introduction to quantum mechanics schrodinger equation and path integral-part 1

INTRODUCTION TO OUHNTUM MECHANICS

Schrddinger Equation and Path Integral

INTRODUCTION TO QUANTUM MECHANICS Schrodinger Equation and Path Integral

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Schrodinger Equation and Path Integn

University of Kaiserslautern, Germany

\(P World Scientific NEW JERSEY LONDON S INGAPORE B E I J I N G S H A N G H A I HONG KONG TAIPEI C H E N N A I

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INTRODUCTION TO QUANTUM MECHANICS: Schrodinger Equation and Path Integral

Copyright 2006 by World Scientific Publishing Co. Pte. Ltd.

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ISBN 981-256-691-0 ISBN 981-256-692-9 (pbk)

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Contents

Preface xv

1 Introduction 1 1.1 Origin and Discovery of Quantum Mechanics 1 1.2 Contradicting Discretization: Uncertainties 7 1.3 Particle-Wave Dualism 12 1.4 Particle-Wave Dualism and Uncertainties 14

1.4.1 Further thought experiments 17 1.5 Bohr's Complementarity Principle 19 1.6 Further Examples 20

2 Hamiltonian Mechanics 23 2.1 Introductory Remarks 23 2.2 The Hamilton Formalism 23 2.3 Liouville Equation, Probabilities 29

2.3.1 Single particle consideration 29 2.3.2 Ensemble consideration 31

2.4 Expectation Values of Observables 34 2.5 Extension beyond Classical Mechanics 38

3 Mathematical Foundations of Quantum Mechanics 41 3.1 Introductory Remarks 41 3.2 Hilbert Spaces 41 3.3 Operators in Hilbert Space 49 3.4 Linear Functionals and Distributions 53

3.4.1 Interpretation of distributions in physics 54 3.4.2 Properties of functionals and the delta distribution . . 55

4 Dirac's Ket- and Bra-Formalism 59 4.1 Introductory Remarks 59 4.2 Ket and Bra States 60

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4.3 Linear Operators, Hermitian Operators 62 4.4 Observables 68 4.5 Representation Spaces and Basis Vectors 71

5 Schrodinger Equation and Liouville Equation 73 5.1 Introductory Remarks 73 5.2 The Density Matrix 73 5.3 The Probability Density p(x, t) 77 5.4 Schrodinger Equation and Liouville Equation 78

5.4.1 Evaluation of the density matrix 80

6 Quantum Mechanics of the Harmonic Oscillator 83 6.1 Introductory Remarks 83 6.2 The One-Dimensional Linear Oscillator 84 6.3 The Energy Representation of the Oscillator 90 6.4 The Configuration Space Representation 91 6.5 The Harmonic Oscillator Equation 98

6.5.1 Derivation of the generating function 98

7 Green's Functions 105 7.1 Introductory Remarks 105

7.2 Time-dependent and Time-independent Cases 105 7.3 The Green's Function of a Free Particle I l l 7.4 Green's Function of the Harmonic Oscillator 113 7.5 The Inverted Harmonic Oscillator 118

7.5.1 Wave packets 118 7.5.2 A particle's sojourn time T at the maximum 123

8 Time-Independent Perturbation Theory 129 8.1 Introductory Remarks 129 8.2 Asymptotic Series versus Convergent Series 130

8.2.1 The error function and Stokes discontinuities 133 8.2.2 Stokes discontinuities of oscillator functions 139

8.3 Asymptotic Series from Differential Equations 143 8.4 Formal Definition of Asymptotic Expansions 146 8.5 Rayleigh-Schrodinger Perturbation Theory 147 8.6 Degenerate Perturbation Theory 152 8.7 Dingle-Miiller Perturbation Method 155

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9 The Density Matrix and Polarization Phenomena 161 9.1 Introductory Remarks 161 9.2 Reconsideration of Electrodynamics 161 9.3 Schrodinger and Heisenberg Pictures 166 9.4 The Liouville Equation 167

10 Quantum Theory: The General Formalism 169 10.1 Introductory Remarks 169 10.2 States and Observables 169

10.2.1 Uncertainty relation for observables A, B 170 10.3 One-Dimensional Systems 173

10.3.1 The translation operator U(a) 176 10.4 Equations of Motion 178 10.5 States of Finite Lifetime 184 10.6 The Interaction Picture 185 10.7 Time-Dependent Perturbation Theory 189 10.8 Transitions into the Continuum 191 10.9 General Time-Dependent Method 195

11 The Coulomb Interaction 199 11.1 Introductory Remarks 199 11.2 Separation of Variables, Angular Momentum 199

11.2.1 Separation of variables 205 11.3 Representation of Rotation Group 206 11.4 Angular Momentum:Angular Representation 210 11.5 Radial Equation for Hydrogen-like Atoms 213 11.6 Discrete Spectrum of the Coulomb Potential 215

11.6.1 The eigenvalues 215 11.6.2 Laguerre polynomials: Various definitions in use! . . 219 11.6.3 The eigenfunctions 223 11.6.4 Hydrogen-like atoms in parabolic coordinates 227

11.7 Continuous Spectrum of Coulomb Potential 234 11.7.1 The Rutherford formula 237

11.8 Scattering of a Wave Packet 239 11.9 Scattering Phase and Partial Waves 243

12 Quantum Mechanical Tunneling 249 12.1 Introductory Remarks 249 12.2 Continuity Equation and Conditions 250 12.3 The Short-Range Delta Potential 251 12.4 Scattering from a Potential Well 254

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12.5 Degenerate Potentials and Tunneling 259

13 Linear Potentials 265 13.1 Introductory Remarks 265 13.2 The Freely Falling Particle: Quantization 265

13.2.1 Superposition of de Broglie waves 266 13.2.2 Probability distribution at large times 270

13.3 Stationary States 272 13.4 The Saddle Point or Stationary Phase Method 276

14 Classical Limit and W K B Method 281 14.1 Introductory Remarks 281 14.2 Classical Limit and Hydrodynamics Analogy 282 14.3 The WKB Method 286

14.3.1 The approximate WKB solutions 286 14.3.2 Turning points and matching of WKB solutions . . . . 290 14.3.3 Linear approximation and matching 293

14.4 Bohr-Sommerfeld-Wilson Quantization 297 14.5 Further Examples 301

15 Power Potentials 307 15.1 Introductory Remarks 307 15.2 The Power Potential 308 15.3 The Three-Dimensional Wave Function 315

16 Screened Coulomb Potentials 319 16.1 Introductory Remarks 319 16.2 Regge Trajectories 322 16.3 The S-Matrix 328 16.4 The Energy Expansion 329 16.5 The Sommerfeld-Watson Transform 330 16.6 Concluding Remarks 336

17 Periodic Potentials 339 17.1 Introductory Remarks 339 17.2 Cosine Potential: Weak Coupling Solutions 341

17.2.1 The Floquet exponent 341 17.2.2 Four types of periodic solutions 350

17.3 Cosine Potential: Strong Coupling Solutions 353 17.3.1 Preliminary remarks 353 17.3.2 The solutions 354

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17.3.3 The eigenvalues 361 17.3.4 The level splitting 363

17.4 Elliptic and Ellipsoidal Potentials 371 17.4.1 Introduction 371 17.4.2 Solutions and eigenvalues 373 17.4.3 The level splitting 375 17.4.4 Reduction to Mathieu functions 377

17.5 Concluding Remarks 378

18 Anharmonic Oscillator Potentials 379 18.1 Introductory Remarks 379 18.2 The Inverted Double Well Potential 382

18.2.1 Defining the problem 382 18.2.2 Three pairs of solutions 384 18.2.3 Matching of solutions 391 18.2.4 Boundary conditions at the origin 393 18.2.5 Boundary conditions at infinity 396 18.2.6 The complex eigenvalues 402

18.3 The Double Well Potential 405 18.3.1 Defining the problem 405 18.3.2 Three pairs of solutions 407 18.3.3 Matching of solutions 412 18.3.4 Boundary conditions at the minima 414 18.3.5 Boundary conditions at the origin 417 18.3.6 Eigenvalues and level splitting 424 18.3.7 General Remarks 427

19 Singular Potentials 435 19.1 Introductory Remarks 435 19.2 The Potential 1/r4 Case of Small h2 436

19.2.1 Preliminary considerations 436 19.2.2 Small h solutions in terms of Bessel functions . . . . 438 19.2.3 Small h solutions in terms of hyperbolic functions . . 441 19.2.4 Notation and properties of solutions 442 19.2.5 Derivation of the S-matrix 446 19.2.6 Evaluation of the S-matrix 455 19.2.7 Calculation of the absorptivity 458

19.3 The Potential 1/r4 Case of Large h2 460 19.3.1 Preliminary remarks 460 19.3.2 The Floquet exponent for large h2 461 19.3.3 Construction of large-h2 solutions 464

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19.3.4 The connection formulas 466 19.3.5 Derivation of the S-matrix 468


20 Large Order Behaviour of Perturbation Expansions 471 20.1 Introductory Remarks 471 20.2 Cosine Potential: Large Order Behaviour 476 20.3 Cosine Potential: Complex Eigenvalues 479

20.3.1 The decaying ground state 479 20.3.2 Decaying excited states 486 20.3.3 Relating the level splitting to imaginary E 493 20.3.4 Recalculation of large order behaviour 494

20.4 Cosine Potential: A Different Calculation 495 20.5 Anharmonic Oscillators 500

20.5.1 The inverted double well 500 20.5.2 The double well 501

20.6 General Remarks 502

21 The Path Integral Formalism 503 21.1 Introductory Remarks 503 21.2 Path Integrals and Green's Functions 504 21.3 The Green's Function for Potential V=0 510

21.3.1 Configuration space representation 510 21.3.2 Momentum space represenation 513

21.4 Including V in First Order Perturbation 514 21.5 Rederivation of the Rutherford Formula 518 21.6 Path Integrals in Dirac's Notation 524 21.7 Canonical Quantization from Path Integrals 533

