Ettore Majorana: Unpublished Research Notes on Theoretical Physics

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Author: Salvatore Esposito | Salvatore Esposito | E. Recami | Alwyn van der Merwe | Roberto Battiston

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Ettore Majorana: Unpublished Research Notes on Theoretical Physics

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

Series Editors: GIANCARLO GHIRARDI, University of Trieste, Italy VESSELIN PETKOV, Concordia University, Canada TONY SUDBERY, University of York, UK ALWYN VAN DER MERWE, University of Denver, CO, USA

Volume 159 For other titles published in this series, go to www.springer.com/series/6001

Ettore Majorana: Unpublished Research Notes on Theoretical Physics Edited by

S. Esposito University of Naples “Federico II” Italy E. Recami University of Bergamo Italy A. van der Merwe University of Denver Colorado, USA R. Battiston University of Perugia Italy

Editors Salvatore Esposito Università di Napoli “Federico II” Dipartimento di Scienze Fisiche Complesso Universitario di Monte S. Angelo Via Cinthia 80126 Napoli Italy Erasmo Recami Università di Bergamo Facoltà di Ingegneria 24044 Dalmine (BG) Italy

Alwyn van der Merwe University of Denver Department of Physics and Astronomy Denver, CO 80208 USA

Roberto Battiston Università di Perugia Dipartimento di Fisica Via A. Pascoli 06123 Perugia Italy

Back cover photo of E. Majorana: Copyright by E. Recami & M. Majorana, reproduction of the photo is not allowed (without written permission of the right holders)

ISBN 978-1-4020-9113-1

e-ISBN 978-1-4020-9114-8

Library of Congress Control Number: 2008935622 c 2009 Springer Science + Business Media B.V. No part of this work 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 the written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for the exclusive use by the purchaser of the work. Printed on acid-free paper 9 8 7 6 5 4 3 2 1 springer.com

“But, then, there are geniuses like Galileo and Newton. Well, Ettore Majorana was one of them...” Enrico Fermi (1938)

CONTENTS

Preface

xiii

Bibliography

xxxvii

Table of contents of the complete set of Majorana’s Quaderni (ca. 1927-1933) xliii

CONTENTS OF THE SELECTED MATERIAL Part I 3

Dirac Theory 1.1 1.2 1.3 1.4

1.5 1.6

Vibrating string [Q02p038] A semiclassical theory for the electron [Q02p039] 1.2.1 Relativistic dynamics 1.2.2 Field equations Quantization of the Dirac ﬁeld [Q01p133] Interacting Dirac ﬁelds [Q02p137] 1.4.1 Dirac equation 1.4.2 Maxwell equations 1.4.3 Maxwell-Dirac theory 1.4.3.1 Normal mode decomposition 1.4.3.2 Particular representations of Dirac operators Symmetrization [Q02p146] Preliminaries for a Dirac equation in real terms [Q13p003] 1.6.1 First formalism 1.6.2 Second formalism 1.6.3 Angular momentum 1.6.4 Plane-wave expansion 1.6.5 Real ﬁelds 1.6.6 Interaction with an electromagnetic ﬁeld

vii

3 4 4 7 22 25 25 27 29 31 32 35 35 36 38 40 44 45 45

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E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

1.7

Dirac-like equations for particles with spin higher than 1/2 [Q04p154]

1.7.1 1.7.2 1.7.3 1.7.4

Spin-1/2 particles (4-component spinors) Spin-7/2 particles (16-component spinors) Spin-1 particles (6-component spinors) 5-component spinors

Quantum Electrodynamics 2.1 2.2

Basic lagrangian and hamiltonian formalism for the electromagnetic ﬁeld [Q01p066] Analogy between the electromagnetic ﬁeld and the Dirac ﬁeld [Q02a101]

2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14 2.15

Electromagnetic ﬁeld: plane wave operators [Q01p068] 2.3.1 Dirac formalism Quantization of the electromagnetic ﬁeld [Q03p061] Continuation I: angular momentum [Q03p155] Continuation II: including the matter ﬁelds [Q03p067] Quantum dynamics of electrons interacting with an electromagnetic ﬁeld [Q02p102] Continuation [Q02p037] Quantized radiation ﬁeld [Q17p129b] Wave equation of light quanta [Q17p142] Continuation [Q17p151] Free electron scattering [Q17p133] Bound electron scattering [Q17p142] Retarded ﬁelds [Q05p065] 2.14.1 Time delay Magnetic charges [Q03p163]

Appendix: Potential experienced by an electric charge [Q02p101]

47 47 48 48 55 57 57 59 64 68 71 78 82 84 94 95 100 101 104 112 116 118 119 121

Part II Atomic Physics 3.1

3.2 3.3 3.4 3.5 3.6

Ground state energy of a two-electron atom [Q12p058] 3.1.1 Perturbation method 3.1.2 Variational method 3.1.2.1 First case 3.1.2.2 Second case 3.1.2.3 Third case Wavefunctions of a two-electron atom [Q17p152] Continuation: wavefunctions for the helium atom [Q05p156] Self-consistent ﬁeld in two-electron atoms [Q16p100] 2s terms for two-electron atoms [Q16p157b] Energy levels for two-electron atoms [Q07p004] 3.6.1 Preliminaries for the X and Y terms

125 125 125 128 129 130 131 133 136 141 144 144 148

ix

CONTENTS

3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 3.15 3.16

3.6.2 Simple terms 3.6.3 Electrostatic energy of the 2s2p term 3.6.4 Perturbation theory for s terms 3.6.5 2s2p 3 P term 3.6.6 X term 3.6.7 2s2s 1 S and 2p2p 1 S terms 3.6.8 1s1s term 3.6.9 1s2s term 3.6.10 Continuation 3.6.11 Other terms Ground state of three-electron atoms [Q16p157a] Ground state of the lithium atom [Q16p098] 3.8.1 Electrostatic potential 3.8.2 Ground state Asymptotic behavior for the s terms in alkali [Q16p158] 3.9.1 First method 3.9.2 Second method Atomic eigenfunctions I [Q02p130] Atomic eigenfunctions II [Q17p161] Atomic energy tables [Q06p026] Polarization forces in alkalies [Q16p049] Complex spectra and hyperﬁne structures [Q05p051] Calculations about complex spectra [Q05p131] Resonance between a p ( = 1) electron and an electron with azimuthal quantum number [Q07p117] 3.16.1 Resonance between a d electron and a p shell I 3.16.2 Eigenfunctions of d 5 , d 3 , p 3 and p 1 electrons 2

3.17 3.18

3.19 3.20

3.21 3.22

2

2

3.22.1 First method 3.22.2 Second method

223 224 225

2

3.16.3 Resonance between a d electron and a p shell II Magnetic moment and diamagnetic susceptibility for a oneelectron atom (relativistic calculation) [Q17p036] Theory of incomplete P triplets [Q07p061] 3.18.1 Spin-orbit couplings and energy levels 3.18.2 Spectral lines for Mg and Zn 3.18.3 Spectral lines for Zn, Cd and Hg Hyperﬁne structure: relativistic Rydberg corrections [Q04p143] Non-relativistic approximation of Dirac equation for a twoparticle system [Q04p149] 3.20.1 Non-relativistic decomposition 3.20.2 Electromagnetic interaction between two charged particles 3.20.3 Radial equations Hyperﬁne structures and magnetic moments: formulae and tables [Q04p165] Hyperﬁne structures and magnetic moments: calculations [Q04p169]

