the field he collectedand presented a self-contained theory with applications to simple systems

(Nord-In most text books on quantum theory, a chapter or two are devoted tothe Heisenberg’s matrix approach, but due to the simplicity of the Schr¨odingerwave 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 Green’s 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 part

of the book particular attention is given to the question of symmetry and servation laws

con-vii

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 lation of quantum theory and the discovery of the fundamental commutationrelations are discussed

formu-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 tions are studied A detailed review of the uncertainty principle for position-momentum, time-energy and angular momentum-angle and some applications

implica-of this principle is given This is followed by an outline implica-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 Wigner’s 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 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 Heisenberg’s equations of motion

mechan-in determmechan-inmechan-ing bound state energies of one-dimensional systems Here Klemechan-in’smethod 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 and

of 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 Runge–Lenz vector

In Chapter 10 methods of integrating Heisenberg’s 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 Heisenberg’s equations of mo-

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 sectionand the scattering amplitude, but then the deteminantal method of Schwinger

is followed to develop the connection between the potential and the scatteringamplitude After this, the time-dependent scattering theory, the scattering ma-trix and the Lippmann–Schwinger equation are studied Other topics reviewed

in this chapter are the impact parameter representation of the scattering plitude, the Born approximation and transition probabilities

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

quan-The motion of a charged particle in electromagnetic field is taken up inChapter 16 with a discussion of the Aharonov–Bohm 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 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 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 ofthe 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

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

xi

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 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 xiii

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

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 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 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

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

16 Motion of a Charged Particle in Electromagnetic Field

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

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 offreedom in terms of Hamilton’s principle This principle states that for everymotion there is a well-defined function of the N coordinates qi and N velocities

takes the least possible value (or extremum) when the system occupies positions

qi(t1) and qi(t2) at the times t1and 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 Euler–Lagrange equation

∂L

∂qi − ddt

trans-Qi= Qi(q1,· · · , qN) , i = 1,· · · N, (1.3)

1

and its inverse transform given by the N equations

qj= qj(Q1,· · · , QN) , j = 1,· · · N (1.4)Now letF (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 ˙qis by Qj s and ˙Qj s using Eq.(1.4) Now by direct differentiation we find that

 ∂L

∂ ˙qi



This result shows that we can express the motion of the system either in terms

of the generalized coordinates qi and generalized velocities ˙qi or in terms of Qjand ˙Qj

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; ˙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 Euler–Lagrangederivative of any total time derivative of a functionF of qi, ˙qii.e d

with-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 Euler–Lagrange equation (1.2), find the total time derivative of ∂ ˙∂Lq

i

and then replace ¨qi by Fi

mi In this way we obtain

is 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 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 S0 which ismoving with a small velocity v with respect to S the velocity is ˙r0, and theLagrangian is

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

and this L2also yields the equations of motion (1.14) However the symmetries

of the two Lagrangians L1and L2are different The Lagrangian L1is 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 from

a total time derivative) but still leaves certain arbitrariness Here we want toinvestigate 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

jmjvj is constant, Eq (1.18),

jmjvj to the Lagrangian without affecting theequations of motion If we denote this new Lagrangian which is found by theaddition of the constant term FP

jmjvj



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

if in L[F ] we replace vi by vi+ v, then L0[F ]− L[F ] will not be a total timederivative unless F is of the form

2µX

i

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

What is interesting about this Lagrangian is that for any two values of theparameter µ, say µ0and µ00, L(µ0)− L(µ00) 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

∂vk

= mkvk+mk

µX

Since the velocities and coordinates are independent, each side of (1.35) must

be equal to zero, and thus we find two “Maxwell” type equations [7]

Noether’s Theorem 7

1.2 Noether’s Theorem

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

When L (qi, ˙qi, t) does not depend on a particular coordinate, say qk, thenclearly the momentum conjugate to this coordinate, pk= ∂L

∂ ˙ q, is conserved Thisparticular coordinate is called ignorable or cyclic coordinate

As an example let us consider the infinitesimal transformation

1.3 The Hamiltonian Formulation

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

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

we find the Newton equations of motion

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

NX

Hamiltonian Formulation 9

to the coordinates and generalized velocities via Eqs (1.45)

q-equivalent Hamiltonians — The complete solution of (1.49) evenfor 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