22 Classical Field Configurations 537 22.1 Introductory Remarks 537 22.2 The Constant Classical Field 539 22.3 Soliton Theories in One Spatial Dimension 544 22.4 Stability of Classical Configurations 549 22.5 Bogomol'nyi Equations and Bounds 554 22.6 The Small Fluctuation Equation 557 22.7 Existence of Finite-Energy Solutions 564 22.8 Ginzburg-Landau Vortices 570 22.9 Introduction to Homotopy Classes 574 22.10The Fundamental Homotopy Group 579

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23 Path Integrals and Instantons 583 23.1 Introductory Remarks 583 23.2 Instantons and Anti-Instantons 583 23.3 The Level Difference 592 23.4 Field Fluctuations 596

23.4.1 The fluctuation equation 596 23.4.2 Evaluation of the functional integral 603 23.4.3 The Faddeev-Popov constraint insertion 609 23.4.4 The single instanton contribution 613 23.4.5 Instanton-anti-instanton contributions 614


24 Path Integrals and Bounces on a Line 619 24.1 Introductory Remarks 619 24.2 The Bounce in a Simple Example 625 24.3 The Inverted Double Well: The Bounce and Complex Energy 631

24.3.1 The bounce solution 631 24.3.2 The single bounce contribution 635 24.3.3 Evaluation of the single bounce kernel 637 24.3.4 Sum over an infinite number of bounces 641 24.3.5 Comments 644

24.4 Inverted Double Well: Constant Solutions 644 24.5 The Cubic Potential and its Complex Energy 645

25 Periodic Classical Configurations 649 25.1 Introductory Remarks 649 25.2 The Double Well Theory on a Circle 650

25.2.1 Periodic configurations 650 25.2.2 The fluctuation equation 659 25.2.3 The limit of infinite period 663

25.3 The Inverted Double Well on a Circle 664 25.3.1 Periodic configurations 664 25.3.2 The fluctuation equation 667 25.3.3 The limit of infinite period 669

25.4 The Sine-Gordon Theory on a Circle 670 25.4.1 Periodic configurations 670 25.4.2 The fluctuation equation 671

25.5 Conclusions 673

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26 Path Integrals and Periodic Classical Configurations 675 26.1 Introductory Remarks 675 26.2 The Double Well and Periodic Instantons 676

26.2.1 Periodic configurations and the double well 676 26.2.2 Transition amplitude and Feynman kernel 678 26.2.3 Fluctuations about the periodic instanton 679 26.2.4 The single periodic instanton contribution 684 26.2.5 Sum over instanton-anti-instanton pairs 688

26.3 The Cosine Potential and Periodic Instantons 690 26.3.1 Periodic configurations and the cosine potential . . . . 690 26.3.2 Transition amplitude and Feynman kernel 693 26.3.3 The fluctuation equation and its eigenmodes 694 26.3.4 The single periodic instanton contribution 696 26.3.5 Sum over instanton-anti-instanton pairs 700

26.4 The Inverted Double Well and Periodic Instantons 702 26.4.1 Periodic configurations and the inverted double well . 702 26.4.2 Transition amplitude and Feynman kernel 705 26.4.3 The fluctuation equation and its eigenmodes 706 26.4.4 The single periodic bounce contribution 708 26.4.5 Summing over the infinite number of bounces 710


27 Quantization of Systems with Constraints 715 27.1 Introductory Remarks 715 27.2 Constraints: How they arise 717

27.2.1 Singular Lagrangians 720 27.3 The Hamiltonian of Singular Systems 723 27.4 Persistence of Constraints in Course of Time 726 27.5 Constraints as Generators of a Gauge Group 727 27.6 Gauge Fixing and Dirac Quantization 734 27.7 The Formalism of Dirac Quantization 736

27.7.1 Poisson and Dirac brackets in field theory 740 27.8 Dirac Quantization of Free Electrodynamics 740 27.9 Faddeev-Jackiw Canonical Quantization 745

27.9.1 The method of Faddeev and Jackiw 745

28 The Quantum-Classical Crossover as Phase Transition 753 28.1 Introductory Remarks 753 28.2 Relating Period to Temperature 755 28.3 Crossover in Previous Cases 756

28.3.1 The double well and phase transitions 757

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28.3.2 The cosine potential and phase transitions 759 28.4 Crossover in a Simple Spin Model 760 28.5 Concluding Remarks 771

29 Summarizing Remarks 773

A Properties of Jacobian Elliptic Functions 775

Bibliography 779

Index 797

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Preface

With the discovery of quantization by Planck in 1900, quantum mechanics is now more than a hundred years old. However, a proper understanding of the phenomenon was gained only later in 1925 with the fundamental Heisenberg commutation relation or phase space algebra and the associated uncertainty principle. The resulting Schrodinger equation has ever since been the the-oretical basis of atomic physics. The alternative formulation by Feynman in terms of path integrals appeared two to three decades later. Although the two approaches are basically equivalent, the Schrodinger equation has found much wider usefulness, particularly in applications, presumably, in view of its simpler mathematics. However, the realization that solutions of classical equations, notably in field theory, play an important role in our understanding of a large number of physical phenomena, intensified the in-terest in Feynman's formulation of quantum mechanics, so that today this method must be considered of equal basic significance. Thus there are two basic approaches to the solution of a quantum mechanical problem, and an understanding of both and their usefulness in respective domains calls for their application to exemplary problems and their comparison. This is our aim here on an introductory level.

Throughout the development of theoretical physics two types of forces played an exceptional role: That of the restoring force of simple harmonic motion proportional to the displacement, and that in the Kepler problem proportional to the inverse square of the distance, i.e. Newton's gravita-tional force like that of the Coulomb potential. In the early development of quantum mechanics again oscillators appeared (though not really those of harmonic type) in Planck's quantization and the Coulomb potential in the Bohr model of the hydrogen atom. Again after the full and proper for-mulation of quantum mechanics with Heisenberg's phase space algebra and Born's wave function interpretation the oscillator and the Coulomb poten-tials provided the dominant and fully solvable models with a large number of at least approximate applications. To this day these two cases of interac-tion with nonresonant spectra feature as the standard and most important

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illustrative examples in any treatise on quantum mechanics and excepting various kinds of square well and rectangular barrier potentials leave the student sometimes puzzled about other potentials that he encounters soon thereafter, like periodic potentials, screened Coulomb potentials and maybe singular potentials, but also about complex energies that he encounters in a parallel course on nuclear physics. Excluding spin, any problem more compli-cated is frequently dispensed with by referring to cumbersome perturbation methods.

Diverse and more detailed quantum mechanical investigations in the sec-ond half of the last century revealed that perturbation theory frequently does permit systematic procedures (as is evident e.g. in Feynman diagrams in quantum electrodynamics), even though the expansions are mostly asymp-totic. With various techniques and deeper studies, numerous problems could, in fact, be treated to a considerable degree of satisfaction perturbatively. With the growing importance of models in statistical mechanics and in field theory, the path integral method of Feynman was soon recognized to offer frequently a more general procedure of enforcing first quantization instead of the Schrodinger equation. To what extent the two methods are actually equivalent, has not always been understood well, one problem being that there are few nontrivial models which permit a deeper insight into their connection. However, the aforementioned exactly solvable cases, that is the Coulomb potential and the harmonic oscillator, again point the way: For scattering problems the path integral seems particularly convenient, whereas for the calculation of discrete eigenvalues the Schrodinger equation. Thus important level splitting formulas for periodic and anharmonic oscillator po-tentials (i.e. with degenerate vacua) were first and more easily derived from the Schrodinger equation. These basic cases will be dealt with in detail by both methods in this text, and it will be seen in the final chapter that poten-tials with degenerate vacua are not exclusively of general interest, but arise also in recently studied models of large spins.

The introduction to quantum mechanics we attempt here could be sub-divided into essentially four consecutive parts. In the first part, Chapters 1 to 14, we recapitulate the origin of quantum mechanics, its mathematical foundations, basic postulates and standard applications. Our approach to quantum mechanics is through a passage from the Poisson algebra of classi-cal Hamiltonian mechanics to the canonical commutator algebra of quantum mechanics which permits the introduction of Heisenberg and Schrodinger pictures already on the classical level with the help of canonical transforma-tions. Then the Schrodinger equation is introduced and the two main exactly solvable cases of harmonic oscillator and Coulomb potentials are treated in detail since these form the basis of much of what follows. Thus this first part

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deals mainly with standard quantum mechanics although we do not dwell here on a large number of other aspects which are treated in detail in the long-established and wellknown textbooks.

In the second part, Chapters 15 to 20, we deal mostly with applica-tions depending on perturbation theory. In the majority of the cases that we treat we do not use the standard Rayleigh-Schrodinger perturbation method but the systematic perturbation procedure of Dingle and Muller which is in-troduced in Chapter 8. After a treatment of power potentials, the chapter thereafter deals with Yukawa potentials, and their eigenvalues. This is fol-lowed by the important case of the cosine or Mathieu potential for which the perturbation method was originally developed, and the behaviour of the eigenvalues is discussed in both weak and strong coupling domains with for-mation of bands and their asymptotic limits. The solution of this case however in nonperiodic form turns out to be a prerequisite for the com-plete solution of the Schrodinger equation for the singular potential 1/r4 in Chapter 19, which is presumably the only such singular case permitting com-plete solution and was achieved only recently. The earlier Chapter 17 also contains a brief description of a similar treatment of the elliptic or Lame po-tential. The following Chapter then deals with Schrodinger potentials which represent essentially anharmonic oscillators. The most prominent examples here are the double well potential and its inverted form. Using perturbation theory, i.e. the method of matched asymptotic expansions with boundary conditions (the latter providing the so-called nonperturbative effects), we derive respectively the level-splitting formula and the imaginary energy part for these cases for arbitrary states. In the final chapter of this part we dis-cuss the large order behaviour of the perturbation expansion with particular reference to the cosine and double well potentials.