151 155 157 158 159 169 170 174 175 176 183 184 184 185 190 191 195 197 201 204 205 211 219

227 229 233 233 237 238 239 242 243 244 245 246 251 251 254

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E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

Molecular Physics 4.1

4.2 4.3

The helium molecule [Q16p001] 4.1.1 The equation for σ -electrons in elliptic coordinates 4.1.2 Evaluation of P2 for s-electrons: relation between W and λ 4.1.3 Evaluation of P1 Vibration modes in molecules [Q06p031] 4.2.1 The acetylene molecule Reduction of a three-fermion to a two-particle system [Q03p176]

Statistical Mechanics 5.1 5.2 5.3 5.4 5.5

Degenerate gas [Q17p097] Pauli paramagnetism [Q18p157] Ferromagnetism [Q08p014] Ferromagnetism: applications [Q08p046] Again on ferromagnetism [Q06p008]

261 261 261 263 275 275 278 282 287 287 288 289 300 307

Part III The Theory of Scattering 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10

Scattering from a potential well [Q06p015] Simple perturbation method [Q06p024] The Dirac method [Q01p106] 6.3.1 Coulomb ﬁeld The Born method [Q01p109] Coulomb scattering [Q01p010] Quasi coulombian scattering of particles [Q01p001] 6.6.1 Method of the particular solutions Coulomb scattering: another regularization method [Q01p008] Two-electron scattering [Q03p029] Compton eﬀect [Q03p041] Quasi-stationary states [Q03p103]

311 311 316 317 318 319 321 324 327 328 330 331 332

Appendix: Transforming a diﬀerential equation [Q03p035]

337

Nuclear Physics

339

7.1 7.2 7.3

Wave equation for the neutron [Q17p129] Radioactivity [Q17p005] Nuclear potential [Q17p006] 7.3.1 Mean nucleon potential 7.3.2 Computation of the interaction potential between nucleons 7.3.3 Nucleon density

339 339 340 340 342 345

CONTENTS

7.4 7.5 7.6

7.3.4 Nucleon interaction I 7.3.4.1 Zeroth approximation 7.3.5 Nucleon interaction II 7.3.5.1 Evaluation of some integrals 7.3.5.2 Zeroth approximation 7.3.6 Simple nuclei I 7.3.7 Simple nuclei II 7.3.7.1 Kinematics of two α particles (statistics) Thomson formula for β particles in a medium [Q16p083] Systems with two fermions and one boson [Q17p090] Scalar ﬁeld theory for nuclei? [Q02p086]

xi 347 351 352 355 358 363 365 367 368 370 370

Part IV Classical Physics

385

8.1 8.2 8.3

Surface waves in a liquid [Q12p054] Thomson’s method for the determination of e/m [Q09p044[ Wien’s method for the determination of e/m (positive charges)

385 387

[Q09p048b]

8.4

Determination of the electron charge [Q09p028] 8.4.1 Townsend eﬀect 8.4.1.1 Ion recombination 8.4.1.2 Ion diﬀusion 8.4.1.3 Velocity in the electric ﬁeld 8.4.1.4 Charge of an ion 8.4.2 Method of the electrolysis (Townsend) 8.4.3 Zaliny’s method for the ratio of the mobility coeﬃcients 8.4.4 Thomson’s method 8.4.5 Wilson’s method 8.4.6 Millikan’s method Electromagnetic and electrostatic mass of the electron

388 390 390 390 392 393 393 394 394 395 396 396

8.5 8.6

[Q09p048] 397 Thermionic eﬀect [Q09p053] 397 8.6.1 Langmuir Experiment on the eﬀect of the electron cloud 399

Mathematical Physics 9.1

Linear partial diﬀerential equations. Complete systems [Q11p087]

9.1.1 9.1.2

9.2

403

Linear operators Integrals of an ordinary diﬀerential system and the partial diﬀerential equation which determines them 9.1.3 Integrals of a total diﬀerential system and the associated system of partial diﬀerential equation that determines them Algebraic foundations of the tensor calculus [Q11p093] 9.2.1 Covariant and contravariant vectors

403 404 405 406 409 409

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E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

9.3

9.4

9.5

9.6

Geometrical introduction to the theory of diﬀerential quadratic forms I [Q11p094] 9.3.1 The symbolic equation of parallelism 9.3.2 Intrinsic equations of parallelism 9.3.3 Christoﬀel’s symbols 9.3.4 Equations of parallelism in terms of covariant components 9.3.5 Some analytical veriﬁcations 9.3.6 Permutability 9.3.7 Line elements 9.3.8 Euclidean manifolds. any Vn can always be considered as immersed in a Euclidean space 9.3.9 Angular metric 9.3.10 Coordinate lines 9.3.11 Diﬀerential equations of geodesics 9.3.12 Application Geometrical introduction to the theory of diﬀerential quadratic forms II [Q11p113] 9.4.1 Geodesic curvature 9.4.2 Vector displacement 9.4.3 Autoparallelism of geodesics 9.4.4 Associated vectors 9.4.5 Remarks on the case of an indeﬁnite ds2 Covariant diﬀerentiation. Invariants and diﬀerential parameters. Locally geodesic coordinates [Q11p119] 9.5.1 Geodesic coordinates 9.5.1.1 Applications 9.5.2 Particular cases 9.5.3 Applications 9.5.4 Divergence of a vector 9.5.5 Divergence of a double (contravariant) tensor 9.5.6 Some laws of transformation 9.5.7 ε systems 9.5.8 Vector product 9.5.9 Extension of a ﬁeld 9.5.10 Curl of a vector in three dimensions 9.5.11 Sections of a manifold. Geodesic manifolds 9.5.12 Geodesic coordinates along a given line Riemann’s symbols and properties relating to curvature [Q11p138]

9.6.1 9.6.2 9.6.3 9.6.4 9.6.5

Index

Cyclic displacement round an elementary parallelogram Fundamental properties of Riemann’s symbols of the second kind Fundamental properties and number of Riemann’s symbols of the ﬁrst kind Bianchi identity and Ricci lemma Tangent geodesic coordinates around the point P0

409 409 409 411 412 413 414 414 415 416 417 418 420 422 422 422 424 424 425 425 425 427 429 430 431 432 433 434 435 435 436 436 437 441 441 443 444 447 447 449

Preface

Without listing his works, all of which are highly notable both for the originality of the methods utilized as well as for the importance of the results achieved, we limit ourselves to the following: In modern nuclear theories, the contribution made by this researcher to the introduction of the forces called ‘Majorana forces’ is universally recognized as the one, among the most fundamental, that permits us to theoretically comprehend the reasons for nuclear stability. The work of Majorana today serves as a basis for the most important research in this ﬁeld. In atomic physics, the merit of having resolved some of the most intricate questions on the structure of spectra through simple and elegant considerations of symmetry is due to Majorana. Lastly, he devised a brilliant method that permits us to treat the positive and negative electron in a symmetrical way, ﬁnally eliminating the necessity to rely on the extremely artiﬁcial and unsatisfactory hypothesis of an inﬁnitely large electrical charge diﬀused in space, a question that had been tackled in vain by many other scholars [4].

