QUANTUM MECHANICS OF ONE- AND TWO-ELECTRON ATOMS

Transcription

1 QUANTUM MECHANICS OF ONE- AND TWO-ELECTRON ATOMS

2 QUANTUM MECHANICS OF ONE- AND TWO-ELECTRON ATOMS HANS A. BETHE AND EDWIN E. SALPETER Cornell University Ithaca, New York A PLENUM/ROSETTA EDITION

3 Library of Congress Cataloging in Publication Data Bethe, Hans Albrecht, Quantum mechanics of one - and two-electron atoms. "A Plenum/Rosetta edition." Bibliography: p. Includes indel(l:s. 1. Problem of many bodies. 2. Atoms. I. Salpeter, Edwin E., joint author. II. Title. QCI74.17.P7B ISBN- I 3: DOl: / e-isbn-i3: This is the authorized paperback edition of Bethe/Salpeter, Quantum Mechanics of One- and Two-Electron Atoms. First edition Springer-Verlag, Berlin-Gottingen-Heidelberg 1977 Plenum Publishing Corporation, New York First paperback printing 1977 A Plenum/Rosetta Edition Published by Plenum Publishing Corporation 227 West 17th Street,New York,N.Y.lOOll 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

4 DEDICATED TO THE MEMORY OF ARNOLD SOMMERFELD

5 Preface to the Plenum/Rosetta Edition. This book was written twenty years ago, but we are leaving it unchanged-not because little has happened to the subject, but because so much has happened that any change would require very major rewriting. On the other hand, no other book on the subject seems to have appeared and we hope this paperback edition will be of some use to a younger generation-if not for another twenty years, at least until someone else has the patience to write a new book. Although we have added no new material we have corrected some misprints. We have added new references only where they correct actual errors. Weare grateful to Drs. G. Feinberg, M. Inokuti, Y. Kim, and A. Petermann for pointing out errors.

6 Preface. Nearly au of this book is taken from an article prepared in 19;6 for a recent volume of the Encyclopedia of Physics. This article, in tum, is partly based on Dr. NORBERT ROSENZWEIG'S translation of an older article on the same subject, written by one of us (HAB) about 25 years ago for the GEIGER-SCHEEL Handbuch der Physik. To the article written last year we have added some Addenda and Errata. These Addenda and Errata refer back to some of the 79 sections of the main text and contain some misprint corrections, additional references and some notes. The aim of this book is two-fold. First, to act as a reference work on calculations pertaining to hydrogen-like and helium-like atoms and their comparison with experiments. However, these calculations involve a vast array of approximation methods, mathematical tricks and physical pictures, which are also useful in the application of quantum mechanics to other fields. In many sections we have given more general discussions of the methods and physical ideas than is necessary for the study of the H- and He-atom alone. We hope that this book will thus at least partly fulfill its second aim, namely to be of some use to graduate students who wish to learn "applied quantum mechanics". A basic knowledge of the principles of quantum mechanics, such as given in the early chapters of SCHIFF'S or BOHM'S book, is presupposed. Like its 1932 predecessor this book mainly gives "low-brow" explicit derivations rather than using more elegant and powerful, but more difficult, formalisms. However, in dealing with angular momentum and sum rules, etc., some operator manipulations are introduced, since these are now commonplace in the treatment of complex atoms. Since it is no longer necessary to "sell quantum mechanics", less space is devoted to the experimental verification of older quantum mechanical result3. A large fraction of the present book is devoted to the Dirac theory of the electron and to radiative effects, including short discussions of the relevant experiments. These topics are also treated from the "low-brow" or "practical" point of view. In particular, no formal derivations of quantum-electrodynamics are presented, but the specific application to atomic systems of general fieldtheoretic results is described in detail. Numbers in square brackets, in the text, e.g. [9J, refer to a short bibliography towards the end of the book, but most references are given as footnotes. References to "this Encyclopedia" refer to the recent Encyclopedia of Physics, edited by S. FLii'GGE and published by Springer-Verlag; "this volume" refers to Vol. XXXV. In many sections atomic units are used, which are defined in the Introduction. The particular definitions of spherical harmonics we have used, as well as useful formulae, are given in an Appendix. We are deeply grateful to many colleagues who have made suggestions, pointed out errors, communicated unpublished results and generally helped with the preparation of the manuscript or the proofreading. We are particularly indebted to Drs. J. F. BIRD and N. ROSENZWEIG who read the proofs patiently and eradicated mistakes impatiently. September H. A. BETHE. E. E. SALPETER.

