QUANTUM THEORY OF TUNNELING 2nd Edition
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QUANTUM THEORY OF TUNNELING 2nd Edition MOHSEN RAZAVY University of Alberta, Canada World Scientific NEW JERSEY LONDON SINGAPORE BEIJING SHANGHAI HONG KONG TAIPEI CHENNAI
Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. QUANTUM THEORY OF TUNNELING 2nd Edition Copyright 2014 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN 978-981-4525-00-8 Printed in Singapore
To the memory of my father M.T. Modarres Razavy
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Preface to the Second Edition Since the appearance of the first edition of this book almost ten years ago, a large number of papers on quantum tunneling and its applications have appeared in print. These recent works have given us a better understanding of certain aspects of the quantum tunneling. Some have clarified our basic ideas, e.g. in multi-dimensional tunneling while others have shown us easier and more accurate computational methods for solving complicated problems. In this second edition, the contents of the text have been expanded and brought up-to-date by including many of the recent developments. In addition, a large number of references have been added and many misprints and mistakes of the first edition have been corrected. The presentation, as in the first edition, remains mathematical and, while lengthy analytical calculations are reproduced when it seemed essential, the results and methods used in the numerical computation are discussed briefly and the reader is referred to the original papers. For me it is not possible to acknowledge and give credit to all those who, in one way or the other, contributed to the subject of discussion in this book. I have borrowed extensively from their papers, methods, tables and graphs, but of course with proper acknowledgements. I am grateful to a number of my colleagues for informing me about the errors and for their valuable suggestions. In particular, I am indebted to Professors W. van Dijk, Y. Nogami, D.W.L. Sprung, J.G. Muga, M.R.A. Shegelski, W. Israel, Y. Sobuti and R. Khajepour for their help and encouragement. Finally, during the years that I have worked on this book, I have invariably received support from my family. In particular, I am indebted to my wife Ghodssi for her patience and to my daughter Maryam who helped me in preparing the manuscript. Mohsen Razavy Edmonton, Canada, April 2013 vii
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Preface to the First Edition The present book grew out of a lecture course given at the Institute for Advanced Studies in Basic Sciences, Zanjan, Iran in the summer of 1999. The intent at the outset was to present some of the basic results and methods of quantum theory of tunneling without concentrating on any particular application. It was difficult to decide what topics should be treated at length and which ones should be omitted from the discussion. Thus my main area of interest, the quantum theory of dissipative tunneling, was left out completely since even an introductory survey of the subject would have nearly doubled the size of the book. I am indebted to my dear friends and colleagues Professors Y. Sobouti and M.R. Khajepour for giving me the opportunity of lecturing to a group of enthusiastic graduate students and also encouraging me to write this monograph. I have benefitted immensely from discussions with my colleague Professor A.Z. Capri and with Mr. Robert Teshima. Above all, I am indebted to my wife who never failed to support me. ix
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Contents Preface to the Second Edition Preface to the First Edition Introduction 1 A Brief History of Quantum Tunneling 1 vii ix xix 2 Some Basic Questions Concerning Quantum Tunneling 8 2.1 Tunneling and the Uncertainty Principle............. 9 2.2 Asymptotic Form of Decay After a Very Long Time....... 11 2.3 Initial Stages of Decay....................... 12 2.4 Solvable Models Exhibiting Different Stages of Decay...... 17 3 Simple Solvable Problems 33 3.1 Confining Double-Well Potentials................. 33 3.2 Tunneling Through Barriers of Finite Extent........... 38 3.3 Tunneling Through a Series of Identical Rectangular Barriers.. 49 3.4 Eckart s Potential.......................... 54 3.5 Double-Well Morse Potential.................... 57 3.6 A Solvable Asymmetric Double-Well Potential.......... 60 4 Time-Dependence of the Wave Function in One-Dimensional Tunneling 64 4.1 Time-Dependent Tunneling for a δ-function Barrier....... 65 4.2 An Asymptotic Expression in Time for the Transmission of a Wave Packet............................. 73 5 Semiclassical Approximations 78 5.1 The WKB Approximation..................... 78 5.2 Method of Miller and Good..................... 88 5.3 Calculation of the Splitting of Levels in a Symmetric Double- Well Potential Using WKB Approximation............ 97 5.4 Energy Eigenvalues for Motion in a Series of Identical Barriers. 100 5.5 Tunneling in Momentum Space.................. 103 5.6 The Bremmer Series......................... 105 xi