In part three the path integral method is introduced and its use is illus-trated by application to the Coulomb potential and to the derivation of the Rutherford scattering formula. Thereafter the concepts of instantons, peri-odic instantons, bounces and sphalerons are introduced and their relevance in quantum mechanical problems is discussed (admittedly in also trespassing the sharp dividing line between quantum mechanics and simple scalar field theory). The following chapters deal with the derivation of level splitting formulas (including excited states) for periodic potentials and anharmonic oscillators and in the one-loop approximation considered are shown to agree with those obtained by perturbation theory with associated boundary conditions. We also consider inverted double wells and calculate with the path integral the imaginary part of the energy (or decay width). The poten-tials with degenerate minima will be seen to re-appear throughout the text, and the elliptic or Lame potential here introduced earlier as a generaliza-

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tion of the Mathieu potential re-appears as the potential in the equations of small fluctuations about the classical configurations in each of the basic cases (cosine, quartic, cubic). All results are compared with those obtained by perturbation theory, and whenever available also with the results of WKB calculations, this comparison on a transparent level being one of the main aims of this text.

The introduction of collective coordinates of classical configurations and the fluctuations about these leads to constraints. Our fourth and final part therefore deals with elementary aspects of the quantization of systems with constraints as introduced by Dirac. We then illustrate the relevance of this in the method of collective coordinates. In addition this part considers in more detail the region near the top of a potential barrier around the configuration there which is known as a sphaleron. The physical behaviour there (in the transition region between quantum and thermal physics) is no longer controlled by the Schrodinger equation. Employing anharmonic oscillator and periodic potentials and re-obtaining these in the context of a simple spin model, we consider the topic of transitions between the quantum and thermal regimes at the top of the barrier and show that these may be classified in analogy to phase transitions in statistical mechanics. These considerations demonstrate (also with reference to the topic of spin-tunneling and large-spin behaviour) the basic nature also of the classical configurations in a vast area of applications.

Comparing the Schrodinger equation method with that of the path inte-gral as applied to identical or similar problems, we can make the following observations. With a fully systematic perturbation method and with ap-plied boundary conditions, the Schrodinger equation can be solved for prac-tically any potential in complete analogy to wellknown differential equations of mathematical physics, except that these are no longer of hypergeometric type. The particular solutions and eigenvalues of interest in physics are as a rule those which are asymptotic expansions. This puts Schrodinger equa-tions with e.g. anharmonic oscillator potentials on a comparable level with, for instance, the Mathieu equation. The application of path integrals to the same problems with the same aims is seen to involve a number of subtle steps, such as limiting procedures. This method is therefore more complicated. In fact, in compiling this text it was not possible to transcribe anything from the highly condensed (and frequently unsystematic) original literature on applications of path integrals (as the reader can see, for instance, from our precise reference to unavoidable elliptic integrals taken from Tables). An expected observation is that ignoring a minor deficiency the WKB ap-proximation is and remains the most immediate way to obtain the dominant contribution of an eigenenergy, it is, however, an approximation whose higher

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order contributions are difficult to obtain. Nonetheless, we also consider at various points of the text comparisons with WKB approximations, also for the verification of results.

In writing this text the author considered it of interest to demonstrate the parallel application of both the Schrodinger equation and the path in-tegral to a selection of basic problems; an additional motivation was that a sufficient understanding of the more complicated of these problems had been achieved only in recent years. Since this comparison was the guide-line in writing the text, other topics have been left out which are usually found in books on quantum mechanics (and can be looked up there), not the least for permitting a more detailed and hopefully comprehensible presentation here. Throughout the text some calculations which require special attention, as well as applications and illustrations, are relegated to separate subsections which lacking a better name we refer to as Examples.

The line of thinking underlying this text grew out of the author's asso-ciation with Professor R. B. Dingle (then University of Western Australia, thereafter University of St. Andrews), whose research into asymptotic ex-pansions laid the ground for detailed explorations into perturbation theory and large order behaviour. The author is deeply indebted to his one-time supervisor Professor R. B. Dingle for paving him the way into this field which though not always at the forefront of current research (including the author's) repeatedly triggered recurring interest to return to it. Thus when instantons became a familiar topic it was natural to venture into this with the intent to compare the results with those of perturbation theory. This endeavour developed into an unforeseen task leading to periodic instan-tons and the exploration of quantum-classical transitions. The author has to thank several of his colleagues for their highly devoted collaboration in this latter part of the work over many years, in particular Professors J.-Q. Liang (Taiyuan), D. K. Park (Masan), D. H. Tchrakian (Dublin) and Jian-zu Zhang (Shanghai). Their deep involvement in the attempt described here is evident from the cited bibliography.*

H. J. W. Miiller-Kirsten

*In the running text references are cited like e.g. Whittaker and Watson [283]. For ease of reading, the references referred to are never cited by mere numbers which have to be identified e.g. at the end of a chapter (after troublesome turning of pages). Instead a glance at a nearby footnote provides the reader immediately the names of authors, e.g. like E. T. Whittaker and G. N. Watson [283], with the source given in the bibliography at the end. As a rule, formulas taken from Tables or elsewhere are referred to by number and/or page number in the source, which is particularly important in the case of elliptic integrals which require a relative ordering of integration limits and parameter domains, so that the reader is spared difficult and considerably time-consuming searches in a source (and besides, shows him that each such formula here has been properly looked up).

Chapter 1

Introduction

1.1 Origin and Discovery of Quantum Mechanics

The observation made by Planck towards the end of 1900, that the formula he had established for the energy distribution of electromagnetic black body radiation was in agreement with the experimentally confirmed Wien- and Rayleigh-Jeans laws for the limiting cases of small and large values of the wave-length A (or AT) respectively is generally considered as the discovery of quantum mechanics. Planck had arrived at his formula with the assump-tion of a distribution of a countable number of infinitely many oscillators. We do not enter here into detailed considerations of Planck, which involved also thermodynamics and statistical mechanics (in the sense of Boltzmann's statistical interpretation of entropy). Instead, we want to single out the vital aspect which can be considered as the discovery of quantum mechanics. Al-though practically every book on quantum mechanics refers at the beginning to Planck's discovery, very few explain in this context what he really did in view of involvement with statistical mechanics.

A "perfectly black body" is defined to be one that absorbs all (thermal) radiation incident on it. The best approximation to such a body is a cavity with a tiny opening (of solid angle dl) and whose inside walls provide a dif-fuse distribution of the radiation entering through the hole with the intensity of the incoming ray decreasing rapidly after a few reflections from the walls. Thermal radiation (with wave-lengths A ~ 10~5 to 10 - 2 cm at moderate temperatures T) is the radiation emitted by a body (consisting of a large number of atoms) as a result of the temperature (as we know today as a result of transitions between a large number of very closely lying energy lev-els). Kirchhoff's law in thermodynamics says that in the case of equilibrium, the amount of radiation absorbed by a body is equal to the amount the body

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2 CHAPTER 1. Introduction

emits. Black bodies as good absorbers are therefore also good emitters, i.e. radiators. The (equilibrium) radiation of the black body can be determined experimentally by sending radiation into a cavity surrounded by a heat bath at temperature T, and then measuring the increase in temperature of the heat bath.

Fig. 1.1 Absorption in a cavity.

Let us look at the final result of Planck, i.e. the formula (to be explained)

u(u,T) = 2*?(-?-)kT, where x = ^ = ^ - . (1.1) y J c3 \ex - l ) kT kXT y '

Here u(v, T)du is the mean energy density (i.e. energy per unit volume) of the radiation (i.e. of the photons or photon gas) in the cavity with both possible directions of polarization (hence the factor "2") in the frequency domain v, v + dv in equilibrium with the black body at temperature T. In Eq. (1.1) c is the velocity of light with c = u\, A being the wave-length of the radiation. The parameters k and h are the constants of Boltzmann and Planck:

k = 1.38 x 1(T23 J K'1, h = 6.626 x 10 - 3 4 J s.

How did Planck arrive at the expression (1.1) containing the constant h by treating the radiation in the cavity as something like a gas? By 1900 two theoretically-motivated (but from today's point of view incorrectly derived) expressions for u(u, T) were known and tested experimentally. It was found that one expression agreed well with observations in the region of small A (or AT), and the other in the region of large A (or AT). These expressions are: (1) Wien's law.

u(u,T) = dv3e-C2U/T, (1.2)

and the

1.1 Origin and Discovery of Quantum Mechanics 3

(2) Rayleigh-Jeans law:

u(i>,T) = 2^C3T, (1.3)

Ci, C2, C3 being constants. Considering Eq. (1.1) in regions of a; "small" (i.e. exp(x) ~ 1+x) and "large" (exp(x) < 1), we obtain:

u(i/, T)

u(i/,T)

2^^kT, {x small),

.47TZ/2

e xhv, (x large).

We see, that the formulas (1.2) and (1.3) are contained in Eq. (1.1) as ap-proximations. Indeed, in the first place Planck had tried to find an expression linking both, and he had succeeded in finding such an expression of the form

u(v,T) = av e 6 " / T - i '

where a and b are constants. When Planck had found this expression, he searched for a derivation. To this end he considered Boltzmann's formula S klnW for the entropy S. Here W is a number which determines the distribution of the energy among a discrete number of objects, and thus over a discrete number of admissible states. This is the point, where the

Fig. 1.2 Distributing quanta (dots) among oscillators (boxes).

discretization begins to enter. Planck now imagined a number TV of oscillators or iV oscillating degrees of freedom, every oscillator corresponding to an eigenmode or eigenvibration or standing wave in the cavity and with mean energy U. Moreover Planck assumed that these oscillators do not absorb or emit energy continuously, but here the discreteness appears properly only in elements (quanta) e, so that W represents the number of possible ways of distributing the number P := NU/e of energy-quanta ("photons", which are indistinguishable) among the N indistinguishable oscillators at


temperature T, U{T) being the average energy emitted by one oscillator. We visualize the iV oscillators as boxes separated by N 1 walls, with the quanta represented schematically by dots as indicated in Fig. 1.2. Then W is given by

{N + p-iy. w = (N - 1)!P! (1.4)

With the help of Stirling's formula*

IniV! ~ JVlniV-iV + O(0), N -* oo,

and the second law of thermodynamics ((dS/dU)v = 1/T), one obtains (cf. Example 1.1)

u = vmrri (L5)

as the mean energy emitted or absorbed by an oscillator (corresponding to the classical expression of 2 x kT/2, as for small values of e). Agreement with Eq. (1.2) requires that e ex is, i.e.

e = his, h = const. (1.6)

Fig. 1.3 Comparing the polarization modes with those of a 2-dimensional oscillator.