7 Contents. Page Introduction. Units THE HYDROGEN ATOM WITHOUT EXTERNAL FIELDS 4 a) Nonrelativistic theory Separation of SCHRODINGER'S equation in spherical polar coordinates. Angularly dependent eigenfunctions and the angular momentum matrix 4 2. Derivation of BALMER'S formula The radial eigenfunctions of the discrete spectrum The eigenfunctions of the continuous spectrum Motion of the nucleus Separation of SCHRODINGER'S equation in parabolic coordinates Methods for the continuous spectrum for a general central potential Wave functions in momentum space. Discrete spectrum Wave functions in momentum space. Continuous spectrum 40 b) DIRAC theory 10. General properties of the DIRAC theory 11. Angular momentum PAULI theory of the spin-electron PAULI theory for a central potential The exact solution of the DIRAC equation 15. DIRAC equation. Continuous spectrum 16. The DIRAC equation in momentum space 17. The fine structure formula... c) Radiative and other corrections Radiative corrections. S-matrix theory 19. Radiative corrections. Bound states 20. Corrections for nuclear motion and structure 21. Fine structure and the LAMB shift 22. Hyperfine structure splitting The fine structure of positronium..... II. THE HELIUM ATOM WITHOUT EXTERNAL FIELDS a) Nonrelativistic theory The SCHRODINGER equation for helium (symmetry) 25. Discussion of variation and perturbation methods. 26. Level scheme of helium Survey of approximations to be used First order HEISENBERG'S method (excited states). 29. Polarization for excited states 30. FocK's method (excited S-states) L HARTREE'S method RITZ variation method (helium ground state) Ground state of helium-like ions with arbitrary Z 34. The negative hydrogen ion Variation method for excited states 36. Miscellaneous calculations 37. Motion of the nucleus b) Relativistic theory Discussion of the BREIT equation 39. The PAULI approximation (low Z)

8 xii Contents. 40. Fine structure splitting of helium Relativistic corrections for the ground state 42. BREIT equation without external field 43. Treatment for large Z Hyperfine structure III. ATOMS IN EXTERNAL FIELDS. 205 a) ZEEMAN effect :W5 45. ZEEMAN effect for a single-electron atom Dependence on magnetic field strength Some corrections to the ZEEMAN effect Extension to many-electron atoms... : ~ Comparison with precision experiments 222 SO. The diamagnetism of helium 227 b) STARK effect in hydrogen Linear STARK effect The quadratic STARK effect STARK effect for strong fields Ionization by the electric field. Quenching of the lines in the STARK effect STARK effect of the fine structure of hydrogen 238 c) STARK effect in helium 56. The STARK effect for weak fields 57. Dependence on field strength The dielectric constant of helium IV. INTERACTION WITH RADIATION a) Discrete spectrum General formulas Selection rules for orbital and magnetic quantum numbers 61. Sum rules Proof of the sum rules The transition probabilities for hydrogen in polar coordinates. 64. Intensity of fine structure lines Intensities in parabolic coordinates (STARK effect). 66. Higher multipole radiation Lifetimes of excited states in hydrogen 68. Alkali and X-ray spectra b) The photoeffect General survey The BORN approximation 71. The absorption coefficient without retardation 72. Angular distribution and retardation Rela ti vistic effects. 74. The optical region 75. Recombination c) Bremsstrahlung 76. General survey 77. Nonrelativistic BORN approximation 78. Calculations for low energies. 79. Relativistic effects Appendix on spherical harmonics Bibliography.... Addenda and errata.... Further addenda and errata Author index. Subject index. Index of Tables Page