xii Contents 6 Generalization of the Bohr-Sommerfeld Quantization Rule and Its Application to Quantum Tunneling 110 6.1 The Bohr-Sommerfeld Method for Tunneling in Symmetric and Asymmetric Wells.......................... 114 6.2 Numerical Examples........................ 117 7 Gamow s Theory, Complex Eigenvalues, and the Wave Function of a Decaying State 120 7.1 Solution of the Schrödinger Equation with Radiating Boundary Condition............................... 120 7.2 Green s Function for the Time-Dependent Schrödinger Equation with Radiating Boundary Conditions............... 124 7.3 The Time Development of a Wave Packet Trapped Behind a Barrier............................... 133 7.4 Method of Auxiliary Potential................... 137 7.5 Determination of the Wave Function of a Decaying State.... 143 7.6 Some Instances Where WKB Approximation and the Gamow Formula Do Not Work....................... 154 8 Tunneling in Symmetric and Asymmetric Local Potentials and Tunneling in Nonlocal and Quasi-Solvable Barriers 159 8.1 Tunneling in Double-Well Potentials................ 160 8.2 Tunneling When the Barrier is Nonlocal............. 165 8.3 Tunneling in Separable Potentials................. 169 8.4 Quasi-Solvable Examples of Symmetric and Asymmetric Double- Wells................................. 171 8.5 Gel fand-levitan Method..................... 174 8.6 Darboux s Method.......................... 176 8.7 Optical Potential Barrier Separating Two Symmetric or Asymmetric Wells............................. 178 9 Classical Descriptions of Quantum Tunneling 186 9.1 Coupling of a Particle to a System with Infinite Degrees of Freedom.............................. 186 9.2 Classical Trajectories with Complex Energies and Quantum Tunneling.............................. 192 10 Tunneling in Time-Dependent Barriers 198 10.1 Multi-Channel Schrödinger Equation for Periodic Potentials.. 199 10.2 Tunneling Through an Oscillating Potential Barrier....... 201 10.3 Separable Tunneling Problems with Time-Dependent Barriers. 210 10.4 Penetration of a Particle Inside a Time-Dependent Potential Barrier................................ 217
Contents xiii 11 Decay Width and the Scattering Theory 221 11.1 One-Dimensional Scattering Theory and Escape from a Potential Well............................... 222 11.2 Scattering Theory and the Time-Dependent Schrödinger Equation............................... 230 11.3 An Approximate Method of Calculating the Decay Widths... 235 11.4 Time-Dependent Perturbation Theory Applied to the Calculation of Decay Widths of Unstable States............. 240 11.5 Early Stages of Decay via Tunneling................ 244 11.6 An Alternative Way of Calculating the Decay Width Using the Second Order Perturbation Theory................ 246 11.7 Tunneling Through Two Barriers................. 249 11.8 R-matrix Formulation of Tunneling Problems.......... 253 11.9 Decay of the Initial State and the Jost Function......... 258 12 The Method of Variable Reflection Amplitude Applied to Solve Multichannel Tunneling Problems 267 12.1 Mathematical Formulation..................... 268 12.2 Variable Partial Wave Phase Method for Central Potentials... 275 12.3 Matrix Equations and Semi-classical Approximation for Many- Channel Problems.......................... 277 13 Path Integral and Its Semiclassical Approximation in Quantum Tunneling 284 13.1 Application to the S-Wave Tunneling of a Particle Through a Central Barrier........................... 288 13.2 Method of Euclidean Path Integral................ 291 13.3 Other Applications of the Path Integral Method in Tunneling. 296 13.4 Complex Time, Path Integrals and Quantum Tunneling..... 302 13.5 Path Integral and the Hamilton-Jacobi Coordinates....... 305 13.6 Path Integral Approach to Tunneling in Nonlocal Barriers... 308 13.7 Remarks About the Semiclassical Propagator and Tunneling Problem............................... 313 14 Heisenberg s Equations of Motion for Tunneling 318 14.1 The Heisenberg Equations of Motion for Tunneling in Symmetric and Asymmetric Double-Wells................... 319 14.2 Heisenberg s Equations for Tunneling in a Symmetric Double-Well............................. 325 14.3 Heisenberg s Equations for Tunneling in an Asymmetric Double- Well.................................. 326 14.4 Tunneling in a Potential Which is the Sum of Inverse Powers of the Radial Distance......................... 327 14.5 Klein s Method for the Calculation of the Eigenvalues of a Confining Double-Well Potential.................. 333