We now obtain the energy density of the radiation, u(i>,T)dv, by multiply-ing U with the number nvdv of modes or oscillators per unit volume with frequency v in the interval v, v + dv, i.e. with

riydu 2 x wdv, (1.7)

*See e.g. I. S. Gradshteyn and I. M. Ryzhik [122], formula 8.343(2), p. 940, there not called Stirling's formula, as in most other Tables, e.g. W. Magnus and F. Oberhettinger [181], p.3. The Stirling formula or approximation will appear frequently in later chapters.

1.1 Origin and Discovery of Quantum Mechanics 5

where the factor 2 takes the two possible mutually orthogonal linear di-rections of polarization of the electromagnetic radiation into account, as indicated in Fig.1.3. We obtain the expression (1.7) for instance, as in elec-trodynamics, where we have for the electric field

E oc elwt \ J eK sin KI^ I sin K2X2 sin K3X3 K

with the boundary condition that at the walls E = 0 at Xi = 0, L for i = 1,2,3 (as for ideal conductors). Then L^j = nrii, rii = 1,2,3,.. . ,

r 2 2 2 2 L K 7T n ,

where^ 2 [2-KUY , (lvL\A 0

KT = I J , so that I I = r r .

The number of possible modes (states) is equal to the volume of the spher-ical octant (where n^ > 0) in the space of n^,i = 1,2,3. The number with frequency v in the interval v, v + dv, i.e. nvdv per unit volume, is given by

, . , dM , . d dj\l -dv = nvdv

3 4*\IL> dv

dv dv |_8 3 \ c / 14 8 2 4TTV

2

= 8 3 ^ ^ = ^ ^ '

as claimed in Eq. (1.7). We obtain therefore

u^T) = Unv = 2^-fJ^i. (1.8)

This is Planck's formula (1.1). We observe that u(v,T) has a maximum which follows from du/dX = 0 (with c = vX). In terms of A we have

u(X,T)dX = ^ehc/*kT_idX,

so that the derivative of u implies (x as in Eq. (1.1))

The solutions of this equation are

^max = 4.965 and xmin = 0.

'''From the equation I -\ JW - V 2 ) E = 0, so that - UJ2/C2 + K? = 0,UJ = 2-KV.


The first value yields

he Amax-T = = Const.

4.965K

This is Wien's displacement law, which had also been known before Planck's discovery, and from which the constant h can be determined from the known value of k.

Later it was realized by H. A. Lorentz and Planck that Eq. (1.8) could be derived much more easily in the context of statistical mechanics. If an os-cillator with thermal weight or occupation probability exp(nx) can assume only discrete energies en = nhu, n = 0,1, 2 , . . . , then (with x = hv/kT) its mean energy is

E n = 0 e nX dx ^0

= /ii/ In = hu-f r%e dx 1 e _ x (1 e~x)z

hv

- v oo) the mean energy vanishes (0 < U < oo). Thus we have a rather complicated system here, that of an oscillation system at absolute temperature T ^ 0. One expects, of course, that it is easier to consider first the case of T = 0, i.e. the behaviour of the system at zero absolute temperature. Since temperature originates through contact with other oscillators, we then have at T = 0 independent oscillators, which can assume the discrete energies en nhv. We are not dealing with the linear harmonic oscillator familiar from mechanics here, but one can expect an analogy. We shall see later that in the case of this linear harmonic oscillator the energies En are given by

En= (n + -jhu= U + I W h=, ra = 0, l ,2 (1.10)

Thus here the so-called zero point energy appears, which did not arise in Planck's consideration of 1900.

One might suppose now, that we arrive at quantum mechanics simply by discretizing the energy and thus by postulating following Planck for the harmonic oscillator the expression (1.10). However, such a procedure leads to contradictions, which can not be eliminated without a different approach. We therefore examine such contradictions next.

1.2 Contradicting Discretization: Uncertainties 7

Example 1.1: Mean energy of an oscillator In Boltzmann's statistical mechanics the entropy S is given by the following expression (which we cite here with no further explanation) S = fcln W, where k is Boltzmann's constant and W is the number of times P indistinguishable elements of energy e can be distributed among TV indistinguishable oscillators, i.e.

W

{N- 1)! and P = UN

(TV-1)!P!

Show with the help of Stirling's formula that the mean energy U of an oscillator is given by

U

exp(e/fcT) - 1

1 + 7 l n ( 1 + 7 ) ~ 7 l n 7

Solution: Inserting W into Boltzmann's formula and using In TV! ~ AT In TV TV, we obtain

S = fc[ln(TV + P - 1)! - ln(TV - 1)! - InP!] ~ kN

The second law of thermodynamics says

\au)v T

For a single oscillator the entropy is s = S/N, so that

1 T

f ds\ \dUjy

, d - k

dU

U\ f U\ U U 1 + - ln(l + - - - i n -

k ( e = - In - + 1

e \U

which for e/kT > 0 becomes

u =

u~

e exp(e/fcT) -

e

- 1 '

- kT

This means U is then the classical expression resulting from the mean kinetic energy per degree of freedom, kT/2, for 2 degrees of freedom.

1.2 Contradicting Discretization: Uncertainties

The far-reaching consequences of Planck's quantization hypothesis were rec-ognized only later, around 1926, with Heisenberg's discovery of the uncer-tainty relation. In the following we attempt to incorporate the above dis-cretizations into classical considerations* and consider for this reason so-called thought experiments (from German "Gedankenexperimente"). We

"This is what was effectively done before 1925 in Bohr's and Sommerfeld's atomic models and is today referred to as "old quantum theory".
file:///dUjy


shall see that we arrive at contradictions. As an example^ we consider the linear harmonic oscillator with energy

E = -mx2 + - w V . (1.11) ZJ Zi

The classical equation of motion

dE n = x(mx + mco x) = 0

permits solutions x = Acos(u>t + S), so that

E = -mco2 A2,

where A is the maximum displacement of the oscillation, i.e. at x 0. We consider first this case of velocity and hence momentum precisely zero, and investigate the possibility to fix the amplitude. If we replace E by the discretized expression (1.10), i.e. by En (n + 1/2)HUJ, we obtain for the amplitude A

A^A-=\[Ef+l- (i-i2) Thus the amplitude can assume only these definite values. We now perform the following thought experiment. We give the oscillator initially an ampli-tude which is not contained in the set (1.12), i.e. for instance an amplitude A with

An


This distance is even less than what one would consider as a certain "di-ameter" of the electron (~ 10~15 meter). Thus it is even experimentally impossible to fix the amplitude A of the oscillator with the required preci-sion. Since A is the largest value of x, where x = 0, we have the problem that for a given definite value of mx, i.e. zero, the value of x = A can not be determined, i.e. given the energy of Eq. (1.10), it is not possible to give the oscillator at the same time at a definite position a definite momentum.

The above expression (1.10) for the energy of the harmonic oscillator, which we have not established so far, has the further characteristic of pos-sessing the "zero-point energy" Hu>/2, the smallest energy the oscillator can assume, according to the formula. Let us now consider the oscillator as a pendulum with frequency u in the gravitational field of the Earth. * Then

" 2 = f , (1-13)

where I is the length of the pendulum. Thus we can vary the frequency cv by varying the length I. This can be achieved with the help of a pivot, attached to a movable frame as indicated in Fig. 1.4. The resultant of the tension in the string of the pendulum, R, always has a nonnegative vertical component. If the pivot is moved downward, work is done against this vertical component of R; in other words, the system receives additional energy. However, there is one case, in which for a very short interval of time, 8t, the pendulum is at angle 0 = 0. Reducing in this short interval of time the length of the pendulum (by an appropriately quick shift of the pivot) by a factor of 4, the frequency of the oscillator is doubled, without supplying it with additional energy. Thus the energy

En= ( n + - ) fojj becomes I n + - IH2co,

without giving it additional energy. This is a self-evident contradiction. This means if the quantum mechanical expression (1.10) is valid we cannot simultaneously fix the energy (with energy conservation), as well as time t to an interval 8t 0.

The source of our difficulties in the considerations of these two examples is that in both cases we try to incorporate the discrete energies (1.10) into the framework of classical mechanics without any changes in the latter. Thus the theory with discrete energies must be very different from classical mechanics with its continuously variable energies.

H. Koppe [152]. See also Example 1.3.


It is illuminating in this context to consider the linear oscillator in phase space (q,p) with

P2 1 29 E 1mco q = const.

2m 2 *

(1.14)

Fig. 1.4 The pendulum with variable length.

This equation is that of an ellipse as a comparison with the Cartesian form

a2 + '" b2

reveals immediately. Evidently the ellipses in the (g,p)-plane have semi-axes of lengths

a = 2E b= V2mE.

mw' (1.15)

Inserting here (1.10), we obtain

2(n + l /2)fr^ hn = ^2m{n + l/2)^. (1.16) mar

We see that for n 0,1, 2 , . . . only certain ellipses are allowed. The area enclosed by such an ellipse is (note A earlier amplitude, now means area)

An = nanbri 2irEn 2Tih{n+ -

UJ

or ,( ' pdq 2irh I n + - ].

(1.17a)

(1.17b)

In the first of the examples discussed above the contradiction arose as a consequence of our assumption that we could put the oscillator initially at


any point in phase space, i.e. at some point which does not belong to one of the allowed ellipses. In the second example we chose n = 0 and thus restricted ourselves to the innermost orbit. However, we also assumed we would know at which point of the orbit the pendulum could be found.

