Topological Effects in Quantum Mechanics

Fundamental Theories of Physics

An International Book Series on The Fundamental Theories of Physics: Their Clarification, Development and Application

Editor: ALWYN VAN DER MERWE, University of Denver, U.S.A.

Editorial Advisory Board: LAWRENCE P. HORWITZ, Tel-Aviv University, Israel BRIAN D. JOSEPHSON, University of Cambridge, U.K. CLIVE KILMISTER, University of London, U.K. PEKKA J. LAHTI, University of Turku, Finland GUNTER LUDWIG, Philipps-Universitiit, Marburg, Germany NATHAN ROSEN, Israel Institute of Technology, Israel ASHER PERES, Israel Institute of Technology, Israel EDUARD PRUGOVECKI, University of Toronto, Canada MENDEL SACHS, State University of New York at Buffalo, U.S.A. ABDUS SALAM, International Centre for Theoretical Physics, Trieste, Italy HANS-JURGEN TREDER, Zentralinstitut for Astrophysik der Akademie der

Wissenschaften, Germany

Volume 107

Topological Effects in Quantum Mechanics

by

G. N. Afanasiev Laboratory QfTh~oretical Physic:r, jain! Institute for Nuclear Research, Dubno. MOSCQw, Russia

.. SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.

A C.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN 978-94-010-5959-6 ISBN 978-94-011-4639-5 (eBook) DOI 10.1007/978-94-011-4639-5

Printed on acid-free paper

AH Rights Reserved © 1999 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1999 Softcover reprint of the hardcover 1 st edition 1999 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permis sion from the copyright owner.

Contents

PREFACE IX

1 INTRODUCTION 1

2 VECTOR POTENTIALS OF STATIC SOLENOIDS 7 2·1 Cylindrical Solenoids . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2·1.1 One Cylindrical Solenoid . . . . . . . . . . . . . . . . 7 2·1.2 Vector Potentials of Two Cylindrical Solenoids with Opposite

Fluxes. . . . . . . . . . . . . . . . 9 2·2 Vector Potential of the Toroidal Solenoid. . . . . . . . . . . . . . . . 13

2·2.1 Toroidal Coordinates. . . . . . . . . . . . . . . . . . . . . . . 13 2·2.2

2·2.3 2·2.4

2·2.5

2·2.6

2·2.7

Vector Potential of a Toroidal Solenoid in the Coulomb Gauge Alternative Representation of the Vector Potentials. Integral Representation of the Vector Potential Thin Toroidal Solenoid. . . . . . . . . . . . . . The Vector Potential in a Non-Standard Gauge Generating Function for the Toroidal Solenoid

15 18 20 23 25

30

3 ELECTROMAGNETIC PROPERTIES OF STATIC SOLENOIDS 35

3·1 Conditions for the Existence of Static Magnetic Solenoids 35 3·2 Currents, Magnetic Dipoles and Monopoles 38 3·3 Electric Static Solenoids . . . . . . . . . . . . . . . . . . . 40

3·3.1 Another Example. . . . . . . . . . . . . . . . . . . 42 3·3.2 How Can One Tell if an Electric Field Exists Inside an Electric

Solenoid? ..... . . . . . . . . . . . . . . . . . . . . . . 43 3·3.3 On the Electric VP Outside an Electric Toroidal Solenoid . . . .. 44

v

VI CONTENTS

3·4 On the Disappearance of the Magnetic Field Inside a Solenoid. . . 44 3·5 Vector Potentials of Toroidal Solenoids with Different Asymptotic

Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3·6 Solenoids with Non-trivial Helicity . . . . . . . . . . . . . . . . . . 50 3·7 Electromagnetic Field of a Cylindrical Solenoid Moving Uniformly in a

Medium. . . . . . . . . . . . . . . . . . . 52 3·8 Generalised Helmholtz Coils and All That . . . . . . . . . . . 55

3·8.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . 56 3·8.2 On Sums and Integrals Involving Legendre Functions. 59 3·8.3 On Helmholtz Coils and Their Generalisations 62 3·8.4 Concluding Remarks on Helmholtz Coils. . . . . . . . 74

4 INTERACTION OF MAGNETISATIONS WITH AN EXTERNAL ELECTROMAGNETIC

FIELD AND A GENERALISATION OF AMPERE'S HYPOTHESIS 77 77 4·1 Introduction ............................. .