xiv Contents 14.6 Finite Difference Method for Tunneling in Confining Potentials. 340 14.7 Finite Difference Method for One-Dimensional Tunneling.... 343 15 Wigner Distribution Function in Quantum Tunneling 349 15.1 Wigner Distribution Function and Quantum Tunneling..... 353 15.2 Wigner Trajectory for Tunneling in Phase Space......... 356 15.3 Entangled Classical Trajectories.................. 361 15.4 Wigner Distribution Function for an Asymmetric Double-Well. 364 15.5 Wigner Trajectory for an Oscillating Wave Packet........ 365 15.6 Margenau-Hill Distribution Function for a Double-Well Potential 365 16 Decay Widths of Siegert States, Complex Scaling and Dilatation Transformation 369 16.1 Siegert Resonant States....................... 370 16.2 A Numerical Method of Determining Siegert Resonances.... 371 16.3 Riccati-Padé Method of Calculating Complex Eigenvalues.... 373 16.4 Complex Rotation or Scaling Method............... 376 16.5 Milne s Method........................... 380 16.6 Complex Energy Resonance States Calculated by Milne s Differential Equation........................ 382 16.7 S-Wave Scattering from a Delta Function Potential....... 384 16.8 Resonant States for Solvable Potentials.............. 386 17 Multidimensional Quantum Tunneling 391 17.1 The Semiclassical Approach of Kapur and Peierls........ 392 17.2 Wave Function for the Lowest Energy State........... 396 17.3 Calculation of the Low-Lying Wave Functions by Quadrature.. 398 17.4 Semiclassical Wave Function.................... 402 17.5 Tunneling of a Gaussian Wave Packet............... 408 17.6 Interference of Waves Under the Barrier.............. 413 17.7 Penetration Through Two-Dimensional Barriers......... 419 17.8 Method of Quasilinearization Applied to the Problem of Multidimensional Tunneling....................... 423 17.9 Solution of the General Two-Dimensional Problems....... 428 17.10 The Most Probable Escape Path.................. 432 17.11 An Extension of the Hamilton-Jacobi Theory and Its Application for Solving Multidimensional Tunneling Problems..... 438 17.12 A Time-Dependent Approach to the Problem of Tunneling in Two Dimensions........................... 444 18 Group and Signal Velocities 453 18.1 Exact Solution of the Problem of Tunneling in a Constant Barrier................................ 459
Contents xv 19 Time-Delay, Reflection Time Operator and Minimum Tunneling Time 468 19.1 Time-Delay Caused by Tunneling................. 469 19.2 Time-Delay for Tunneling of a Wave Packet........... 473 19.3 Landauer and Martin Criticism of the Definition of the Time- Delay in Quantum Tunneling.................... 482 19.4 Other Approaches to the Tunneling Time Problem........ 485 19.5 Time-Delay in Multichannel Tunneling.............. 488 19.6 Reflection Time in Quantum Tunneling.............. 491 19.7 Minimum Tunneling Time..................... 496 19.8 Traversal-Time Wave Function................... 498 20 More About Tunneling Time 505 20.1 Dwell and Phase Tunneling Times................. 506 20.2 Büttiker and Landauer Time.................... 516 20.3 Larmor Clock for Measuring Tunneling Times.......... 520 20.4 Tunneling Time and Its Determination Using the Internal Energy of a Simple Molecule...................... 524 20.5 Intrinsic Time............................ 526 20.6 Measurement of Tunneling Time by Quantum Clocks...... 529 20.7 A Critical Study of the Tunneling Time Determination by a Quantum Clock........................... 531 20.8 Tunneling Time According to Low and Mende.......... 537 21 Tunneling of a System with Internal Degrees of Freedom 545 21.1 Lifetime of Coupled-Channel Resonances............. 545 21.2 Two-Coupled Channel Problem with Spherically Symmetric Barriers................................ 547 21.3 Tunneling of a Simple Molecule.................. 551 21.4 Tunneling of a Homonuclear Molecule in a Symmetric Double- Well Potential............................ 554 21.5 Tunneling of a Molecule in Asymmetric Double-Wells...... 556 21.6 Tunneling of a Molecule Through a Potential Barrier...... 561 21.7 Tunneling of Composite Systems in Nuclear Reactions...... 573 21.8 Antibound State of a Molecule................... 580 22 Motion of a Particle in a Waveguide with Variable Cross Section and in a Space Bounded by a Dumbbell-Shaped Object 584 22.1 An Exactly Solvable Quantum Waveguide............ 587 22.2 Motion of a Particle in a Space Bounded by a Surface of Revolution.............................. 594 22.3 Testing the Accuracy of the Present Method........... 598 22.4 Calculation of the Eigenvalues................... 600 22.5 Quantum Wires........................... 603
