LONG-RANGE CASIMIR FORCES. Theory and Recent Experiments on Atomic Systems

Transcription

1 LONG-RANGE CASIMIR FORCES Theory and Recent Experiments on Atomic Systems

2 Finite Systems and Multiparticle Dynamics Series Editors Frank S. Levin, Brown University, Providence, Rhode Island David A. Micha, University of Florida, Gainesville, Florida LONG-RANGE CASIMIR FORCES: Theory and Recent Experiments on Atomic Systems Edited by Frank S. Levin and David A. Micha A Continuation Order Plan is available for this series. A continuation order will bring delivery of each new volume immediately upon publication. Volumes are billed only upon actual shipment. For further information, please contact the publisher.

3 LONG-RANGE CASIMIR FORCES Theory and Recent Experiments on Atomic Systems Edited by FRANK S. LEVIN Brown University Providence, Rhode Island and DAVID A. MICHA University of Florida Gainesville, Florida Springer Science+Business Media, LLC

4 Library of Congress Cataloging-in-Publication Data Long-range Casimir forces : theory and recent experiments on atomic systems / edited by Frank S. Levin and David A. Micha. p. cm. (Finite systems and mu 11ipartic 1 e dynamics) Includes bibliographical references and index. ISBN Casimir effect. I. Levin, F. S. (Frank S.), 1933 II. Micha, David. III. Series. QC680.L '7 dc CIP ISBN ISBN (ebook) DOI / Springer Science+Business Media New York Originally published by Plenum Press, New York in 1993 Softcover reprint of the hardcover 1st edition 1993 All rights reserved No part of this book may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher

5 CONTRIBUTORS RICHARD J. DRACHMAN Laboratory for Astronomy and Solar Physics, NASA-Goddard Space Flight Center, Greenbelt, Maryland G. W. F. DRAKE Department of Physics, University of Windsor, Windsor, Ontario, Canada N9B 3P4 GERALD FEINBERG Department of Physics, Columbia University, New York, New York S. R. LUNDEEN Department of Physics, University of Notre Dame, Notre Dame, Indiana LARRY SPRucn Department of Physics, New York University, New York, New York JOSEPH SUCHER Center for Theoretical Physics, Department of Physics, University of Maryland, College Park, Maryland 20742

6 EDITORS' FOREWORD This series, Finite Systems and Multiparticle Dynamics, is intended to provide timely reviews of current research topics, written in a style sufficiently pedagogic so as to allow a nonexpert to grasp the underlying ideas as well as understand technical details. The series is an outgrowth of our involvement with three interdisciplinary activities, namely, those arising from the American Physical Society's Topical Group on Few Body Systems and Multiparticle Dynamics, the series of Gordon Research Conferences first known by the title "Few Body Problems in Chemistry and Physics" and later renamed "Dynamics of Simple Systems in Chemistry and Physics," and the series of Sanibel Symposia, sponsored in part by the University of Florida. The vitality of these activities and the enthusiastic response to them by researchers in various subfields of physics and chemistry have convinced us that there is a place--even a need-for a series of timely reviews on topics of interest not only to a narrow band of experts but also to a broader, interdisciplinary readership. It is our hope that the emphasis on pedagogy will permit at least some of the books in the series to be useful in graduate-level courses. Rather than use the adjective "Few-Body" or "Simple" to modify the word "Systems" in the title, we have chosen "Finite." It better expresses the wide range of systems with which the reviews of the series may deal. On the "few-body" side these could include nucleons or deuterons, light nuclei, low-z atoms, and small molecules; on the "many-body" side, possibilities include polyatomic molecules, heavier nuclei, clusters, and "few-body" dynamics in larger systems and at interlaces, although infinite systems like nuclear matter or the electron gas are excluded. The series will emphasize theoretical, computational, and experimental methods. It is clear that researchers in fields such as molecular and nuclear sciences, for example, may develop theoretical methods similar to those of another field, often without either group being aware of it. These parallel developments are the result of efforts made to vii

7 viii EDITORS' FOREWORD analyze complex physical systems in terms of their simpler constituents, and their complex dynamical properties in terms of the dynamics of subsystems and of collective modes. A similar comment holds with respect to experimental techniques. The pedagogic aspect of the reviews commissioned for Finite Systems and Multiparticle Dynamics is designed specifically to promote both learning and cross-disciplinary exchanges. To give some notion of the scope we envisage for the series, we list below the general area of each of the first three volumes: Vol I. Long-Range (Casimir) Forces Vol II. Coulomb Forces in Three-Particle Atomic and Nuclear Collision Systems Vol III. Recent Developments in Molecular and Nuclear Reaction Dynamics Topics for further volumes include time-dependent methods for collisions and numerical and computational techniques for bound states of finite systems. It is our hope and expectation that this series will encourage and enhance the interdisciplinary activities already fostered by the organizations and meetings noted at the start of this foreword.