4·2 Magnetisation, Toroidalisation, and a Generalisation of Ampere's Hypothesis ......................... 77

4·3 Interaction with an External Electromagnetic Field. . . 81 4·4 Magnetisations and the Debye Potential Representation 83 4·5 Physical Meaning of the W Functions . . . . . . . . . . . 84 4·6 Transition to Point-Like Sources ............. 89 4·7 The Interaction of Charge Densities with an External Field 90 4·8 Motion of a Toroidal Solenoid in an External Electromagnetic Field 91 4·9 On the Inversion of the Debye Parametrisation 96 4·10 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . 97

5 ELECTROMAGNETIC PROPERTIES OF TIME-DEPENDENT SOLENOIDS

5·1 Cylindrical Solenoids ............... . 5·1.1 Particular Time Dependences of Currents

5·2 Toroidal Solenoids ......... . 5·2.1 Historical Introduction ......... .

99 99 99

103 103

5·2.2 Concrete Time Dependences ...... 105 5·3 Elementary Time-Dependent Toroidal Sources. 110

5·3.1 The Radiation of Elementary Toroidal Sources 110 5·3.2 Finite Toroidal-Like Configurations. . . . . . . 116

5·4 Vector Solutions of the Laplace Equation ....... 118 5·4.1 The Main Properties of Elementary Vector Potentials and Vector

Spherical Harmonics . . . . . . . . . . . 118 5·4.2 Vector Solutions of Laplace's Equation. 121 5·4.3 Toroidal Multipole Moments 126 5·4.4 A Concrete Example . . . . . . . . . . . 129

6 RADIATIONLESS TIME-DEPENDENT CHARGE-CURRENT SOURCES

6·1 Introduction................ 6·1.1 6·1.2 6·1.3 6·1.4

Conditions for Radiationlessness ... . . . . Particular Cases ............... . Application to the Vavilov-Cherenkov Effect Non-Static Electric Capacitors ....... .

131 131 131 132 135 136

Contents VII

6·2 On Radiationless Topologically Non-Trivial Sources of Electromagnetic Fields .................. . . . . . . . . . . . . . 139 6·2.1 The Simplest Non-Trivial Non-Radiating Sources. . . 139

6·3 On Current Configurations Generating a Static Electric Field 142 6·3.1 On Current Electrostatics 145

6·4 On Electric Vector Potentials . . . . 148 6·4.1 Magnetic Analogue . . . . . . 150

6·5 More General Radiationless Sources 151 6·6 Concluding Remarks Toroidal Radiationless Sources 153

7 SELECTED TOPOLOGICAL EFFECTS OF QUANTUM MECHANICS 155 7·1 An Elementary Exposition of the Bohm-Aharonov Effect 155

7·1.1 B-A Scattering on a Single Finite Cylindrical Solenoid 155 7·1.2 Feinberg VB Bohm-Aharonov . . . . . . . . . . . . . . 157 7·1.3 Resolution of the Controversy. . . . . . . . . . . . . . 159 7·1.4 Application of a Generating Function to Scattering by a Cylindrical

Solenoid . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 7·1.5 Scattering for the Exceptional Value of a Magnetic Flux 162 7·1.6 On the Single-Valuedness of the B-A Wave Function 164 7·1.7 On the Physical Nature of the Dirac Phase Factor . . . 165

7·2 Attempts at Proving the Non-Existence of the B-A Effect . . . 169 7·3 Scattering of Charged Particles on Two Infinite Cylindrical Solenoids. 173

7·3.1 The Born Approximation . . . . . . . . . . . . . . . . . . 174 7·3.2 The High Energy Approximation . . . . . . . . . . . . . . 176 7·3.3 Scattering for an Unusual Orientation of the Wave Vector 177 7·3.4 Fraunhofer Diffraction on Two Cylindrical Solenoids . . . 178 7·3.5 Fresnel Diffraction on Two Cylindrical Solenoids . . . . . 181 7·3.6 Uncertainties in the Interpretation of the Experimental Data 184 7 ·3. 7 Measurement of the Intensity in the Direct Beam . 185