xvi Contents 23 Relativistic Formulation of Quantum Tunneling 611 23.1 One-Dimensional Tunneling of the Electrons........... 611 23.2 Relativistic Effects in Time-Dependent Tunneling........ 616 23.3 Tunneling of Spinless Particles in One Dimension........ 621 23.4 Tunneling Time in Special Relativity............... 624 23.5 Quantum Tunneling Times for Relativistic Particles....... 630 24 Inverse Problems of Quantum Tunneling 641 24.1 A Method for Finding the Potential from the Reflection Amplitude.............................. 642 24.2 Determination of the Shape of the Potential Barrier in One- Dimensional Tunneling....................... 644 24.3 Construction of a Symmetric Double-Well Potential from the Known Energy Eigenvalues..................... 649 24.4 The Inverse Problem of Tunneling for Gamow States...... 652 24.5 Prony s Method for Determination of Complex Energy Eigenvalues.............................. 655 25 Some Examples of Quantum Tunneling in Atomic and Molecular Physics 660 25.1 Torsional Vibration of a Molecule................. 660 25.2 Electron Emission from the Surface of Cold Metals....... 663 25.3 Ionization of Atoms in Very Strong Electric Field........ 667 25.4 A Time-Dependent Formulation of Ionization in an Electric Field............................. 670 25.5 Energy Levels of the Ammonia Molecule and the Ammonia Maser................................. 674 25.6 Optical Isomers........................... 678 25.7 Three-Dimensional Tunneling in the Presence of a Constant Field of Force................................ 680 26 Some Examples in Condensed Matter Physics 688 26.1 The Band Theory of Solids and the Kronig-Penney Model... 688 26.2 Tunneling in Metal-Insulator-Metal Structures.......... 692 26.3 Many-Electron Formulation of the Current............ 693 26.4 Excitation of Closely Spaced Energy Levels in Heterostructures: The Time-Dependent Formulation................. 700 26.5 Electron Tunneling Through Heterostructures.......... 706 26.6 The Josephson Effect........................ 711 27 Alpha Decay 722 27.1 The Time-Independent Formulation of the α Decay....... 725 27.2 The Time-Dependent Formulation of the α Decay........ 729 27.3 The WKB Approximation..................... 734 27.4 Electromagnetic Radiation by a Charged Particle While Tunneling Through a Barrier.................... 739
Contents xvii 27.5 Perturbation Theory Applied to the Problem of Bremsstrahlung in α-decay.............................. 749 Index 759
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Introduction Quantum tunneling is a microscopic phenomenon where a particle can penetrate and in most cases pass through a potential barrier. The maximum height of the barrier is assumed to be higher than the kinetic energy of the particle, therefore such a motion is not allowed by the laws of classical dynamics. The simplest problems in quantum tunneling are one-dimensional and most of the research is done on these problems. But the extension of one-dimensional tunneling to higher dimensions is not straightforward. In addition there are certain characteristics that appear in two- or three-dimensional tunneling problems which do not show up in the one-dimensional motion. In most of the one-dimensional systems we study the motion of a particle in a potential V (x), where V (x) has a finite (or infinite) number of maxima. As long as the height of the barriers remain finite, the motion of the particle will not be restricted and we may choose the energy of the particle E to be greater than the asymptotic value of the potential say at x =. Then the simplest case will be that of a particle with energy E > V ( ) approaching the barrier from the left and then penetrating the barrier. Now depending on whether E > V ( ) or E < V ( ) the particle can pass through the potential or be reflected back and move to x =. The value of E is arbitrary as long as these conditions or inequalities are met. The other possibility is the one where the potential is finite on one side and tends to infinity on the other, e.g. V ( ) = and V ( ) is finite. In addition the potential has at least one local maximum say at x = a. Then depending on the boundary conditions of the problem we can have two different possibilities: (i) - If the particle moves from x = in the direction of x =, then there is the possibility that by tunneling the particle can enter the region between a and and stays there for a finite time (a metastable state). (ii) - If the initial condition states that the particle is in the region < x < a, then in the course of time by the process of tunneling the particle passes through the barrier and goes to x =. This initial state has also a finite lifetime. In both of these cases there are characteristic energies for which the tunneling probability is large, whereas for other energies it is small. xix