 ∂H

∂p

2+ V (q))

We note that the quantity inside the curly bracket in (1.52) is the Hamiltonian

H0which is also the total energy of the particle Thus if G(H0) is an arbitraryfunction of H0, we have

 ∂H

∂p

2+ V (q)

#

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

∂p

∂ ˙q =dG

p =

where g(q) is an arbitrary function of q For systems with two or more degrees

of 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

Let us consider two simple examples:

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

1

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

H(p, q) = A (V (q))1cosh

r2

These equations can be integrated to yield

12

"

 ∂H

∂p

2+ ω2q2

#

12

"

 ∂H

∂q

2+ ω2p2

#

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

j 0

where ε is a constant with the dimension of energy

Galilean Invariant Hamiltonians — The same type of ambiguity which

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

kpk· vk− L(µ) is

2 k2mk −

P

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

to the Hamilton principle Eq (1.44)

δZLdt = δ

Z



X

A different but more useful case is when the generating function depends

on old coordinates qi and the new momenta Pi By writing (1.80) as

+



n +12

m2ρ(t) + Ω¨ 2(t)ρ(t)− 1

depends explicitly on time, and therefore the new Hamiltonian K(P, Q, t) isgiven by

K(P, Q, t) = H(p, q, t) + ∂F2

1m

ρ2(t)



same whether expressed in terms of (p, q) or (P, Q), Eq (1.76) Thus

Similarly we calculate ˙P by differentiating P (p, q) with respect to time and

Action-Angle Variables 15where J is the Jacobian of the transformation

where in getting the last step we have used (1.95) Therefore Eq (1.97) with

J = 1 shows that under canonical transformation the area (or volume) in phasespace is preserved

Definition of the Action-Angle Variables — We define the angle variables which we denote by{θ, 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

action-A(E) =

Ip(q, E)dq = 2

Z 2π 0

Equation (1.102) defines the action variable The conjugate of action variable is

θ which varies between−π and π This relation can be generalized to a systemwith N degrees of freedom where for the k-th degree of freedom we have

Ik = 12π

I

pk(E, q1,· · · , qN)dqk (1.103)

This action is conjugate to the angle variable θk, where θk changes between

−π < θk≤ π Thus all angle variables are periodic with period 2π

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, θ) =(I , I ,· · · I ; θ , θ ,· · · θ ), where θ is an N -dimensional angle variable,

−π ≤ θk ≤ π By this transformation the Hamiltonian becomes a function of I,

H = H(I), i.e all θk s become cyclic coordinates Thus we have

In this relation ωk is a characteristic angular frequency of the motion and δk is

a phase shift For a completely separable system ωk depends on Ik alone Sincethe motion is periodic in θk, therefore θk(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 θ is a periodic function of the angle variables with each variablehaving a period 2π Hence we can expand the function A(I, θ) of the dynamicalvariables, Ii, θi in terms of the Fourier series

A(I, θ) =

∞X

j1= −∞

· · ·

∞X

j N = −∞

Aj1j2···j Nexp

it



j1

∂E

∂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

NX

l=1

jl∂E

but these frequencies are not generally commensurable, therefore the sum is not

a 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 tain conditions we can still express p and q in terms of θk s The parametricrepresentation of p and q as functions of N -dimensional angle variable θ

Action-Angle Variables 17

Figure 1.1: The phase space point (θ 1 , θ 2 , I 1 , I 2 ) for a system with two degrees of freedom is periodic in each θ 1 and θ 2 and this point moves in a region which is a toroid The two curves

C 1 and C 2 define the action integrals I 1 and I 2

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

Ik=I

C k

where p and q are the N -dimensional vectors andCks 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

con-dI

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

NX

{u, v}p i ,q i=

NX

j=1{u, v}P j ,Q j{Qj, Pj}p i ,q i (1.119)

This result shows that the Poisson bracket {u, v}p i ,q i remains unchanged by

a 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}

where u, v and w are functions of piand qi.

(4) - Jacobi identity

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

and Φ is called an integral of motion

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

1

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 pkand vk involves the velocity of the other particles For simplicity let

us consider the case of two interacting particles in one dimension, where from(1.24) we have