Thus in attempting to incorporate the discrete quantization condition into the context of classical mechanics we see, that a system cannot be lo-calized with arbitrary precision in phase space, in other words the area AA, in which a system can be localized, is not nought. We can write this area

(1.17a) AA > An+1 -An

y'= 2TT/L

since the system cannot be "between" An+i and An. Since A A represents an element of area of the (q, p)-plane, we can write more precisely

ApAq > 2irh. (1.18)

This relation, called the Heisenberg uncertainty relation, implies that if we wish to make q very precise by arranging Aq to be very small, the comple-mentary uncertainty in momentum, Ap, becomes correspondingly large and extends over a large number of quantum states, as for instance in the second example considered above and illustrated in Fig. 1.5.

Fig. 1.5 Precise q implying large uncertainty in p.

Thus we face the problem of formulating classical mechanics in such a way that by some kind of extension or generalization we can find a way to quantum mechanics. Instead of the deterministic Newtonian mechanics which for a given precise initial position and initial momentum of a system yields the precise values of these for any later time we require a formulation answering the question: If the system is at time t = 0 in the area defined by


the limits 0 < q < q + Aq, 0

1.3 Particle-Wave Dualism 13

the number of such photo-electrons on the intensity of the incoming light. This is true even for very weak light. Einstein concluded from this effect, that the energy in a light ray is transported in the form of localized packets, called wave packets, which are also described as photons or quanta. Indeed the Compton effect, i.e. the elastic scattering of light, demonstrates that photons can be scattered off electrons like particles. Thus whereas Planck postulated that an oscillator emits or absorbs radiation in units of hv = hu>, Einstein went further and postulated that radiation consists of discrete quanta.

Thus light can be attributed a wave nature but also a corpuscular, i.e. particle-like, nature. In the interference experiment light behaves like a wave, but in the photoelectric effect like a stream of particles. One could try to play a trick, and use radiation which is so weak that it can transport only very few photons. What does the interference pattern then look like? Instead of bands one observes a few point-like spots. With an increasing number of photons these spots become denser and produce bands. Thus the interference experiment is always indicative of the wave nature of light, whereas the photoelectric effect is indicative of its particle-like nature. Without going into further historical details we add here, that it was Einstein in 1905 who attributed a momentum p to the light quantum with energy E = hv, and both he and Planck attributed to this the momentum

The hypothesis that every freely moving nonrelativistic microscopic particle with energy E and momentum p can be attributed a plane harmonic matter wave ip(r,t) was put forward much later, i.e. in 1924, by de Broglie.t This wave can be written as a complex function

ij)(T,t) =Aeik-r-iut,

where r is the position vector, and to and k are given by

E hio, p = /ik.

Thus particles also possess a wave-like nature. It is wellknown that this was experimentally verified by Davisson and Germer [64], who demonstrated the existence of electron waves by the observation of diffraction fringes instead of intensity distributions in appropriate experiments.

fL. de Broglie [39].


1.4 Particle-Wave Dualism and Uncertainties

We saw above that we can observe the wave nature of light in one type of experiment, and its particle-like nature in another. We cannot observe both types simultaneously, i.e. the wave-like nature together with the particle-like nature. Thus these wave and particle aspects are complementary, and show up only under specific experimental situations. In fact, they exclude each other. Every a t tempt to single out either of these aspects, requires a mod-ification of the experiment which rules out every possibility to observe the other aspect.* This becomes particularly clear, if in a double-slit experiment the detectors which register outcoming photons are placed immediately be-hind the diaphragm with the two slits: A photon is registered only in one detector, not in both hence it cannot split itself. Applying the above uncertainty principle to this situation, we identify the at tempt to determine which slit the photon passes through with the observation of its position coordinate q. On the other hand the observation of the interference fringes corresponds to the observation of its momentum p. Since the reader will ask himself what happens in the case of a single slit, we consider this case in Example 1.2.

Example 1.2: The Single-Slit Experiment Discuss the uncertainties of the canonical variables in relation to the diffraction fringes observed in a single-slit experiment.

Solution: Let light of wave-length A fall vertically on a diaphragm Si with slit AB as shown schematicaly in Fig. 1.7.

^ y

Ax

Fig. 1.7 Schematic arrangement of the single-slit experiment.

On the screen S2 one then observes a diffraction pattern of alternately bright and dark fringes, in the

See, for instance, the discussion in A. Messiah [195], Vol. I, Sec. 4.4.4. Considerable discussion can be found in A. Rae [234].

1.4 Particle-Wave Dualism and Uncertainties 15

figure indicated by maxima and minima of the light intensity I. As remarked earlier, the fringes are formed by interference of rays traversing different paths from the source to the observation screen. Before we enter into a discussion of uncertainties, we derive an expression for the intensity I. Since the derivation is not of primary interest here, we resort to a (still somewhat cunbersome) trick justification, which however can also be obtained in a rigorous way." We subdivide the distance AB = Ax into N equal pieces AP\, P1P2,..., as indicated in Fig. 1.8.

t Ax

\

Si

*

p-3

v "^^J

B ^ ^

w

Q

Fig. 1.8 The wave-front WW.

We consider rays deflected by an angle 9 with wave-front WW' and bundled by a lense L and focussed at a point Q on the screen S2. Since WW1 is a wave-front, all points on it have the same phase, so that light sent out from a source at Q reaches every point on WW1 at the same time and across equal distances. Hence a phase difference at Q can be attributed to different path lengths from Pi,P2,... to WW'. Considering two paths from neighbouring points Pi,Pj along AB, the difference in their lengths is Axsva6/N. In the case of a wave having the shape of the function

sin kr = sin -2?r

this implies a phase difference given by

2n Ax |r |exp(iS), we can similarly imagine the wave at Q, and this means its amplitude and phase, as represented by a vector, and similarly the wave of any component of the ray passing through AP\, P1P2, If we represent their effects at Q by vectors of equal moduli but different directions, their sum is the resultant OPN as indicated in Fig. 1.9. In the limit N > 00 the N vectors produce the arc of a circle. The angle 5 between the tangents at the two ends is the phase difference of the rays from the edges of the slit:

27T

5 = 2a = lim NSN = Aa;s in0 . (1.21)

If all rays were in phase, the amplitude, given by the length of the arc OQ, would be given by the chord OQ. Hence we obtain for the amplitude A at Q if AQ is the amplitude of the beam at the slit:

. length of chord OQ , 2a sin a , sin a A0 , , , ; 7^-=A0 =A0 . (1.22)

length of arc OQ a2a

"S. G. Starling and A. J. Woodall [260], p. 664. For other derivations see e.g. A. Brachner and R. Fichtner [32], p. 52.


The intensity at the point Q is therefore

where from Eq. (1.21)

h = h

K , . . k a = - f l i s inB = - A i s i n t

A 2

Fig. 1.9 The resultant OPM of N equal vectors with varying inclination.

Thus the intensity at the point Q is

Ie=Io sin2 (fcAx sin6(/2)

(fcAx sin 0/2)2

The maxima of this distribution are obtained for

fcAxsinfl = (2n + 1 ) - , i.e. for Ax sin0 = (2n + 1 ) - = (2n + 1) A

and minima for 1 fcAx sin # = 7171", i.e. for Ax i : TlA.

(1.23)

(1.24a)

(1.24b)

The maxima are not exactly where only the numerator assumes extremal values, since the variable also occurs in the denominator, but nearby.

We return to the single-slit experiment. Let the light incident on the diaphragm Si have a sharp momentum p = h/\. When the ray passes through the slit the position of the photon is fixed by the width of the slit Ax, and afterwards the photon's position is even less precisely known. We have a situation which for the observation on the screen S2 is a past (the uncertainty relation does not refer to this past with px = 0, rather to the position and momentum later; for the situation of the past A x A p is less than h). The above formula (1.23) gives the probability that after passing through the slit the photon appears at some point on the screen 52. This probability says, that the photon's momentum component px after passing through the slit is no longer zero, but indeterminate. It is not possible to predict at which point on S2 the photon will appear (if we knew this, we could derive px from this). The momentum uncertainty in the direction x can be estimated from the geometry of Fig. 1.10, where 6 is the angle in the direction to the first minimum:

Apx = 2px =2psin6 = sing. (1-25) A

From Eq. (1.24b) we obtain for the angle 9 in the direction of the first minimum

Ax sin 6 = A,

1.4 Particle-Wave Dualism and Uncertainties 17

Fig. 1.10 The components of momentum p.

so that Ax Apx = 2h.

If we take the higher order minima into account, we obtain AxApx = 2nh, or

Ax Apx > h.

We see that as a consequence of the indeterminacy of position and momentum, one has to introduce probability considerations. The limiting value of the uncertainty relation does not depend on how we try to measure position and momentum. It does also not depend on the type of particle (what applies to electromagnetic waves, applies also to particle waves).

1.4.1 Further thought experiments

Another experiment very similar to that described above is the attempt to localize a particle by means of an idealized microscope consisting of a single lense. This is depicted schematically in Fig. 1.11.

light

Fig. 1.11 Light incident as shown.

The resolving power of a lense L is determined by the separation Aa; of the first two neighbouring interference fringes, i.e. the position of a particle is at best determinable only up to an uncertainty Ax. Let 9 be one half of the angle as shown in Fig. 1.11, where P is the particle. We allow light to fall in the direction of x on the particle, from which it is scattered. We assume a quantum of light is scattered from P through the lense L to S where it


is focussed and registered on a photographic plate. For the resolving power Ax of the lense one can derive a formula like Eqs. (1.24a), (1.24b) . This is derived in books on optics, and hence will not be verified here, i.eJ

Ax~--. (1.26a) 2 sm 0

The precise direction in which the photon with momentum p = h/X is scat-tered is not known. However, after scattering of the photon, for instance along PA in Fig. 1.11, the uncertainty in its x-component is

1h Apx = 2psin0 = sine (1.26b) A

(prior to scattering the x-components of the momenta of the particle and the photon may be known precisely). From Eqs. (1.26a), (1.26b) we obtain again

Ax Apx ~ h.

The above considerations lead to the question of what kind of physical quantities obey an uncertainty relation. For instance, how about momentum and kinetic energy T? Apparently there are "compatible!'1 and "incompatible" quantities, the latter being those subjected to an uncertainty relation. If the momentumpx is "sharp", meaning Apx = 0, then also T = px

2/2m is sharp, i.e. T and px are compatible. In the case of angular momentum L = r x p, we have

|L| = |r | |p ' | = rp',

where p' = p sin 0. As one can see, r and p' are perpendicular to each other and thus can be sharp simultaneously. If p' lies in the direction of x, we have

Ax Ap' > h,

where now Ax = rAip, ip being the azimuthal angle, i.e.

rAipAp'>h, i.e. ALA

h.