xx Introduction (iii) -The third possibility appears in the case where the potential V (x) is a confining potential, i.e. V (x) tends to infinity on both sides of the central maximum (or maxima), V (x) as x ±. In this case the motion of the particle will be restricted to a part of the x-axis. Depending on the number of maxima of V (x), we can have the eigenvalue problem for a double- or multi-well potential. The motion of the wave packet for double-wells, due to their importance in applied physics have received extensive treatment and will be discussed at length in this book. A double-well can be symmetric or asymmetric. The motion of a wave packet which represents the particle in a symmetric double-well, under certain conditions, can be obtained from the superposition of the eigenfunctions of the two lowest states of the system. When this happens then the wave packet oscillates between the two wells with a well defined frequency, and it also preserves its shape after successive back and forth tunneling. This very important case is called quantum coherence. For an asymmetric double-well, if the motion from one well to the other takes place by tunneling, then we have a situation which we call quantum hopping. In two or three dimensions when the barrier is only a function of the distance from the origin of the coordinate system, i.e. ρ = x 2 + y 2 in cylindrical coordinates or r = x 2 + y 2 + z 2, in spherical polar coordinates we can separate the variables in the Schrödinger equation and thus reduce the problem to a one-dimensional motion but now with the boundary conditions imposed at ρ = 0 or r = 0 and at ρ or r going to infinity. For instance in three dimensions if we assume that V (r ) 0 and that V (r) has a maximum at r = a, then if the particle is originally confined in the region 0 r < a, it can tunnel through the barrier and go to infinity. These special cases of two- or three-dimensional tunneling can be regarded as one-dimensional but the potentials V (ρ) and V (r) are replaced by V eff (ρ) and V eff (r), with the boundary condition that the reduced radial wave function must vanish at the origin. A Brief Review of the Contents of This Book In the first chapter of this book we present a brief history of the subject of quantum tunneling and the role that a number of pioneers played in its development. In the second chapter we discuss the physics of tunneling and the solution to the problem of local kinetic energy provided by the uncertainty principle. Furthermore we show that the principles of wave mechanics imply that, in general, the decay of the system, either because of tunneling or by some other mechanism, is nonexponential. Following this argument, we consider a special solvable problem to show that the exponential decay law is a very good approximation except for very short initial time and also after a very long time, and only at these extremes there are departures from the exponential decay. In the next chapter (Chapter 3) we solve a number of simple problems for which analytic solutions are known, and we determine, in appropriate cases, the lifetime of the quasistationary states, and or the motion of the wave packets. Of special interest is the resonant tunneling from either two identical barriers or a group of barriers. The time-dependence of the wave function in one-dimensional tunneling is
Quantum Theory of Tunneling xxi studied in Chapter 4, where time evolutions of a plane wave and a wave packet are formulated and their asymptotic forms are obtained. In Chapter 5 we study the semiclassical or WKB approximation and the conditions under which this approximation is valid. In addition we discuss another approach proposed by Miller and Good. We apply the WKB technique to calculate the energy separation between the two lowest levels of a symmetric double-well potential. Higher order corrections for WKB and Miller-Good approximations are also obtained in this chapter. In this connection we also consider a series known as Bremmer series in which the first term is the WKB wave function, but the sum of all of the terms converges to the exact wave function. Another important semiclassical approximation is the quantization rule of Bohr and Sommerfeld which is of great historical significance in atomic physics. In Chapter 6 we generalize this rule to the problems of quantum tunneling and we find the well-known Gamow formula for the decay of a system by means of tunneling. The same Bohr-Sommerfeld rule can also be used to determine the energy levels of symmetric and asymmetric double-wells. Gamow found his well-known formula by employing the complex eigenvalues and the Gamow states. In Chapter 7 we show that even though Gamow s approach is in apparent contradiction with the principles of quantum theory and this is a result of the approximate nature of his approach, nonetheless it is a useful approximation. Realizing that Gamow s approach is an approximation, we find that there are certain systems for which this approximation breaks down. In addition to discussing these systems, we find that the method has another shortcoming, viz, for the