7·4 Scattering of Charged Particles by a Toroidal Solenoid 189 7·4.1 The Fraunhofer Approximation. . . . . . . . . . . 189 7·4.2 The Fresnel Approximation . . . . . . . . . . . . . 192 7·4.3 Numerical Investigation of Electron Diffraction by a Toroidal

Solenoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 7·4.4 Tonomura's Experiments .................... 194 7·4.5 Scattering for an Unusual Direction of the Initial Wave Vector 201 7·4.6 Scattering of Charged Particles by a Toroidal Solenoid with

Non-Zero Helicity .................. 202 7·5 On the Super-Current Arising in a Superconducting Ring . . . . . 202 7·6 The Time-Dependent Bohm-Aharonov Effect . . . . . . . . . . . . 205 7·7 Scattering of Charged Particles in an Ideal Multiply Connected Spacial

Region. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 7·8 On the Alternative Interpretation of the Bohm-Aharonov Effect 208

7·8.1 An Infinite Cylindrical Solenoid. . . . . . 211 7·8.2 Two Cylindrical Solenoids with <PI = -<P2 . . . . . . 212

7·9 The B-A Effect for Bound States. . . . . . . . . . . . . . . 213 7·10 Quantum Impenetrability and Bohm-Aharonov Scattering. 215

7·10.1 Non-Relativistic Impenetrability Conditions 216 7·10.2 Relativistic Impenetrability Conditions 224

VIII

7·11 The Aharonov-Casher Effect ................. . 7·11.1 The Toroidal Aharonov-Casher Effect ....... . 7·11.2 On the Experimental Verification of the A-C Effect

CONTENTS

234 234 239

8 TOPOLOGICAL EFFECTS FOR THE FREE ELECTROMAGNETIC FIELD 241 8·1 Introduction ................................... 241 8·2 Relativistic Helicity and Its Physical Meaning . . . . . . . . . . . . . . . . 243 8·3 Gauge-Invariant Representation for the Energy of a Weak Gravitational

Field. . . . . . . . . . . . . . . 246 8·4 Photon Densities . . . . . . . . . . . . ... . 249

249 250

252 255

8·4.1 Historical Introduction. . . . . . .. . 8·4.2 Preliminaries about Photon Densities 8·4.3 The Conservation Laws 8·4.4 External Currents .....

8·5 Numerical Results ......... 256 8·6 On the Ehrenfest-Pauli Objections 264 8·7 On the Hegerfeldt Theorem and All That 265

8·7.1 The Klein-Gordon Equation 265 8·7.2 The Wave Equation . . . . . . . . 266 8·7.3 Maxwell's Equations . . . . . . . . 268 8·7.4 On the Transformations of Photon Densities. 270 8·7.5 Causality and Positive Definiteness of the Probability Density. 272

8·8 Electromagnetic Waves versus Photons 274 8·9 Discussion.......................... 277

9 TOPOLOGY OF THE VAVILOV-CHERENKOV RADIATION FIELD 9·1 Introduction.........................

281 281 282 9·2 Mathematical Preliminaries ............... .

9·3 Electromagnetic Field of a Uniformly Moving Charge on Finite and Infinite Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 9·3.1 Uniformly Moving Charge with Velocity v < en . . . . . . 283 9·3.2 Uniform Motion of a Charge with Velocity v > en 283 9·3.3 Uniform Motion with v < en on a Finite Spacial Interval. 286 9·3.4 Uniform Motion with v> en on a Semi-Finite Spacial Interval 288 9·3.5 Uniform Motion at v > e" on a Finite Time Interval . . . . .. 291 9·3.6 Discussion of the Results Obtained . . . . . . . . . . . . . . .. 295

9·4 Vavilov-Cherenkov Effect and Accelerated Motion of a Charged Particle 296

9·4.1 The Motion of Charge under Constant Acceleration 297 9·4.2 Charge Motion in a Constant Electric Field . . . . . . . . . . .. 310

REFERENCES 321

INDEX 337

Preface

Classical electromagnetism and quantum mechanics form the foundation of modern theo­retical physics. In my opinion the best expositions of them are the ten volume course on theoretical physics by Landau and Lifshitz, Jackson's Classical Electrodynamics and Schiff's Quantum Mechanics. These books were published at least twenty years ago and there is some gap between their scope and the subject matter which at present is treated in a variety of articles and reviews published in scientific journals. This book is aimed at filling this gap at least in part.