Thus the angular momentum L is not simultaneously exactly determinable with the angle 0 the

"See, for instance, N. F. Mott, [199], p. 111. In some books the factor of "2" is missing; see, for instance, S. Simons [251], p. 12.

1.5 The Complementarity Primciple 19

quantities Ax, Apx are uncertainties at one and the same instant of time, and x and px cannot assume simultaneously precisely determined values. If, however, we consider a wave packet, such as we consider later, which spreads over a distance Ax and has group velocity VQ = p/m, the situation is different. The energy E of this wave packet (as also its momentum) has an uncertainty given by

AE -T^Ap = vGAp. op

The instant of time t at which the wave packet passes a certain point x is not unique in view of the wave packet's spread Ax. Thus this time t is uncertain by an amount

Ax At w .

vG

It follows that AtAE^AxAp>h. (1.27)

Thus if a particle does not remain in some state of a number of states for a period of time longer than At, the energy values in this state have an indeterminacy of |Ai|.

1.5 Bohr's Complementarity Principle

Vaguely expressed the complementarity principle says that two canonically conjugate variables like position coordinate x and the the associated canoni-cal momentum p of a particle are related in such a way that the measurement of one (with uncertainty Ax) has consequences for the measurement of the other. But this is essentially what the uncertainty relation expresses. Bohr's complementarity principle goes further. Every measurement we are inter-ested in is performed with a macroscopic apparatus at a microscopic object. In the course of the measurement the apparatus interferes with the state of the microscopic object. Thus really one has to consider the combined system of both, not a selected part alone. The uncertainty relation shows: If we try to determine the position coordinate with utmost precision all information about the object's momentum is lost precisely as a consequence of the disturbance of the microscopic system by the measuring instrument. The so-called Kopenhagen view, i.e. that of Bohr, is expressed in the thesis that the microscopic object together with the apparatus determine the result of a measurement. This implies that if a beam of light or electrons is passed through a double-slit (this being the apparatus in this case) the photons or


electrons behave like waves precisely because under these observation condi-tions they are waves, and that on the other hand, when observed in a counter, they behave like a stream of particles because under these conditions they are particles. In fact, without performance of some measurement (e.g. at some electron) we cannot say anything about the object's existence. The Kopenhagen view can also be expressed by saying that a quantity is real, i.e. physical, only when it is measured, or put differently the properties of a quantum system (e.g. whether wave-like or corpuscular) depend on the method of observation. This is the domain of conceptual difficulties which we do not enter into in more detail here.*

1.6 Further Examples

Example 1.3: The oscillator with variable frequency Consider an harmonic oscillator (i.e. simple pendulum) with time-dependent frequency w(t). (a) Considering the case of a monotonically increasing frequency w(t), i.e. dui/dt > 0, from LUQ to u>', show that the energy E' satisfies the following inequality

Eo < E' < y-Eo, (1.28) wo

where Eo is its energy at deflection angle 6 = 0Q. Compare the inequality with the quantum mechanical zero point energy of an oscillator. (b) Considering the energy of the oscillator averaged over one period of oscillation (for slow, i.e. adiabatic, variation of the frequency) show that the energy becomes proportional to ur. What is the quantum mechanical interpretation of the result?

Solution: (a) The equation of motion of the oscillator of mass m and with variable frequency co(t) is

mx + mui (t)x = 0,

where, according to the given conditions,

dui . _ > 0, u> = u>o a,t t = 0, w = ui at t = T, dt

i.e. io{t) grows monotonically. Multiplying the equation of motion by x we can rewrite it as

dt

The energy of the oscillator is

1 2 , 1 2 w 2 -mx -\mui (t)x 2 2 W

1 2 ^ 2 n mx = 0. 2 dt

l l 0 , l 9 dE 1 9 dJ1

E -mx1 + -mu}z(t)x2, so that - = -mxz > 0, (1.29) 2 2 y ' dt 2 dt ~ v '

where we used the given conditions in the last step. On the other hand, dividing the equation of motion by UJ2 and proceeding as before, we obtain

- [mx + mur (t)x\ = 0, i.e. dt

1 1 -2 1 2 --mx Hmx u22 2

1 2d mx .

2 dt\u)2

"See e.g. A. Rae [234]; P. C. W. Davies and J. R. Brown [65].

1.6 Farther Examples 21

2

d ( E\ 1 , d / 1 \ mx2 dw ) = -mx2 ( = < 0, (1.30) dt\uJ2J 2 dt\u>2) UJ3 dt ~ v

where the inequality again follows as before. We deduce from the last relation that

1 dE E dw2 1 dE 1 dw2 < 0, i.e. < . (1.31) u}2 dt UJ4 dt ~ E dt ~ u)2 dt

Integrating we obtain

fE' dE ^ f"'2 dw2 . n _ E , _, n 2,a,'2 E' ^ u'2

/ < / 7T, i-e. [InE f < lna;2 u 2 , i.e. < -JEo E - Jui u,2 ' [ >E-[ J"o2 E0 ~ UJ2

or

E' < -^-EQ.

Next we consider the case of the harmonic oscillator as a simple pendulum in the gravitational field of the Earth with

e + wge-o, o = f.

and we assume that as explained in the foregoing the length of the pendulum is reduced by one half so that

J2 = 2 - =2u;2.

Then the preceding inequality becomes E' < 2E0.

In shortening the length of the pendulum we apply energy (work against the tension in the string), maximally however EQ . Only in the case of the instantaneous reduction of the length at 6 = 0 (the pivot does not touch the string!) no energy is added, so that in this case E' = EQ, i.e.

E0 < E' < 2E0.

We can therefore rewrite the earlier inequality as

, u'2 E0 0 and error A E = 0.

(b) The classical expression for E contains u> quadratically, the quantum mechanical expression is linear in OJ. We argue now that we can obtain an expression for E c i a s s i c a l by assuming that w(t) varies very little (i.e. "adiabatically") within a period of oscillation of the oscillator, T. Classical mechanics is deterministic (i.e. the behaviour at time t follows from the equation of motion and


the initial conditions); hence for the consideration of a single mass point there is no reason for an averaging over a period, unless we are not interested in an exact value but, e.g. in the average

(lmX/ = ^I0 \mx2{P>dt- (L 3 2)

If u> is the frequency of x(t), i.e. x(t) oc cosujt or sinu>t depending on the initial condition, then x2(t) = UJ2X2 and hence

l-mw2x2\ = (-mx2\ = -E

(as follows also from the virial theorem). If we now insert in the equation for dE/dt, i.e. in Eq. (1.29), for mx2 /2 the mean value

/ I 2 \ 1 ( - mx ) = , \ 2 / 2 u 2 '

we obtain dE_/l 2\dw

2_Edw2 dE _ 1 dw2 _ du ~dt ~ \2mX / ~dT ~ 2w2~dT' r ~E ~ ~iU> ~~ ~u7'

and hence E = const. w

In quantum mechanics with E = hw{n + 1/2) this implies H(n + 1/2) = const., i.e. n = const. This means, with slow variation of the frequency the system remains in state n. This is an example of the so-called adiabatic theorem of Ehrenfest, which formulates this in a general formJ

Example 1.4: Angular spread of a beam A dish-like aerial of radius R is to be designed which can send a microwave beam of wave-length A = 2irh/p from the Earth to a satellite. Estimate the angular spread 6 of the beam.

Solution: Initially the photons are restricted to a transverse spread of length Ax = 2R. From the uncertainty relation we obtain the uncertainty /\px of the transverse momentum px as Apx ^ h/2R. Hence the angle 0 is given by

~~ p 2R\2nh) ~ A-KR'

See e.g. L. Schiff [243], pp. 25 - 27.

Chapter 2

Hamiltonian Mechanics

2.1 Introductory Remarks

In this chapter we first recapitulate significant aspects of the Hamiltonian formulation of classical mechanics. In particular we recapitulate the con-cept of Poisson brackets and re-express Hamilton's equations of motion in terms of these. We shall then make the extremely important observation that these equations can be solved on the basis of very general properties of the Poisson bracket, i.e. without reference to the original definition of the latter. This observation reveals that classical mechanics can be formulated in a framework which permits a generalization by replacing the c-number valued functions appearing in the Poisson brackets by a larger class of quan-tities, such as matrices and operators. Thus in this chapter we attempt to approach quantum mechanics as far as possible within the framework of classical mechanics. We shall see that we can even define such concepts as Schrodinger and Heisenberg pictures in the purely classical context.

2.2 The Hamilton Formalism

In courses on classical mechanics it is shown that Hamilton's equations can be derived in a number of ways, e.g. from the Lagrangian with a Legendre transform or with a variational principle from the Hamiltonian H(qi,Pi), i.e.

rt2 r 6 / ^2PiQi-H(qi,Pi) dt = 0,

where now (different from the derivation of the Euler-Lagrange equations) the momenta pi and coordinates qi are treated as independent variables. As

23

24 CHAPTER 2. Hamiltonian Mechanics

is wellknown, one obtains the Hamilton equations*

OH . dH

In this Hamilton formalism it is wrong to consider the momentum pi as mqi, i.e. as mass times velocity. Rather, Pi has to be considered as an independent quantity, which can be observed directly at time t, whereas the velocity requires observations of space coordinates at different times, since

.. qi(t + 6t) - qi(t) qi = hm f .

6t->0 5t

Real quantities which are directly observable are called observables. A sys-tem consisting of several mass points is therefore described by a number of such variables, which all together describe the state of the system. All functions u(qi,pi) of qi,p% are therefore again observables. Compared with an arbitrary function f(qi,Pi,t), the entire time-dependence of observables u(qi,Pi) is contained implicitly in the canonical variables q^ and pi. The total time derivative of u can therefore be rewritten with the help of Eqs. (2.1) as

d . . x^fdu. du \ x^/dudH du dH\ ,n n.