wave functions corresponding to the complex eigenvalues, the integral ψ 2 dx is divergent. A re-examination and resolution of this difficulty is also given in Chapter 7. At this point we also introduce the concepts of survival and nonescape probabilities and how they are related to the radiation boundary condition. In Chapter 8, we discuss the question of tunneling in nonlocal and separable potentials for which little is known. We also return to the problem of tunneling in double-wells. Here we will discuss the possibility of tunneling of a wave packet which originally is localized in one of the wells to the second well. While for two symmetric wells, the tunneling is always possible, for asymmetric wells tunneling is possible only when certain conditions are satisfied, we discuss these conditions in this chapter. Chapter 9 deals with the interesting question of the different classical descriptions of tunneling. First we show that this can be achieved by coupling the motion of the tunneling particle to a specific system with infinite degrees of freedom. Then we consider classical systems with non-hermitian Hamiltonians and show that such a classical system allows for trajectories in the classically forbidden regions similar to the penetration of waves in quantal mechanics. We assumed, up to this point, that the potential is time-independent. If the potential depends on space as well as on time, only for special cases the problem is exactly solvable. Because of the importance of this type of tunneling
xxii Introduction in the physics of layered semiconductors, we need a general method of formulating and solving the problem. In Chapter 10 we will investigate the simplest types of tunneling involving time-dependent potentials. When quantum tunneling is three-dimensional with an effective potential V (r) + h2 l(l+1) r, we can calculate the decay width or the lifetime from scattering 2 theory. In Chapter 11 we first establish the connection between the quantum scattering theory and the width of the decaying states, and we formulate two parallel approaches to this subject. We show how for different potentials we can calculate the decay width exactly or approximately. This formulation also enables us to study the time-dependence of the decay of initial state by tunneling, both for early times and in the exponential regime. When the tunneling particle (or system of particles) has internal degrees of freedom, like a simple molecule, and also in some cases where the timedependence of the potential is sinusoidal, we can decompose the Schrödinger equation into an infinite set of coupled equations. A simple and accurate method for solving a set of equations of this type (which can only be solved numerically) is the method of variable reflection coefficient. Chapter 12 is devoted to a study of this technique and its applications. In three subsequent chapters 13-15 we study alternative ways of formulating the tunneling problem starting with Feynman s method of path integration. Most of the techniques based on Feynman s method such as instantons, Euclidean path integration and the introduction of complex time in the formulation are developed for use in subatomic and particle physics. However they can be applied directly to the simpler cases like systems with few degrees of freedom. The difficulty of generalizing these ideas to multi-dimensional systems is a serious limitation of the method, and at present it is not clear whether the introduction of complex time would enable one to overcome this difficulty or not. Near the end of Chapter 13 we discuss an interesting method of using the Hamilton-Jacobi coordinates and path integration to solve a simple tunneling problem. But again it is not known whether this approach can be extended to other more complicated systems or not. Continuing our discussion of the alternative methods of solving different tunneling situations, we investigate the solution of Heisenberg s equations of motion for quantum tunneling of a single particle. Up to now only few problems, namely those involving potentials expressible as polynomials in x (or in 1 r ) and of the form n a nx n or n b nr n have been studied. The advantage of using operator equations for solving tunneling problems lies in the facts that (i) - the initial wave packet does not change in time and (ii) - since Heisenberg s equations are similar to the classical equations of motion therefore the definition of some of the dynamical quantities such as tunneling time is clearer in this formulation. On practical side, Heisenberg s equations have been used (a) - to calculate eigenvalues for confining double-well potentials and (b) - by replacing the operator differential equations by difference equations one can calculate tunneling in confining potentials and also for tunneling of a wave packet through a barrier.