This book has a threefold aim. First, to teach a graduate student how to solve concrete problems in classical electrodynamics and quantum mechanics. Second, to in­troduce him to topics of mathematical physics and show, for example, how different solutions of the same physical problem can yield new relations between special functions of mathematical physics. Third, and perhaps this is the most important goal, is to intro­duce the reader to modern problems of electromagnetic theory and quantum mechanics. We mean, for example, the electromagnetic fields of static and non-static solenoids, the Bohm-Aharonov and Aharonov-Casher effects, the Tonomura experiments testing the foundations of quantum mechanics, localisability of electromagnetic waves and photons, new aspects of Vavilov-Cherenkov radiation, and all that.

The book begins with the discussion of the properties of cylindrical and toroidal solenoids, in Chapter 2. It is shown that the use of different gauges of electromagnetic potentials permits one to solve more effectively the concrete problem being treated. By comparing vector potentials obtained in different ways we obtain new relations between special functions of mathematical physics.

Chapter 3 is devoted to the solution of concrete problems related to solenoids. The conditions for their existence are explicitly formulated.

Electric solenoids are introduced having the property that their electric field strength

IX

x PREFACE

is confined entirely within the solenoid, whilst outside it there is an electric vector pot­ential which can not be removed by a gauge transformation.

The vector potential of the ordinary toroidal solenoid decreases as 1/r3 at large distances. We find distributions of currents (or magnetisations) inside a torus which (for the same magnetic flux inside the torus) generate vector potentials falling off as 1/r2n+I as r -t 00.

For a static magnetic solenoid at rest the magnetic field strength is confined to its interior. The same is true for a solenoid moving uniformly in vacuo (because it is possible to use a Lorentz transformation to pass to the reference frame in which the solenoid is at rest). However, the electromagnetic field of a solenoid moving in a medium extends beyond its boundary (the previous argument does not work, because in the rest frame mentioned above the medium moves relative to the solenoid). This is explicitly shown for the case of a cylindrical solenoid moving uniformly in a medium.

In the same chapter the generalized Helmholtz coils are considered. They have many attractive properties. In particular, they can be placed on the surface of a sphere in such a way as to cancel any set of multipoles of the magnetic field inside the sphere S. This permits one to create a magnetic field inside the sphere with the prescribed properties.

Ampere's hypothesis states that the magnetic field of a current in an infinitely small circular coil is equivalent to that of magnetic dipole orthogonal to the plane of the circular current. In Chapter 4 we generalize Ampere's hypothesis to more general current con­figurations and show how these configurations interact with an external electromagnetic field.

The electromagnetic properties of time-dependent solenoids are discussed in Chap­ter 5. Time-dependent current configurations are found which generate zero electromag­netic field strengths and time-dependent potentials outside them. Similarly to the vector solutions of the Helmholtz equation, we find vector solutions of the Laplace equation and apply them to clarifying the physical meaning of the so called toroidal moments.

Radiationless time-dependent charge-current configurations are discussed in Chap­ter 6. We prove the existence of time-dependent charge-current configurations corre­sponding to zero electromagnetic field strengths and non-vanishing time-dependent elec­tromagnetic potentials (unable to be removed by a gauge transformation) outside these configurations. These radiationless configurations have many practical applications.

Selected topological effects of quantum mechanics are discussed in Chapter 7. It begins with an elementary exposition of the Bohm-Aharonov effect. The controversies in its interpretation are discussed. The B-A effect is discussed for one cylindrical solenoid, for two cylindrical solenoids with opposite magnetic fluxes, and for the toroidal solenoid. The validity of different approximations (Fraunhofer, Fresnel, Kirchhoff, high energy, etc.) for the description of the B-A effect is analysed. A non-standard direction is found for the wave-vector of incoming electrons so that the electron wave function disappears in some region of space for the particular value of the magnetic flux inside the solenoid. This particular property may be used as an unambigious test of the influence of the enclosed fields.