S(,P.) = [wm + WiK) = [WiWi - ^ ^ j . (2.2) If we have only one degree of freedom (i = 1), this expression is simply a functional determinant. One now defines as (nonrelativistic) Poisson bracket the expression^

With this definition we can rewrite Eq. (2.2) as

This equation is, in analogy with Eqs. (2.1), the equation of motion of the observable u. One can verify readily that Eq. (2.4) contains as special cases the Hamilton Eqs. (2.1). We can therefore consider Eq. (2.4) as the general-ization of Eqs. (2.1). It suggests itself therefore to consider more closely the properties of the symbols (2.3). The following properties can be verified:

*See e.g. H. Goldstein [114], chapter VII. ^As H. Goldstein [114] remarks at the end of his chapter VIII, the standard reference for the

application of Poisson brackets is the book of P. A. M. Dirac [75], chapter VIII. It was only with the development of quantum mechanics by Heisenberg and Dirac that Poisson brackets gained widespread interest in modern physics.

2.2 The Hamilton Formalism 25

(1) Antisymmetry: {A,B} = -{B,A}, (2.5a)

(2) linearity:

{A, a i S i + a2B2} = ax{A, Bx} + a2{A, B2}, (2.5b)

(3) complex conjugation (note: observables are real, but could be multiplied by a complex number):

{A,B}* = {A*,B*}, (2.5c)

(4) product formation:

{A,BC} = {A,B}C + B{A,C}, (2.5d)

(5) Jacobi identity:

{A, {B, C}} + {B, {C, A}} + {C, {A, B}} = 0. (2.5e)

The first three properties are readily seen to hold. Property (2.5d) is useful in calculations. As long as we are concerned with commuting quantities, like here, it is irrelevant whether we write

{A,B}C or C{A,B}.

Later we shall consider noncommuting quantities, then the ordering is taken as in (2.5d) above.

If we evaluate the Poisson brackets for qi,Pi, we obtain the fundamental Poisson brackets. These are

{Qi,Qk} = 0, {qi,Pk} = 5ik, {pi,Pk} = 0. (2.6)

We can now show, that the very general Eq. (2.4), which combines the Hamil-ton equations, can be solved solely with the help of the properties of Poisson brackets and the fundamental Poisson brackets (2.6), in other words without any reference to the original definition (2.3) of the Poisson bracket. If, for example, we wish to evaluate {A, B}, where A and B are arbitrary observ-ables, we expand A and B in powers of qi and pi and apply the above rules until only the fundamental brackets remain. Since Eqs. (2.6) give the values of these, the Poisson bracket {A, B} is completely evaluated. As an example we consider a case we shall encounter again and again, i.e. that of the linear harmonic oscillator. The original definition of the Poisson bracket will not


be used at all. In the evaluation one should also note that the fact that g, and Pi are ordinary real number variables and that H(q,p) is an ordinary function is also irrelevant. Since constants are also irrelevant in this context, we consider as Hamiltonian the function

H(q,p) = (p2 + q2). (2.7)

According to Eq. (2.4) we have for u = q,p,

q = {q,H}, (2.8a)

and P={p,H}. (2.8b)

We insert (2.7) into (2.8a) and use the properties of the Poisson bracket and Eqs. (2.6). Then we obtain:

q = [q,$p2 + q2)} = l({q,P2} + {q,q2})

= 2i{q,p}p + p{q,p$

= P. (2.9)

Similarly we obtain from Eq. (2.8b)

p=-q. (2.10)

From Eqs. (2.9) and (2.10) we deduce

q = p = -q, q + q = o,

and so q = -q, 'q' = -q, "4' = q,...,

from which we infer that

q(t) qocost + posint, (2.11a)

or

q(t) = qo+ Pot - -qot2 - yPot3 + (2.11b)

In classical mechanics one studies also canonical transformations. These are transformations

qi>Qi = Qi(q,p,t), Pi> Pi = Pi(q,p,t), (2.12)

2.2 The Hamilton Formalism 27

for which the new coordinates are also canonical, which means that a Hamil-ton function K(Q, P) exists, for which Hamilton's equations hold, i.e.

. _ a x p__dK_

We write the reversal of the transformation (2.12)

Qi>qi = qi{Q,P,t), Pi^Pi=Pi(Q,P,t). (2.14)

With the help of the definition (2.3) we can now express the Poisson bracket { 4, B} of two observables A and B in terms of either set of canonical vari-ables, i.e. as

{A,B}q,P r a S {AiB}Q,P-

One can then show that

{A,B}q,p = {A,B}Q>P, (2.15a)

provided the transformation q,p Q,P is canonical in the sense defined above. The proof requires the invariance of the fundamental Poisson brackets under canonical transformations, i.e. that (dropping the subscripts therefore)

{Pi,Pk} = 0, {qi,pk} = 5ik, {qi,qk} = 0,

{PhPk} = 0, {QhPk} = Sik, {Qi,Qk} = 0. (2.15b)

The proof of the latter invariance is too long to be reproduced in detail here but can be found in the book of Goldstein.* Hence in Example 2.1 we verify only Eq. (2.15a). Example 2.2 below contains a further illustration of the use of Poisson brackets, and Example 2.3 deals with the relativistic extension.

In classical mechanics we learned yet another important aspect of the Hamilton formalism: We can inquire about that particular canonical trans-formation, which transforms qi,Pi back to their constant initial values, i.e. those at a time t 0. Of course, this transformation is described precisely by the equations of motion but we shall not consider this in more detail here.

Example 2.1: Canonical invariance of Poisson bracket Assuming the invariance of the fundamental Poisson brackets under canonical transformations Qj = Qj(Q'P)>Pj = Pj{Q>P), verify that the Poisson bracket of two observables A and B is invariant, i.e. Eq. (2.15a).

*H. Goldstein [114], chapter VIII.

(2.13)


S o l u t i o n : Us ing t h e defini t ion of t h e Po i s son b racke t app l i ed t o ^4 a n d B we have

r , v^ (9A dB dA 8B {A,B}q,p = ^ | _ _

*-? V dqj dpj dpj dqj

E

E k

dAL/'^B_dQk 8B_dI\\ _ 8A_f^B_dQk dB 8Pk

dqj \ dQk dpj dPk dpj J dpj \ 8Qk dqj dPk dqj

{A, Qk}q,p^ + {A, Pk}q,p-^ OQk "Pk.

Replacing here A by Q and B by A, we obtain

{Qk,A}q,p = J2 dA dA

{Qk,Qj}q,P-^pr- + {Qk,Pj}q,p-^-dQ, vdP,

(2.15b) dA

d~Fk

Replacing in the above A by P and B by A, we obtain analogously

dA {Pk,A}q,p =

dQk'

Inserting both of these results into the first equation, we obtain as claimed by Eq. (2.15a)

r v - ( 9A dB dA dB \ r A 1

{ A ' B ^ = Ebrs r - sF57r ={A,B}QIP. V V Qk dPk dPk dQk J

Example 2.2: Solution of Galilei problem wi th Poisson brackets Consider the Hamiltonian for the free fall of a mass point mo in the gravitational field (linear potential),

1 H = p + m0gq

and solve the canonical equations with Poisson brackets for initial conditions q(0) = qo,p(0) = po-

Solution: The solution can be looked up in the literature.

Example 2.3: Relativist ic Poisson brackets By extending qi,pt to four-vectors (in a (1 + 3)-dimensional space) define relativistic Poisson brackets.

Solution: Relativistically we have to treat space and time on an equal footing. Thus we extend q and p to space-time vectors (t,q/c) and (E,pc), their product Et qp being relativistically invariant. Thus whenever q and p are multiplied, we have Et. The relativistic Poisson bracket (subscript r) therefore becomes

{u,F}r du dF du dF

dq dp dp dq

du &F _ du dF '

~di d~E ~ d~E~8t

Consider

F = H(q,p) - E(t).

See P. Mittelstaedt [197], p. 236.

2.3 Liouville Equation, Probabilities 29

(This is, of course, numerically zero, but partial derivatives of F do not vanish, since H is expressed as a function of q and p, and E as a function of t). Then

, rrr s ,_,,,., dudH dudH du dE(t) du du . du . du lu , H(q,p) - E(t)}r = 1 = 1 q-\ P = .

Hence

at

Relativistically we really should have clu/dr, where dr is the difference of proper time given by

(*.)'=(*)*-*, =j1-u^.y, w^ du-du

c2 ' dt V c2\dtj' dr dt y/l - qZ/c2'

2.3 Liouville Equation, Probabilities

2.3.1 Single particle consideration

We continue to consider classical mechanics in which the canonical coordi-nates qi,Pi are the coordinates of some mass point m;, and the space spanned by the entire set of canonical coordinates is described as its phase space. But now we consider a system whose phase space coordinates are not known pre-cisely. Instead we assume a case in which we know only that the system is located in some particular domain of phase space. Let us assume that at some initial time to the system may be found in a domain Go(q,p) around some point qo,po, and at time t > to in a domain G\(q,p). Of course, it is the equations of motion which lead from Go(q,p) to G\{q,p). Since Hamilton's equations give a continuous map of one domain onto another, also boundary points of one domain are mapped into boundary domains of the other, so that Go(qo,po) = Gi(q,p), i.e. if qo,po is a point on Go, one obtains Gi with

q = q((lo,Po,to;t), p = p(qo,Po,t0;t).

We distinguish in the following between two kinds of probabilities. We consider first the a priori weighting or a priori probability, g, which is the probability of a particle having a coordinate q between q and q + Aq and a momentum p between p and p + Ap. This probability is evidently propor-tional to AqAp, i.e.

g oc AqAp. (2.16)

For example, in the case of the linear oscillator with energy E given by Eq. (1.14) and area A of the phase space ellipse given by Eq. (1.17a), we have

. I , , 2TTE , , dE A = (p apaq and hence g oc .