8 PREFACE The topic of this first volume in the series Finite Systems and Multiparticle Dynamics is long-range (Casimir) forces. Although the system treated in Chapters 2, 3, and 4 is the two-electron atom, in particular He, the general topic is one for which it is possible to provide a broad, qualitative introduction where the basic physics can be stressed. We feel very fortunate that Larry Spruch not only agreed to write such an introductory chapter but also that he succeeded so well in conveying the ideas, unburdened by extensive mathematical detail. The pedagogic tone set in Spruch's chapter is, happily for us and for the readers of this volume, matched by those of the other contributors. All the authors are experts on the subject and have made major contributions to our understanding of it. And each responded to our request for a comprehensive review written in a manner comprehensible to nonexperts with a chapter that exceeded our expectations. Strangers to this field of research should have little difficulty, if any, in grasping not only the broad picture but many of the details as well. Chapter 2, by Stephen Lundeen, describes the experimental challenges and advances involved in measurements from which the effects of long-range Casimir forces can be extracted. The results of the state-ofthe-art experimental work of Lundeen and collaborators is compared with a brief overview of the theoretical situation. This theoretical situation is described in detail in the next two chapters. Gordon Drake shows in Chapter 3 how results of extremely high accuracy, including a variety of relativistic corrections, can be obtained via a specific variational procedure applied to the He atom. Both low-lying energies and Rydberg levels relevant to Casimir forces are calculated to many significant figures and compared to the experimental results. Chapter 4, written by Richard Drachman, describes the optical potential method for determining the energies of arbitrary two-electron atoms, in particular their Rydberg states, appropriate to an evaluation of Casimir force effects. The method here involves the systematic use of high-order perturbation expansions. ix

9 x PREFACE Finally, in Chapter 5, Gerald Feinberg and Joseph Sucher develop the theoretical foundation underlying the existence and general nature of Casimir forces, using Feynman diagram methods. To make the derivations as transparent as pedagogically possible, they emphasize the simplest cases of two spin-o particles and of a spin-o and a spin-l/2 particle. Our pleasure at having received chapters we regard so highly is mitigated, however, by the death of Gerald Feinberg, co-author of Chapter 5 and a major figure in the establishment of the modern theoretical foundation of this subject. This volume is dedicated to his memory. His contributions to atomic physics, including the investigations on Casimir forces described in Chapter 5, are only a part of the rich intellectual legacy of his accomplishments, which range from work on symmetry principles and elementary particle physics to cosmology and the philosophy of science.

10 CONTENTS CHAPTER 1 AN OVERVIEW OF LoNG-RANGE CASIMIR INTERACTIONS Larry Spruch 1. Introductory Remarks Some Remarkable Aspects of Casimir Interactions Some Criteria for Choosing Systems to Be Studied A Comparison of Lamb Shifts and Casimir Shifts The Asymptotic Domain Shifts versus Broadening Effects The Significance of Casimir Effects The Goal of This Introductory Chapter Summary: Some Aspects of Casimir Effects A Digression Preliminaries: Some Relevant Aspects of the Uncertainty Principle for Particles and for Fields The Existence of a Lower Bound on the Energy of a Hydrogen Atom The Nonrelativistic Long-Range Interaction of Two Hydrogen Atoms Vacuum Fluctuations of the Electromagnetic Field The Force per Unit Area between Two Infinite Plane Parallel Walls Ideal Walls Nonideal Walls The Strange Case of the Nonrelativistic Limit A Simple Physical Derivation of the Asymptotic Form of the Retarded Potential for Two Finite Polarizable Systems Some Introductory Comments A Semiquantitative Discussion Applications: Atom-Atom, Electron-Atom and Electron- Ion, and Electron-Electron Interactions Gravitational Effects xi