With this justiﬁcation, the judging committee of the 1937 competition for a new full professorship in theoretical physics at Palermo, chaired by Enrico Fermi (and including Enrico Persico, Giovanni Polvani and Antonio Carrelli), suggested the Italian Minister of National Education should appoint Ettore Majorana “independently of the competition rules, as full professor of theoretical physics in a university of the Italian kingdom1 because of his high and well-deserved reputation” [4]. Evidently, to gain such a high reputation the few papers that the Italian scientist had chosen to publish were enough. It is interesting to note that proper light was shed by Fermi on Majorana’s symmetrical approach to electrons and antielectrons (today climaxing in its application to neutrinos and antineutrinos) and on its ability to eliminate the hypothesis 1 Which

happened to be the University of Naples.

xiii

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E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

known as the “Dirac sea”, a hypothesis that Fermi deﬁned as “extremely artiﬁcial and unsatisfactory”, despite the fact that in general it had been uncritically accepted. However, one of the most important works of Majorana, the one that introduced his “inﬁnite-components equation” was not mentioned: it had not been understood yet, even by Fermi and his colleagues. Bruno Pontecorvo [2], a younger colleague of Majorana at the Institute of Physics in Rome, in a similar way recalled that “some time after his entry into Fermi’s group, Majorana already possessed such an erudition and had reached such a high level of comprehension of physics that he was able to speak on the same level with Fermi about scientiﬁc problems. Fermi himself held him to be the greatest theoretical physicist of our time. He often was astounded ....” Majorana’s fame rests solidly on testimonies like these, and even more on the following ones. At the request of Edoardo Amaldi [1], Giuseppe Cocconi wrote from CERN (18 July 1965): In January 1938, after having just graduated, I was invited, essentially by you, to come to the Institute of Physics at the University of Rome for six months as a teaching assistant, and once I was there I would have the good fortune of joining Fermi, Gilberto Bernardini (who had been given a chair at Camerino University a few months earlier) and Mario Ageno (he, too, a new graduate) in the research of the products of disintegration of μ “mesons” (at that time called mesotrons or yukons), which are produced by cosmic rays.... A few months later, while I was still with Fermi in our workshop, news arrived of Ettore Majorana’s disappearance in Naples. I remember that Fermi busied himself with telephoning around until, after some days, he had the impression that Ettore would never be found. It was then that Fermi, trying to make me understand the signiﬁcance of this loss, expressed himself in quite a peculiar way; he who was so objectively harsh when judging people. And so, at this point, I would like to repeat his words, just as I can still hear them ringing in my memory: ‘Because, you see, in the world there are various categories of scientists: people of a secondary or tertiary standing, who do their best but do not go very far. There are also those of high standing, who come to discoveries of great importance, fundamental for the development of science’ (and here I had the impression that he placed himself in that category). ‘But then there are geniuses like Galileo and Newton. Well, Ettore was one of them. Majorana had what no one else in the world had ...’.

Fermi, who was rather severe in his judgements, again expressed himself in an unusual way on another occasion. On 27 July 1938 (after

PREFACE

xv

Majorana’s disappearance, which took place on 26 March 1938), writing from Rome to Prime Minister Mussolini to ask for an intensiﬁcation of the search for Majorana, he stated: “I do not hesitate to declare, and it would not be an overstatement in doing so, that of all the Italian and foreign scholars that I have had the chance to meet, Majorana, for his depth of intellect, has struck me the most” [4]. But, nowadays, some interested scholars may ﬁnd it diﬃcult to appreciate Majorana’s ingeniousness when basing their judgement only on his few published papers (listed below), most of them originally written in Italian and not easy to trace, with only three of his articles having been translated into English [9, 10, 11, 12, 28] in the past. Actually, only in 2006 did the Italian Physical Society eventually publish a book with the Italian and English versions of Majorana’s articles [13]. Anyway, Majorana has also left a lot of unpublished manuscripts relating to his studies and research, mainly deposited at the Domus Galilaeana in Pisa (Italy), which help to illuminate his abilities as a theoretical physicist, and mathematician too. The year 2006 was the 100th anniversary of the birth of Ettore Majorana, probably the brightest Italian theoretician of the twentieth century, even though to many people Majorana is known mainly for his mysterious disappearance, in 1938, at the age of 31. To celebrate such a centenary, we had been working—among others—on selection, study, typographical setting in electronic form and translation into English of the most important research notes left unpublished by Majorana: his so-called Quaderni (booklets); leaving aside, for the moment, the notable set of loose sheets that constitute a conspicuous part of Majorana’s manuscripts. Such a selection is published for the ﬁrst time, with some understandable delay, in this book. In a previous volume [15], entitled Ettore Majorana: Notes on Theoretical Physics, we analogously published for the ﬁrst time the material contained in diﬀerent Majorana booklets—the so-called Volumetti, which had been written by him mainly while studying physics and mathematics as a student and collaborator of Fermi. Even though Ettore Majorana: Notes on Theoretical Physics contained many highly original ﬁndings, the preparation of the present book remained nevertheless a rather necessary enterprise, since the research notes publicited in it are even more (and often exceptionally) interesting, revealing more fully Majorana’s genius. Many of the results we will cover on the hundreds of pages that follow are novel and even today, more than seven decades later, still of signiﬁcant importance for contemporary theoretical physics.

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E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