J UJ to


If g depended on time t it would be dynamical and would involve known in-formation about the particle. Thus g must be independent of t, as is demon-strated by Liouville's theorem in Example 2.4; in view of this independence it can be expressed in terms of the conserved energy E. Example 2.5 there-after provides an illustration of the a priori weighting expressed in terms of energy E.

Example 2.4: Liouville's theorem Show that AqAp is independent of time i, which means, this has the same value at a time to, as at a time t'0 ^ to

Solution: We consider

- d l n ( A 9 A P ) -r f ( A < Z ) ' . d ( A p ) '

dt dt Aq dt Ap

Here d(Aq)/dt is the rate at which the q-walls of the phase space element move away from the centre of the element,

dq Aq dq Aq q H to the right and q to the left.

dq 2 dq 2

Hence from the difference:

- = Aq, and similarly = Ap, dt dq dt dp

and with with Hamilton's equations (2.1):

d , , A A s dq dp d2H d2H

In(AqAp) = + = = 0. dt dq dp dqdp dpdq

Example 2.5: A priori weighting of a molecule If the rotational energy of a diatomic molecule with moment of inertia / is

1 / 2 , Pi , ? + 2 / \ sin26>/ ' 21E 2IEsin2e'

in spherical polar coordinates, the (pg,p^,)-curve for constant E and $ is as may be seen by comparison with Eqs. (1.14) to (1.15) an ellipse of area dpgdp,/, = 2TrIEsm6. Show that the total volume of phase space covered for constant E is 8n2IE, and hence g oc %ir2IdE.

Solution: Integrating over the angles we have

J0=0 Je=0

=2TT f e = 7 r

2frEsin.eded = 8TT2IE.

Hence g oc 8n2IdE.


2.3.2 Ensemble consideration

We now assume a large number of identical systems the entire collection is called an ensemble all of whose initial locations are possible locations of our system in the neighbourhood of the point qo,Po- Thus we assume a large number of identical sytems, whose positions in phase space are characterized by points. We consider the totality of these systems which is described by a density of points p (number dn of points per infinitesimal volume) in phase space, i.e. by

-j-r = P(9P.*). d1dP = JJdqidpi. (2.17)

F

0

^ G T ^ ^

Fig. 2.1 The system moving from domain Go to domain G\.

Thus dn is that number of systems which at time t are contained in the domain q,q + dq;p,p + dp. The total number of systems N is obtained by integrating over the whole of phase space, i.e.

dn p(q,p,t)dqdp = N.

With a suitable normalization we can write this

/ W(q,p,t)dqdp= 1, W=-p(q,p,t).

(2.18)

(2.19)

Thus W is the probability to find the system at time t at q,p. Since W has the dimension of a reciprocal action, it is suggestive to introduce a factor 2-KK with every pair dpdq without, however, leaving the basis of classical mechanics! Hence we set

/ w ) ^ = . (2.20)


We can consider 2irh as a unit of area in (here the (1 + l)-dimensional) phase space.

- * q

Fig. 2.2 The ensemble in phase space.

We are now interested in how n ov W changes in time, i.e. how the system moves about in phase space. The equation of motion for n or W is the so-called Liouville equation. In order to derive this equation, we consider the domain G in Fig. 2.3 and establish an equation for the change of the number of points or systems in G in the time interval dt. In doing this, we take into account, that in our consideration no additional points are created or destroyed.

p + dp

P

O q

G

qn -dq

Fig. 2.3 The region G.

The number of points at time t + dt in domain G, i.e. p(q,p, t + dt)dqdp, is equal to the number in G at time t plus the number that went into G in the time interval dt minus the number that left G in the time interval dt, i.e. if vq(q,p) and vp(q,p) denote the velocities in directions q and p

p{q,p, t + dt)dqdp p(q,p, t)dqdp

p(q,p,t)dp{ -jt dt Q,P

p(q + dq,p,t)dp( I dt + . a t / q+dq,p


and thus

p(q, p,t + dt)dqdp p(q, p, t)dqdp

= p(q,p,t)vq(q,p)dtdp - p(q + dq,p,t)vq(q + dq,p)dtdp

+p(q,p,t)vp(q,p)dtdq - p(q,p + dp,t)vp(q,p + dp)dtdq.

Dividing both sides by dqdpdt this becomes

p(q,p,t + dt) - p(q,p,t) dt

= p(q,P,t)vq{q,p) - p(q + dq,p, t)vq(q + dq,p) dq

p(q,p,t)vp(q,p) - p(q,p + dp,t)vp(q,p + dp) dp

or

However,

so that

| = - | K ( 9 , P ) ] - | K ( Q , P ) ] .

, . . dH . 3H Mq,p) = q = -g^, vp(q,p)=p = -,

dt ~~dq\P dp) + dp\P ~dq)~ ~~dq~~dp~ + dp~~dq~ ~ { ,P*'

Hence

% = iH>P}- (2-21)

This is the Liouville equation which describes the motion of the ensemble or, put differently, the probable motion of the system under consideration. Comparison of Eq. (2.21) with Eq. (2.4) shows that p and u satisfy very similar equations. With Eqs. (2.19), (2.20) and (2.21) we can also write

dW(q,p,t) dt

{H(q,p),W(q,p,t)} with JW(q,p,t)^- = 1. (2.22)

The generalization to n degrees of freedom is evident: The volume element of phase space is


dtjv

where

is the probability for the system to be at time t in the volume q, q+dq; p, p+dp. We deduce from the Liouville equation the important consequence that

^ M = 0, (2.24)

since the total derivative is made up of precisely the partial derivatives con-tained in Eq. (2.24). Equation (2.24) implies that p is a constant in time, and hence that equal phase space volumes contain the same number of systems, and this means since these systems are contained in a finite part V of phase space that

\y dt

We have in particular, since no systems are created or destroyed, that

if qo,po are the initial values of q,p (cf. Example 2.4). Thus in Fig. 2.1 the area Go is equal to the area G\.

2.4 Expectation Values of Observables

Let u = u(q,p) be an observable. We define as expectation value of u(q,p) the following expression:

(u)=Ju(q,p)W(q,p,t)(^-J. (2.26)

With Eq. (2.4), i.e.

we described the time variation of the observable u(q,p). We now inquire about the time variation of the expectation value (it) of u. We shall see that we have two possibilities for this, i.e. for

i-!/"w>(^)". (-, The first and most immediate possibility is as indicated - that the density or probability W(q,p, t) depends explicitly on time t (if determined at a fixed

2.4 Expectation Values, Observables 35

point in phase space), and the time variation d(u)/dt is attributed to the fact that it is this probability (that u(q,p) assumes certain values) that depends explicitly on time. Then Eq. (2.27) becomes

! - / ( * P > ! " W > ( ^ ) "

= Ju(q,p){H(q,p),W(q,p,t)}(^y, (2.28)

where we used Eq. (2.22). However, we can also employ a more complicated consideration.^ Solving

the equations of motion for q,p, we can express these in terms of their initial values qo,Po, i-e. at t = 0, so that

Q = g(qo,Po,t), p = f(qo,Po,t), (2.29)

and hence

u(q,p) = u(q,p,0) =u(g(qo,po,t),f(qo,po,t),0) = u0{qo,p0,t). (2.30)

The distribution of the canonical variables is given by W(q,p,t). Thus we can write, since W oc p is constant in time according to Eq. (2.24):

W(q,p,t) = W(g(q0,po,t),f(q0,po,t),t)

= W(q0,po,0) = W0(q0,Po) at time t = 0, (2.31)

i.e. W is the density in the neighbourhood of a given point in phase space and has an implicit dependence on time t. With these expressions we obtain for the expectation value (U)Q:

(u)o = Ju0(qo,Po,t)W0(q0,p0)(^^-J. (2.32)

In this expression the time t is contained explicitly in the observable u(q,p) = uo{qo,Po,i). We expect, of course, that

(u) = (u)0. (2.33)

We verify this claim as follows. Reversing Eq. (2.29), we have

Qo = g(q,p,t), po = f(q,p,t), (2.34)

1See also H. Goldstein [114], Sec. 8.8.


so that on the other hand with Eq. (2.29)

q = g(g(q,P,t)J(q,P,t),t), p = f(

2.4 Expectation Values, Observables 37

Here we perform the transformation (2.34) and use (2.25) and (2.31), so that

dqdp\n

~d~t 0 = -J{H(q,p),u(q,p)}W(q,p,t)

2-irhJ (2.41)

This expression contains {H, u}W instead of u{H,W} in Eq.(2.28). How-ever, from the properties of the Poisson bracket, we obtain

{H, uW} = {H, u}W + u{H, W}.

The phase-space integral of a Poisson bracket like

(2.42)

I w-xtf)" vanishes under certain conditions. Consider

~dHd(uW) OHd(uW) {H,uW} = J2

i

= dqi dpi

_9_(dEuW dpi V dqi

dpi dqi

d fdH

dqi V dpi uW

If for all i:

lim uW = 0, lim uW 0, Pi->oo Oqi i j i^ ioo dpi

(2.43)

(2.44)

(which is reasonable since the density vanishes at infinity), we obtain zero after partial integration of I and hence from Eqs. (2.41) and (2.42) the rela-tion

d , v f , x r , ^ , , . , , / d q d p \n

dt


In the other case, called u Heisenberg picture", the probability of the initial values Wo(qo,po) is assumed," and the explicit time-dependence is transferred into the correspondingly transformed observables Uo(qo,po,t). The equation of motion is then that of an observable, i.e. (cf. Eq. (2.4))

^ = {(,,rt,F(.p)},

the reason being that since qo,po are constant initial values we have

du0(q0,p0,t) __ du0(qo,po,t) dt ~ dt

(2J30) du(q,p) It

(=4) {u(q,p),H(q,p)}. (2.46)

We thus also recognize the connection between the Liouville equation, as the equation of motion of an ensemble or of a probability distribution on the one hand, and the equation of motion (2.4) of an observable on the other.

2.5 Extension beyond Classical Mechanics

With

introduction to quantum mechanics schrodinger equation and path integral-part 1

Documents

liouville equation

introductory remarks

t schrodinger equation

harmonic oscillator

discovery of quantum

classical mechanics

hamiltonian mechanics

path integral introduction