11 xii CONTENI'S 4.5. Dimensional Analysis A Quantitative Analysis Magnetic Effects and Catch A Simple Physical Derivation of the Asymptotic Form of the Retarded Potential for a Finite Polarizable System and a Plane Wall Some Introductory Comments A Finite Polarizable System and an Ideal Wall: A Semiquantitative Analysis Applications: Atom-Wall and Electron-Wall 5.4. Dimensional Analysis A Quantitative Analysis The Lamb Shift of Hydrogen versus the Retardation Energy Shift ~ E rof emurium t (an Electron Bound to a Wall) 6.1. The Determination of ~ E rof e Murium t for an Ideal Wall 6.2. The Physical Basis of the Lamb Shift of Hydrogen versus That of ~ E rof emurium t for an Ideal Wall Notes... ' Casimir, the Determination of the Value of the Fine Structure Constant, and the Sign of Retardation Potentials 7.2. On the Verification of a Casimir Effect The Relative Powers of l/r or liz of (Relativistic) Retardation Interactions and (Nonrelativistic) Dynamic Correlation Interactions A Semiclassical Estimation of the Radiative Mean Life of a Hydrogenlike State Advantages and Disadvantages of Heliumlike Ions versus Helium Atoms On Referencing and Accreditation, and a Summary, All Very Biased... References CHAPTER 2 EXPERIMENTAL STUDIES OF HIGH-L RYDBERG STATES IN HEUUM S. R. Lundeen 1. Introduction Experiments Discussion of Results 4. Conclusions References

12 CONTENTS xiii CHAPTER 3 HIGH-PRECISION CALCULATIONS FOR THE RYDBERG STATES OF HELIUM G. W. F. Drake 1. Introduction Variational Calculations for Helium 2.1. Construction of Basis Sets Optimization of Nonlinear Parameters Nonrelativistic Eigenvalues Small Corrections Finite Nuclear Mass Corrections 3.2. Relativistic Corrections Quantum Electrodynamic Corrections 3.4. Asymptotic Limits Results for Energy Levels and Discussion 4.1. Description of the Tables Comparisons with Experiment-Low-Lying States 4.3. Comparisons with Experiment-Rydberg States 5. Concluding Remarks... Appendix on Jacobi Coordinates References CHAPTER 4 HIGH RYDBERG STATES OF Two-ELECTRON ATOMS IN PERTURBATION THEORY Richard J. Drachman 1. Introduction The Optical Potential The Feshbach Projection Operator Formalism 2.2. The Hamiltonian Asymptotic Expansions Some Mathematical Techniques The Dalgarno-Lewis Method The Recursion Relation for (l/xk) 3.3. Nonadiabatic Terms Evaluation of the Nonrelativistic Energies 4.1. Perturbations of Second Order in V 4.2. Perturbations of Third Order in V 4.3. Perturbations of Fourth Order in V 4.4. The Effect of L\2 and VI

13 xiv CoNTENTS 5. Nonrelativistic Energy Shifts and Their Accuracy Tabulation of Nonrelativistic Energy Shifts Short-Range Corrections-When Are They Negligible? Relativistic Corrections: Application of the Breit-Pauli Operator Relativistic Effects on the Rydberg Electron Relativistic Effects on the Core Electron Relativistic Interaction between the Two Electrons: "Retardation Effects" Comparison with Experiment and Future Developments 269 References CHAPTER 5 LoNG-RANGE ELECTROMAGNETIC FORCES IN QUANTUM THEORY: ThEORETICAL FOUNDATIONS Joseph Sucher and Gerald Feinberg 1. Introduction Preliminary Remarks QED, Atomic Physics, and the Electron-Electron Potential Aim of This Chapter; Outline of Contents Time-Ordered and Feynman Diagrams in Field Theory Review: The Free Spin-O Field Interactions of Spin-O Particles: A Simple Model Brief Review of Scattering Theory Transition Amplitude and Feynman Amplitude Two-Body Scattering: Time-Ordered Tree Graphs Two-Body Scattering: Feynman Tree Graphs Beyond Tree Graphs Off-Shell Feynman Amplitudes and Their Uses The Techniques of Dispersion Theory Lorentz-Invariant Variables Analyticity and "Crossing": First Examples Crossing Symmetry and Bose Symmetry Analyticity of Higher-Order Amplitudes Spectral Representations: Mathematical Aspects Unitarity; Discontinuity Rules Potentials from One-Quantum and Two-Quantum Exchange Potentials in Quantum Field Theory: Part Potential from One-Quantum Exchange Potentials in Quantum Field Theory: Part Potential from Two-Quantum Exchange.. 314

14 CoNTENTS XV 5. Scalar QED, Feynman Rules, and Compton Amplitudes Scalar QED Feynman Rules for Scalar QED Photons and Gauge Invariance Compton Amplitude and Tensor for a Spin-O Particle Examples The Long-Range Force between Two Spin-O Particles General Analysis Two Neutral Particles A Neutral and a Charged Particle Two Charged Particles The Long-Range Force between a Spin-O and a Spin-l/2 Particle General Considerations Dirac Form of the Long-Range Potential Schrodinger-Pauli Form of the Potential Summary, Extensions, and Remarks on Applications Summary of Results Extensions Remarks on Applications 343 References INDEX