Historical prelude For nonspecialists, the name of Ettore Majorana is frequently associated with his mysterious disappearance from Naples, on 26 March 1938, when he was only 31; afterwards, in fact, he was never seen again. But the myth of his “disappearance” [4] has contributed to nothing but the fame he was entitled to, for being a genius well ahead of his time. Ettore Majorana was born on 5 August 1906 at Catania, Sicily (Italy), to Fabio Majorana and Dorina Corso. The fourth of ﬁve sons, he had a rich scientiﬁc, technological and political heritage: three of his uncles had become vice-chancellors of the University of Catania and members of the Italian parliament, while another, Quirino Majorana, was a renowned experimental physicist, who had been, by the way, a former president of the Italian Physical Society. Ettore’s father, Fabio, was an engineer who had founded the ﬁrst telephone company in Sicily and who went on to become chief inspector of the Ministry of Communications. Fabio Majorana was responsible for the education of his son in the ﬁrst years of his school-life, but afterwards Ettore was sent to study at a boarding school in Rome. Eventually, in 1921, the whole family moved from Catania to Rome. Ettore ﬁnished high school in 1923 when he was 17, and then joined the Faculty of Engineering of the local university, where he excelled, and counted Giovanni Gentile Jr., Enrico Volterra, Giovanni Enriques and future Nobel laureate Emilio Segr`e among his friends. In the spring of 1927 Orso Mario Corbino, the director of the Institute of Physics at Rome and an inﬂuential politician (who had succeeded in elevating to full professorship the 25-year-old Enrico Fermi, just with the intention of enabling Italian physics to make a quality jump) launched an appeal to the students of the Faculty of Engineering, inviting the most brilliant young minds to study physics. Segr`e and Edoardo Amaldi rose to the challenge, joining Fermi and Franco Rasetti’s group, and telling them of Majorana’s exceptional gifts. After some encouragement from Segr`e and Amaldi, Majorana eventually decided to meet Fermi in the autumn of that year. The details of Majorana and Fermi’s ﬁrst meeting were narrated by Segr´e [3], Rasetti and Amaldi. The ﬁrst important work written by Fermi in Rome, on the statistical properties of the atom, is today known as the Thomas–Fermi method. Fermi had found that he needed the solution to a nonlinear diﬀerential equation characterized by unusual boundary conditions, and in a week of assiduous work he had calculated the solution with a little hand calculator. When Majorana met Fermi for the ﬁrst time, the latter spoke about his equation, and showed his

PREFACE

xvii

numerical results. Majorana, who was always very sceptical, believed Fermi’s numerical solution was probably wrong. He went home, and solved Fermi’s original equation in analytic form, evaluating afterwards the solution’s values without the aid of a calculator. Next morning he returned to the Institute and sceptically compared the results which he had written on a little piece of paper with those in Fermi’s notebook, and found that their results coincided exactly. He could not hide his amazement, and decided to move from the Faculty of Engineering to the Faculty of Physics. We have indulged ourselves in the foregoing anecdote since the pages on which Majorana solved Fermi’s diﬀerential equation were found by one of us (S.E.) years ago. And recently [22] it was explicitly shown that he followed that night two independent paths, the ﬁrst of them leading to an Abel equation, and the second one resulting in his devising a method still unknown to mathematics. More precisely, Majorana arrived at a series solution of the Thomas–Fermi equation by using an original method that applies to an entire class of mathematical problems. While some of Majorana’s results anticipated by several years those of renowned mathematicians or physicists, several others (including his ﬁnal solution to the equation mentioned) have not been obtained by anyone else since. Such facts are further evidence of Majorana’s brilliance.

Majorana’s published articles Majorana published few scientiﬁc articles: nine, actually, besides his sociology paper entitled “Il valore delle leggi statistiche nella ﬁsica e nelle scienze sociali” (“The value of statistical laws in physics and the social sciences”), which was, however, published not by Majorana but (posthumously) by G. Gentile Jr., in Scientia (36:55–56, 1942), and much later was translated into English. Majorana switched from engineering to physics studies in 1928 (the year in which he published his ﬁrst article, written in collaboration with his friend Gentile) and then went on to publish his works on theoretical physics for only a few years, practically only until 1933. Nevertheless, even his published works are a mine of ideas and techniques of theoretical physics that still remain largely unexplored. Let us list his nine published articles, which only in 2006 were eventually reprinted together with their English translations [13]: 1. Sullo sdoppiamento dei termini Roentgen ottici a causa dell’elettrone rotante e sulla intensit`a delle righe del Cesio, Rendiconti Accademia Lincei 8, 229–233 (1928) (in collaboration with Giovanni Gentile Jr.)

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2. Sulla formazione dello ione molecolare di He, Nuovo Cimento 8, 22–28 (1931) 3. I presunti termini anomali dell’Elio, Nuovo Cimento 8, 78–83 (1931) 4. Reazione pseudopolare fra atomi di Idrogeno, Rendiconti Accademia Lincei 13, 58–61 (1931) 5. Teoria dei tripletti P’ incompleti, Nuovo Cimento 8, 107–113 (1931) 6. Atomi orientati in campo magnetico variabile, Nuovo Cimento 9, 43–50 (1932) 7. Teoria relativistica di particelle con momento intrinseco arbitrario, Nuovo Cimento 9, 335–344 (1932) ¨ 8. Uber die Kerntheorie, Zeitschrift f¨ ur Physik 82, 137–145 (1933); Sulla teoria dei nuclei, La Ricerca Scientiﬁca 4(1), 559–565 (1933) 9. Teoria simmetrica dell’elettrone e del positrone, Nuovo Cimento 14, 171–184 (1937) While still an undergraduate, in 1928 Majorana published his ﬁrst paper, (1), in which he calculated the splitting of certain spectroscopic terms in gadolinium, uranium and caesium, owing to the spin of the electrons. At the end of that same year, Fermi invited Majorana to give a talk at the Italian Physical Society on some applications of the Thomas–Fermi model [23] (attention to which was drawn by F. Guerra and N. Robotti). Then on 6 July 1929, Majorana was awarded his master’s degree in physics, with a dissertation having as a subject “The quantum theory of radioactive nuclei”. By the end of 1931 the 25-year-old physicist had published two articles, (2) and (4), on the chemical bonds of molecules, and two more papers, (3) and (5), on spectroscopy, one of which, (3), anticipated results later obtained by a collaborator of Samuel Goudsmith on the “Auger eﬀect” in helium. As Amaldi has written, an in-depth examination of these works leaves one struck by their quality: they reveal both deep knowledge of the experimental data, even in the minutest detail, and an uncommon ease, without equal at that time, in the use of the symmetry properties of the quantum states to qualitatively simplify problems and choose the most suitable method for their quantitative resolution. In 1932, Majorana published an important paper, (6), on the nonadiabatic spin-ﬂip of atoms in a magnetic ﬁeld, which was later extended by Nobel laureate Rabi in 1937, and by Bloch and Rabi in 1945. It established the theoretical basis for the experimental method used to reverse the spin also of neutrons by a radio-frequency ﬁeld, a method that

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is still practised today, for example, in all polarized-neutron spectrometers. That paper contained an independent derivation of the well-known Landau–Zener formula (1932) for nonadiabatic transition probability. It also introduced a novel mathematical tool for representing spherical functions or, rather, for representing spinors by a set of points on the surface of a sphere (Majorana sphere), attention to which was drawn not long ago by Penrose and collaborators [29] (and by Leonardi and coworkers [30]). In the present volume the reader will ﬁnd some additions (or modiﬁcations) to the above-mentioned published articles. However, the most important 1932 paper is that concerning a relativistic ﬁeld theory of particles with arbitrary spin, (7). Around 1932 it was commonly believed that one could write relativistic quantum equations only in the case of particles with spin 0 or 1/2. Convinced of the contrary, Majorana—as we have known for a long time from his manuscripts, constituting a part of the Quaderni ﬁnally published here— began constructing suitable quantum-relativistic equations for higher spin values (1, 3/2, etc.); and he even devised a method for writing the equation for a generic spin value. But still he published nothing,2 until he discovered that one could write a single equation to cover an inﬁnite family of particles of arbitrary spin (even though at that time the known particles could be counted on one hand). To implement his programme with these “inﬁnite-components” equations, Majorana invented a technique for the representation of a group several years before Eugene Wigner did. And, what is more, Majorana obtained the inﬁnitedimensional unitary representations of the Lorentz group that would be rediscovered by Wigner in his 1939 and 1948 works. The entire theory was reinvented in a Soviet series of articles from 1948 to 1958, and ﬁnally applied by physicists years later. Sadly, Majorana’s initial article remained in the shadows for a good 34 years until Fradkin [28], informed by Amaldi, realized what Majorana many years earlier had accomplished. All the scientiﬁc material contained in (and in preparation for) this publication of Majorana’s works is illuminated by the manuscripts published in the present volume. At the beginning of 1932, as soon as the news of the Joliot–Curie experiments reached Rome, Majorana understood that they had discovered the “neutral proton” without having realized it. Thus, even before the oﬃcial announcement of the discovery of the neutron, made soon afterwards by Chadwick, Majorana was able to explain the structure and stability of light atomic nuclei with the help of protons and neutrons, 2 Starting