A numerical investigation of some famous Tonomura experiments testing the foun­dations of quantum mechanics is given.

The appearance of the electric current inside a circular tum embracing the magnetic flux (but lying entirely within the region where the magnetic field is equal to zero) when this circular turn becomes superconducting is also discussed in the same section. The appearance of the above electric current leads in tum to the appearance of a magnetic field in the surrounding space. This magnetic field is evaluated in a closed form.

Preface xi

The insufficiency of some of the alternative interpretations of the B-A effect is also discussed.

Ordinary solenoids are characterised by the sole topological invariant the magnetic flux. There exist other topological invariants (in addition to the magnetic flux) which characterise more subtle distributions of a magnetic field. The simplest of them is helicity, which characterises how much a particular magnetic line is twisted (or knotted). We estimate how the value of the helicity affects the B-A cross-section.

The different definitions of quantum impenetrability are also discussed. They lead to physically different B-A scattering cross-sections, and this should be kept in mind when analysing experimental results.

It is known that to the B-A effect there corresponds a dual effect. We refer to the so called Aharonov-Casher effect. For a cylindrical geometry the B-A effect is the scattering of electric charges by an impenetrable cylindrical solenoid. The corresponding A-C effect is the scattering of magnetic dipoles by a charged infinite filament. For toroidal geometry the B-A effect is the scattering of electric charges by a toroidal solenoid. It is shown in the same chapter that the A-C effect dual to the toroidal B-A effect is the scattering of toroidal moments by a Coulomb centre.

The localisability problem for electromagnetic waves and photons is discussed in Chapter 8. After a brief historical introduction we discuss the difference between electro­magnetic waves and photons. An electromagnetic wave is able to be localisable because it contains both positive and negative frequency components. Photons are not able to be localisable because their wave function (i. e., E and H) contains only positive fre­quency components. The notion of helicity for a free electromagnetic field is analysed. The generalised helicity is introduced, which is a conserved quantity coinciding with the difference between the right and left circularly polarised photons comprising the electro­magnetic field. It seems that it completes the list of the zilch type of invariants found by Lipkin and Ragusa. A gauge invariant expression for the energy of the free gravita­tional field is obtained which strongly resembles the well known bilinear expression for the total number of photons comprising the electromagnetic field. Numerical investiga­tions of different photon densities and the corresponding conservation laws are presented. The Ehrenfest-Pauli objections against the use of such densities are analysed and partly removed. It is shown how the non-Iocalisibility of a single photon can be reconciled with the localisability of the electromagnetic wave.

New aspects of Vavilov-Cherenkov radiation are considered in Chapter 9. Exact expressions are found for the electromagnetic field arising from the instant acceleration of a charged particle, its subsequent motion at a velocity exceeding the velocity of light in the medium, and the instant transition into the state of rest (Tamm's problem). It turns out that these expressions have definite advantages over the usual expressions founded upon the use of the Fourier transform (Tamm's approximate solution). In particular, they clearly show when and where the Cherenkov radiation should be observed in order to discriminate it from the bremsstrahlung.

The effects arising from accelerated and decelerated motion of a charged point part­icle inside a medium are also studied. This motion is manifestly relativistic, and may be produced by a constant uniform electric field. It is shown that in addition to the bremsstrahlung and Cherenkov shock wave a previously unknown electromagnetic shock wave arises when the velocity of the charge coincides with the velocity of light in that medium. For accelerated motion this shock wave, forming an indivisible entity with the Cherenkov shock wave, arrives after the arrival of the bremsstrahlung shock wave. For decelerated motion the above shock wave detaches from the charge at the moment when

xii PREFACE

its velocity coincides with the velocity of light in the medium. This wave, existing even after the termination of the motion of the charge, propagates with the velocity of light in the medium and arrives before the arrival the bremsstrahlung shock wave. This shock wave, having the same singularity as the Cherenkov shock wave, is more singular than the bremsstrahlung shock wave. The space-time regions in which these shock waves exist, and conditions under which they can be observed, are determined.