Quantum Theory of Tunneling xxiii The fourth approach which we study in Chapter 15 is a method based on the Wigner distribution function. Using this distribution function we can follow the motion of a wave packet in phase space. As an example we present the case of tunneling through two rectangular barriers and show how this can be used to determine a tunneling time. A related and interesting question is that of finding the Wigner trajectory for a wave packet formed from the superposition of the two lowest eigenfunctions. As we mentioned earlier this is a coherent (shape preserving) tunneling with a period T = 2π E 1 E 0, where E 0 and E 1 are the two lowest eigenvalues of the system. Preliminary result suggests that the Wigner trajectory in the phase space for this motion does not have a fixed period and this is a strange result. Methods for determining Siegert resonant states numerically and also calculating the decay widths using complex scaling and the dilatation transformation are subjects for discussion in Chapter 16. In Chapter 17 we study the important and difficult problem of multi-dimensional tunneling when the wave equation is not separable. There is considerable amount of published works about this subject, but as yet a satisfactory and reliable technique for determining the motion of the particle has not been found. In particular, questions concerning the most probable escape path and the tunneling time need more careful investigation. In addition to the direct methods of solving the wave equation, we also discuss the interesting idea of extending the Hamilton-Jacobi theory to solve the wave equation for multi-dimensional problems. The three following chapters 18-20 deal with different aspects of the tunneling time. As a way of introducing the subject, we start by defining the physical concepts of group and signal velocities for a wave. Then by following the works of Sommerfeld, Brillouin and Stevens, we show how we can calculate the tunneling time for simple potentials. In Chapter 19 we continue our discussion of the tunneling time with an examination of the idea of time-delay. Starting with the classically well-defined concept of the time of flight over a barrier, we first develop a semiclassical formulation and later present a definition of the quantal time-delay and show its relationship to the Wigner inequality. We observe that the definition of the time-delay can be extended to the case of confining potentials. To this end we make use of the Schwinger work which relates the phase shift to the splitting of the energy levels and we find a connection between the time of oscillation from one well to the other and the derivative of phase shift with respect to energy. These formulations are mostly based on the form of classical travel time of the particle and do not have a proper definition in terms of Hermitian operators for observables. However if we follow the works of Keleber et al., we can show two cases where rigorous definitions of tunneling times are possible. These two cases are reflection time and the minimum tunneling time (Chapter 18). In Chapter 20 we discuss two methods suggested by Büttiker and Landauer for the measurement of tunneling time using the motion of the particle as a quantum clock. We also study the possibility of the determination of the tunneling time from the excitation of a molecule. A critical examination of the
xxiv Introduction role of a quantum clock for the measurement of the tunneling time, and the difficulties that one encounters in reading this typical quantum clock are also studied in this chapter. We conclude Chapter 20 by pointing out an interesting, but controversial observation of Low and Mende. In Chapter 21 we consider the question of tunneling of a simple diatomic molecule. We observe that the time of oscillation of a molecule in a symmetric double-well potential is dependent on the harmonic force which binds the two atoms of the molecule. When the double-well is asymmetric, only for cases of resonance the tunneling of the molecule from one well to the other is possible. We show that for those cases where the molecule is initially trapped behind a barrier, and then by means of tunneling escapes to infinity, the method of variable reflection coefficient offers a convenient way of finding the solution. If a particle is constrained to move within an impenetrable surface, e.g. in a waveguide or two cavities connected by a narrow passage, then tunneling plays an important role in determining stationary and propagating modes in waveguides and the energy levels in the cavities (Chapter 22). Chapter 23 is devoted to relativistic quantum tunneling. In the first part we obtain the approximate solution of the Dirac equation for a fermion tunneling through a barrier and in the second part we calculate the decay width for tunneling of a boson. In Chapter 24 we study the problem of construction of the potential barrier from one of the following sets of data: (i) -The reflection amplitude for all energies. (ii) - The transmission coefficient for the energies below the maximum height of the potential. (iii) - From the discrete complex eigenvalues for Gamow (or Siegert) states. In Chapter 25 we consider the application of the concept of tunneling in a number of problems of atomic and molecular physics where tunneling is an essential mechanism for the observed phenomena. Among these we discuss the torsional oscillations of a molecule, cold emission of electrons from a metal, ionization of atoms in a strong electric field, ammonia maser and the three-dimensional tunneling in a linear potential. Among the many applications to the condensed matter physics, we discuss the band theory of solids and tunneling through the metal-insulator-metal structures, the theory of electron tunneling through heterostructures and the Josephson effect (Chapter 26). Finally in the last chapter (Chapter 27) we study the α-decay of a radioactive nucleus and the verification of the Geiger-Nuttall empirical formula. Since in α decay the α particle is moving above or under the barrier, and in both cases it experiences acceleration resulting in the emission of photon, this bremsstrahlung can provide additional information about the nuclear structure and about the interaction causing decay. This aspect of the α decay will be discussed in detail.