in 1974, some of us [21] published and revaluated only a few of the pages devoted in Majorana’s manuscripts to the case of a Dirac-like equation for the photon (spin-1 case).

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antedating in this way also the pioneering work of D. Ivanenko, as both Segr´e and Amaldi have recounted. Majorana’s colleagues remember that even before Easter he had concluded that protons and neutrons (indistinguishable with respect to the nuclear interaction) were bound by the “exchange forces” originating from the exchange of their spatial positions alone (and not also of their spins, as Heisenberg would propose), so as to produce the α particle (and not the deuteron) as saturated with respect to the binding energy. Only after Heisenberg had published his own article on the same problem was Fermi able to persuade Majorana to go for a 6-month period, in 1933, to Leipzig and meet there his famous colleague (who would be awarded the Nobel prize at the end of that year); and ﬁnally Heisenberg was able to convince Majorana to publish his results in ¨ the paper “Uber die Kerntheorie”. Actually, Heisenberg had interpreted the nuclear forces in terms of nucleons exchanging spinless electrons, as if the neutron were formed in practice by a proton and an electron, whereas Majorana had simply considered the neutron as a “neutral proton”, and the theoretical and experimental consequences were quickly recognized by Heisenberg. Majorana’s paper on the stability of nuclei soon became known to the scientiﬁc community—a rare event, as we know—thanks to that timely “propaganda” made by Heisenberg himself, who on several occasions, when discussing the “Heisenberg–Majorana” exchange forces, used, rather fairly and generously, to point out more Majorana’s than his own contributions [33]. The manuscripts published in the present book refer also to what Majorana wrote down before having read Heisenberg’s paper. Let us seize the present opportunity to quote two brief passages from Majorana’s letters from Leipzig. On 14 February 1933, he wrote to his mother (the italics are ours): “The environment of the physics institute is very nice. I have good relations with Heisenberg, with Hund, and with everyone else. I am writing some articles in German. The ﬁrst one is already ready ...” [4]. The work that was already ready is, naturally, the cited one on nuclear forces, which, however, remained the only paper in German. Again, in a letter dated 18 February, he told his father (our italics): “I will publish in German, after having extended it, also my latest article which appeared in Il Nuovo Cimento” [4]. But Majorana published nothing more, either in Germany—where he had become acquainted, besides with Heisenberg, with other renowned scientists, including Ehrenfest, Bohr, Weisskopf and Bloch—or after his return to Italy, except for the article (in 1937) of which we are about to speak. It is therefore important to know that Majorana was engaged in writing other papers: in particular, he was expanding his article about the inﬁnite-components equations. His research activity during the years 1933–1937 is testiﬁed by the documents presented in this volume, and

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particularly by a number of unpublished scientiﬁc notes, some of which are reproduced here: as far as we know, it focused mainly on ﬁeld theory and quantum electrodynamics. As already mentioned, in 1937 Majorana decided to compete for a full professorship (probably with the only desire to have students); and he was urged to demonstrate that he was still actively working in theoretical physics. Happily enough, he took from a drawer3 his writing on the symmetrical theory of electrons and antielectrons, publishing it that same year under the title “Symmetric theory of electrons and positrons”. This paper—at present probably the most famous of his—was initially noticed almost exclusively for having introduced the Majorana representation of the Dirac matrices in real form. But its main consequence is that a neutral fermion can be identical with its antiparticle. Let us stress that such a theory was rather revolutionary, since it was at variance with what Dirac had successfully assumed in order to solve the problem of negative energy states in quantum ﬁeld theory. With rare daring, Majorana suggested that neutrinos, which had just been postulated by Pauli and Fermi to explain puzzling features of radioactive β decay, could be particles of this type. This would enable the neutrino, for instance, to have mass, which may have a bearing on the phenomena of neutrino oscillations, later postulated by Pontecorvo. It may be stressed that, exactly as in the case of other writings of his, the “Majorana neutrino” too started to gain prominence only decades later, beginning in the 1950s; and nowadays expressions such as Majorana spinors, Majorana mass and even “majorons” are fashionable. It is moreover well known that many experiments are currently devoted the world over to checking whether the neutrinos are of the Dirac or the Majorana type. We have already said that the material published by Majorana (but still little known, despite everything) constitutes a potential gold mine for physics. Many years ago, for example, Bruno Touschek noticed that the article entitled “Symmetric theory of electrons and positrons” implicitly contains also what he called the theory of the “Majorana oscillator”, described by the simple equation q + ω 2 q = εδ(t), where ε is a constant and δ is the Dirac function [4]. According to Touschek, the properties of the Majorana oscillator are very interesting, especially in connection with its energy spectrum; but no literature seems to exist on it yet.

3 As

we said, from the existing manuscripts it appears that Majorana had formulated also the essential lines of his paper (9) during the years 1932–1933.

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An account of the unpublished manuscripts The largest part of Majorana’s work was left unpublished. Even though the most important manuscripts have probably been lost, we are now in possession of (1) his M.Sc. thesis on “The quantum theory of radioactive nuclei”; (2) ﬁve notebooks (the Volumetti) and 18 booklets (the Quaderni); (3) 12 folders with loose papers; and (4) the set of his lecture notes for the course on theoretical physics given by him at the University of Naples. With the collaboration of Amaldi, all these manuscripts were deposited by Luciano Majorana (Ettore’s brother) at the Domus Galilaeana in Pisa. An analysis of those manuscripts allowed us to ascertain that they, except for the lectures notes, appear to have been written approximately by 1933 (even the essentials of his last article, which Majorana proceeded to publish, as we already know, in 1937, seem to have been ready by 1933, the year in which the discovery of the positron was conﬁrmed). Besides the material deposited at the Domus Galilaeana, we are in possession of a series of 34 letters written by Majorana between 17 March 1931 and 16 November 1937, in reply to his uncle Quirino—a renowned experimental physicist and a former president of the Italian Physical Society—who had been pressing Majorana for help in the theoretical explanation of his experiments:4 such letters have recently been deposited at Bologna University, and have been published in their entirety by Dragoni [8]. They conﬁrm that Majorana was deeply knowledgeable even about experimental details. Moreover, Ettore’s sister, Maria, recalled that, even in those years, Majorana—who had reduced his visits to Fermi’s institute, starting from the beginning of 1934 (that is, just after his return from Leipzig)—continued to study and work at home for many hours during the day and at night. Did he continue to dedicate himself to physics? From one of those letters of his to Quirino, dated 16 January 1936, we ﬁnd a ﬁrst answer, because we learn that Majorana had been occupied “for some time, with quantum electrodynamics”; knowing Majorana’s love for understatements, this no doubt means that during 1935 he had performed profound research at least in the ﬁeld of quantum electrodynamics. This seems to be conﬁrmed by a recently retrieved text, written by Majorana in French [25], where he dealt with a peculiar topic in quantum electrodynamics. It is instructive, as to that topic, to quote directly from Majorana’s paper.

4 In

the past, one of us (E.R.) was able to publish only short passages of them, since they are rather technical; see [4].

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Let us consider a system of p electrons and set the following assumptions: 1) the interaction between the particles is suﬃciently small, allowing us to speak about individual quantum states, so that one may regard the quantum numbers deﬁning the conﬁguration of the system as good quantum numbers; 2) any electron has a number n > p of inner energy levels, while any other level has a much greater energy. One deduces that the states of the system as a whole may be divided into two classes. The ﬁrst one is composed of those conﬁgurations for which all the electrons belong to one of the inner states. Instead, the second one is formed by those conﬁgurations in which at least one electron belongs to a higher level not included in the above-mentioned n levels. We shall also assume that it is possible, with a suﬃcient degree of approximation, to neglect the interaction between the states of the two classes. In other words, we will neglect the matrix elements of the energy corresponding to the coupling of diﬀerent classes, so that we may consider the motion of the p particles, in the n inner states, as if only these states existed. Our aim becomes, then, translating this problem into that of the motion of n − p particles in the same states, such new particles representing the holes, according to the Pauli principle.

Majorana, thus, applied the formalism of ﬁeld quantization to Dirac’s hole theory, obtaining a general expression for the quantum electrodynamics Hamiltonian in terms of anticommuting “hole quantities”. Let us point out that in justifying the use of anticommutators for fermionic variables, Majorana commented that such a use “cannot be justiﬁed on general grounds, but only by the particular form of the Hamiltonian. In fact, we may verify that the equations of motion are better satisﬁed by these relations than by the Heisenberg ones.” In the second (and third) part of the same manuscript, Majorana took into consideration also a reformulation of quantum electrodynamics in terms of a photon wavefunction, a topic that was particularly studied in his Quaderni (and is reproduced here). Majorana, indeed, reformulated quantum electrodynamics by introducing a real-valued wavefunction for the photon, corresponding only to directly observable degrees of freedom. In some other manuscripts, probably prepared for a seminar at Naples University in 1938 [24], Majorana set forth a physical interpretation of quantum mechanics that anticipated by several years the Feynman approach in terms of path integrals. The starting point in Majorana’s notes was to search for a meaningful and clear formulation of the concept of quantum state. Afterwards, the crucial point in the Feynman formulation of quantum mechanics (namely that of considering not only the paths corresponding to classical trajectories, but all the possible paths joining an initial point with the ﬁnal point) was really introduced by Majorana, after a discussion about an interesting example of a harmonic oscillator. Let us also emphasize the key role played by the

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symmetry properties of the physical system in the Majorana analysis, a feature quite common in his papers. Do any other unpublished scientiﬁc manuscripts of Majorana exist? The question, raised by his answer to Quirino and by his letters from Leipzig to his family, becomes of greater importance when one reads also his letters addressed to the National Research Council of Italy (CNR) during that period. In the ﬁrst one (dated 21 January 1933), he asserts: “At the moment, I am occupied with the elaboration of a theory for the description of arbitrary-spin particles that I began in Italy and of which I gave a summary notice in Il Nuovo Cimento ....” [4]. In the second one (dated 3 March 1933) he even declares, referring to the same work: “I have sent an article on nuclear theory to Zeitschrift f¨ ur Physik. I have the manuscript of a new theory on elementary particles ready, and will send it to the same journal in a few days” [4]. Considering that the article described above as a “summary notice” of a new theory was already of a very high level, one can imagine how interesting it would be to discover a copy of its ﬁnal version, which went unpublished. (Is it still, perhaps, in the Zeitschrift f¨ ur Physik archives? Our search has so far ended in failure.) A few of Majorana’s other ideas which did not remain concealed in his own mind have survived in the memories of his colleagues. One such reminiscence we owe to Gian-Carlo Wick. Writing from Pisa on 16 October 1978, he recalls: The scientiﬁc contact [between Ettore and me], mentioned by Segr´e, happened in Rome on the occasion of the ‘A. Volta Congress’ (long before Majorana’s sojourn in Leipzig). The conversation took place in Heitler’s company at a restaurant, and therefore without a blackboard ...; but even in the absence of details, what Majorana described in words was a ‘relativistic theory of charged particles of zero spin based on the idea of ﬁeld quantization’ (second quantization). When much later I saw Pauli and Weisskopf’s article [Helv. Phys. Acta 7 (1934) 709], I remained absolutely convinced that what Majorana had discussed was the same thing ... [4, 26].

Teaching theoretical physics As we have seen, Majorana contributed signiﬁcantly to theoretical research which was among the frontier topics in the 1930s, and, indeed, in the following decades. However, he deeply thought also about the basics, and applications, of quantum mechanics, and his lectures on theoretical physics provide evidence of this work of his.

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As realized only recently [34], Majorana had a genuine interest in advanced physics teaching, starting from 1933, just after he obtained, at the end of 1932, the degree of libero docente (analogous to the German Privatdozent title). As permitted by that degree, he requested to be allowed to give three subsequent annual free courses at the University of Rome, between 1933 and 1937, as testiﬁed by the lecture programmes proposed by him and still present in Rome University’s archives. Such documents also refer to a period of time that was regarded by his colleagues as Majorana’s “gloomy years”. Although it seems that Majorana never delivered these three courses, probably owing to lack of appropriate students, the topics chosen for the lectures appear very interesting and informative. The ﬁrst course (academic year 1933–1934) proposed by Majorana was on mathematical methods of quantum mechanics.5 The second course (academic year 1935–1936) proposed was on mathematical methods of atomic physics.6 Finally, the third course (academic year 1936–1937) proposed was on quantum electrodynamics.7 Majorana could actually lecture on theoretical physics only in 1938 when, as recalled above, he obtained his position as a full professor in Naples. He gave his lectures starting on 13 January and ending with his disappearance (26 March), but his activity was intense, and his interest in teaching was very high. For the beneﬁt of his students, and perhaps

5 The

programme for it contained the following topics: (1) unitary geometry, linear transformations, Hermitian operators, unitary transformations, and eigenvalues and eigenvectors; (2) phase space and the quantum of action, modiﬁcations of classical kinematics, and general framework of quantum mechanics; (3) Hamiltonians which are invariant under a transformation group, transformations as complex quantities, noncompatible systems, and representations of ﬁnite or continuous groups; (4) general elements on abstract groups, representation theorems, the group of spatial rotations, and symmetric groups of permutations and other ﬁnite groups; (5) properties of the systems endowed with spherical symmetry, orbital and intrinsic momenta, and theory of the rigid rotator; (6) systems with identical particles, Fermi and Bose–Einstein statistics, and symmetries of the eigenfunctions in the centre-of-mass frames; (7) Lorentz group and spinor calculus, and applications to the relativistic theory of the elementary particles. 6 The corresponding subjects were matrix calculus, phase space and the correspondence principle, minimal statistical sets or elementary cells, elements of quantum dynamics, statistical theories, general deﬁnition of symmetry problems, representations of groups, complex atomic spectra, kinematics of the rigid body, diatomic and polyatomic molecules, relativistic theory of the electron and the foundations of electrodynamics, hyperﬁne structures and alternating bands, and elements of nuclear physics. 7 The main topics were relativistic theory of the electron, quantization procedures, ﬁeld quantities deﬁned by commutability and anticommutability laws, their kinematic equivalence with sets with an undetermined number of objects obeying Bose–Einstein or Fermi statistics, respectively, dynamical equivalence, quantization of the Maxwell–Dirac equations, study of relativistic invariance, the positive electron and the symmetry of charges, several applications of the theory, radiation and scattering processes, creation and annihilation of opposite charges, and collisions of fast electrons.

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also for writing a book, he prepared careful lecture notes [17, 18]. A recent analysis [36] showed that Majorana’s 1938 course was very innovative for that time, and this has been conﬁrmed by the retrieval (in September 2004) of a faithful transcription of the whole set of Majorana’s lecture notes (the so-called Moreno document) comprising the six lectures not included in the original collection [19]. The ﬁrst part of his course on theoretical physics dealt with the phenomenology of atomic physics and its interpretation in the framework of the old Bohr–Sommerfeld quantum theory. This part has a strict analogy with the course given by Fermi in Rome (1927–1928), attended by Majorana when a student. The second part started, instead, with classical radiation theory, reporting explicit solutions to the Maxwell equations, scattering of solar light and some other applications. It then continued with the theory of relativity: after the presentation of the corresponding phenomenology, a complete discussion of the mathematical formalism required by that theory was given, ending with some applications such as the relativistic dynamics of the electron. Then, there followed a discussion of important eﬀects for the interpretation of quantum mechanics, such as the photoelectric eﬀect, Thomson scattering, Compton eﬀects and the Franck–Hertz experiment. The last part of the course, more mathematical in nature, treated explicitly quantum mechanics, both in the Schr¨ odinger and in the Heisenberg formulations. This part did not follow the Fermi approach, but rather referred to personal previous studies, getting also inspiration from Weyl’s book on group theory and quantum mechanics.

A brief sketch of Ettore Majorana: Notes on Theoretical Physics In Ettore Majorana: Notes on Theoretical Physics we reproduced, and translated, Majorana’s Volumetti: that is, his study notes, written in Rome between 1927 and 1932. Each of those neatly organized booklets, prefaced by a table of contents, consisted of about 100−150 sequentially numbered pages, while a date, penned on its ﬁrst blank page, recorded the approximate time during which it was completed. Each Volumetto was written during a period of about 1 year. The contents of those notebooks range from typical topics covered in academic courses to topics at the frontiers of research: despite this unevenness in the level of sophistication, the style is never obvious. As an example, we can recall Majorana’s study of the shift in the melting point of a substance when it is placed in a magnetic ﬁeld, or his examination of heat propagation

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using the “cricket simile”. As to frontier research arguments, we can recall two examples: the study of quasi-stationary states, anticipating Fano’s theory, and the already mentioned Fermi theory of atoms, reporting analytic solutions of the Thomas–Fermi equation with appropriate boundary conditions in terms of simple quadratures. He also treated therein, in a lucid and original manner, contemporary physics topics such as Fermi’s explanation of the electromagnetic mass of the electron, the Dirac equation with its applications and the Lorentz group. Just to give a very short account of the interesting material in the Volumetti, let us point out the following. First of all, we already mentioned that in 1928, when Majorana was starting to collaborate (still as a university student) with the Fermi group in Rome, he had already revealed his outstanding ability in solving involved mathematical problems in original and clear ways, by obtaining an analytical series solution of the Thomas–Fermi equation. Let us recall once more that his whole work on this topic was written on some loose sheets, and then diligently transcribed by the author himself in his Volumetti, so it is contained in Ettore Majorana: Notes on Theoretical Physics. From those pages, the contribution of Majorana to the relevant statistical model is also evident, anticipating some important results found later by leading specialists. As to Majorana’s major ﬁnding (namely his methods of solutions of that equation), let us stress that it remained completely unknown until very recently, to the extent that the physics community ignored the fact that nonlinear diﬀerential equations, relevant for atoms and for other systems too, can be solved semianalytically (see Sect. 7 of Volumetto II). Indeed, a noticeable property of the method invented by Majorana for solving the Thomas–Fermi equation is that it may be easily generalized, and may then be applied to a large class of particular diﬀerential equations. Several generalizations of his method for atoms were proposed by Majorana himself: they were rediscovered only many years later. For example, in Sect. 16 of Volumetto II, Majorana studied the problem of an atom in a weak external electric ﬁeld, that is, the problem of atomic polarizability, and obtained an expression for the electric dipole moment for a (neutral or arbitrarily ionized) atom. Furthermore, he also started applying the statistical method to molecules, rather than single atoms, by studying the case of a diatomic molecule with identical nuclei (see Sect. 12 of Volumetto II). Finally, he considered the second approximation for the potential inside the atom, beyond the Thomas–Fermi approximation, by generalizing the statistical model of neutral atoms to those ionized n times, the case n = 0 included (see Sect. 15 of Volumetto II). As recently pointed out by one of us (S.E.) [23], the approach used by Majorana to this end is

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rather similar to the one now adopted in the renormalization of physical quantities in modern gauge theories. As is well documented, Majorana was among the ﬁrst to study nuclear physics in Rome (we already know that in 1929 he defended an M.Sc. thesis on such a subject). But he continued to do research on similar topics for several years, till his famous 1933 theory of nuclear exchange forces. For (α,p) reactions on light nuclei, whose experimental results had been interpreted by Chadwick and Gamov, in 1930 Majorana elaborated a dynamical theory (in Sect. 28 of Volumetto IV) by describing the energy states associated with the superposition of a continuous spectrum and one discrete level [35]. Actually, Majorana provided a complete theory for the artiﬁcial disintegration of nuclei bombarded by α particles (with and without α absorption). He approached this question by considering the simplest case, with a single unstable state of a nucleus and an α particle, which spontaneously decays by emitting an α particle or a proton. The explicit expression for the total cross-section was also given, rendering his approach accessible to experimental checks. Let us emphasize that the peculiarity of Majorana’s theory was the introduction of quasi-stationary states, which were considered by U. Fano in 1935 (in a quite diﬀerent context), and widely used in condensed matter physics about 20 years later. In Sect. 30 of Volumetto II, Majorana made an attempt to ﬁnd a relation between the fundamental constants e, h and c. The interest in this work resides less in the particular mechanical model adopted by Majorana (which led, indeed, to the result e2 hc far from the true value, as noticed by the Majorana himself) than in the interpretation adopted for the electromagnetic interaction, in terms of particle exchange. Namely, the space around charged particles was regarded as quantized, and electrons interacted by exchanging particles; Majorana’s interpretation substantially coincides with that introduced by Feynman in quantum electrodynamics after more than a decade, when the space surrounding charged particles would be identiﬁed with the quantum electrodymanics vacuum, while the exchanged particles would be assumed to be photons. Finally, one cannot forget the pages contained in Volumetti III and V on group theory, where Majorana showed in detail the relationship between the representations of the Lorentz group and the matrices of the (special) unitary group in two dimensions. In those pages, aimed also at extending Dirac’s approach, Majorana deduced the explicit form of the transformations of every bilinear quantity in the spinor ﬁelds. Certainly, the most important result achieved by Majorana on this subject is his discovery of the inﬁnite-dimensional unitary representations

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of the Lorentz group: he set forth the explicit form of them too (see Sect. 8 of Volumetto V, besides his published article (7)). We have already recalled that such representations were rediscovered by Wigner only in 1939 and 1948, and later, in 1948–1958, were eventually studied by many authors. People such as van der Waerden recognized the importance, also mathematical, of such a Majorana result, but, as we know, it remained unnoticed till Fradkin’s 1966 article mentioned above.

This volume: Majorana’s research notes The material reproduced in Ettore Majorana: Notes on Theoretical Physics was a paragon of order, conciseness, essentiality and originality, so much so that those notebooks can be partially regarded as an innovative text of theoretical physics, even after about 80 years, besides being another gold mine of theoretical, physical and mathematical ideas and hints, stimulating and useful for modern research too. But Majorana’s most remarkable scientiﬁc manuscripts—namely his research notes—are represented by a host of loose papers and by the Quaderni: and this book reproduces a selection of the latter. But the manuscripts with Majorana’s research notes, at variance with the Volumetti, rarely contain any introductions or verbal explanations. The topics covered in the Quaderni range from classical physics to quantum ﬁeld theory, and comprise the study of a number of applications for atomic, molecular and nuclear physics. Particular attention was reserved for the Dirac theory and its generalizations, and for quantum electrodynamics. The Dirac equation describing spin-1/2 particles was mostly considered by Majorana in a Lagrangian framework (in general, the canonical formalism was adopted), obtained from a least action principle (see Chap. 1 in the present volume). After an interesting preliminary study of the problem of the vibrating string, where Majorana obtained a (classical) Dirac-like equation for a two-component ﬁeld, he went on to consider a semiclassical relativistic theory for the electron, within which the Klein–Gordon and the Dirac equations were deduced starting from a semiclassical Hamilton–Jacobi equation. Subsequently, the ﬁeld equations and their properties were considered in detail, and the quantization of the (free) Dirac ﬁeld was discussed by means of the standard formalism, with the use of annihilation and creation operators. Then, the electromagnetic interaction was introduced into the Dirac equation, and the superposition of the Dirac and Maxwell ﬁelds was studied in a very personal and original way, obtaining the expression for the quantized

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Hamiltonian of the interacting system after a normal-mode decomposition. Real (rather than complex) Dirac ﬁelds, published by Majorana in his famous paper, (9), on the symmetrical theory of electrons and positrons, were considered in the Quaderni in various places (see Sect. 1.6), by two slightly diﬀerent formalisms, namely by diﬀerent decompositions of the ﬁeld. The introduction of the electromagnetic interaction was performed in a quite characteristic manner, and he then obtained an explicit expression for the total angular momentum, carried by the real Dirac ﬁeld, starting from the Hamiltonian. Some work, as well, at the basis of Majorana’s important paper (7) can be found in the present Quaderni (see Sect. 1.7 of this volume). We have already seen, when analysing the works published by Majorana, that in 1932 he constructed Dirac-like equations for spin 1, 3/2, 2, etc. (discovering also the method, later published by Pauli and Fiertz, for writing down a quantum-relativistic equation for a generic spin value). Indeed, in the Quaderni reproduced here, Majorana, starting from the usual Dirac equation for a four-component spinor, obtains explicit expressions for the Dirac matrices in the cases, for instance, of six-component and 16-component spinors. Interestingly enough, at the end of his discussion, Majorana also treats the case of spinors with an odd number of components, namely of a ﬁve-component ﬁeld. With regard to quantum electrodynamics too, Majorana dealt with it in a Lagrangian and Hamiltonian framework, by use of a least action principle. As is now done, the electromagnetic ﬁeld was decomposed in plane-wave operators, and its properties were studied within a full Lorentz-invariant formalism by employing group-theoretical arguments. Explicit expressions for the quantized Hamiltonian, the creation and annihilation operators for the photons as well as the angular momentum operator were deduced in several diﬀerent bases, along with the appropriate commutation relations. Even leaving aside, for a moment, the scientiﬁc value those results had especially at the time when Majorana achieved them, such manuscripts have a certain importance from the historical point of view too: they indicate Majorana’s tendency to tackle topics of that kind, nearer to Heisenberg, Born, Jordan and Klein’s, than to Fermi’s. As we were saying, and as already pointed out in previous literature [21], in the Quaderni one can ﬁnd also various studies, inspired by an idea of Oppenheimer, aimed at describing the electromagnetic ﬁeld within a Dirac-like formalism. Actually, Majorana was interested in describing the properties of the electromagnetic ﬁeld in terms of a real wavefunction for the photon (see Sects. 2.2, 2.10), an approach that

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went well beyond the work of contemporary authors. Other noticeable investigations of Majorana concerned the introduction of an intrinsic time delay, regarded as a universal constant, into the expressions for electromagnetic retarded ﬁelds (see Sect. 2.14), or studies on the modiﬁcation of Maxwell’s equations in the presence of magnetic monopoles (see Sect. 2.15). Besides purely theoretical work in quantum electrodynamics, some applications as well were carefully investigated by Majorana. This is the case of free electron scattering (reported in Sect. 2.12), where Majorana gave an explicit expression for the transition probability, and the coherent scattering, of bound electrons (see Sect. 2.13). Several other scattering processes were also analysed (see Chap. 6) within the framework of perturbation theory, by the adoption of Dirac’s or of Born’s method. As mentioned above, the contribution by Majorana to nuclear physics which was most known to the scientiﬁc community of his time is his theory in which nuclei are formed by protons and neutrons, bound by an exchange force of a particular kind (which corrected Heisenberg’s model). In the present Quaderni (see Chap. 7), several pages were devoted to analysing possible forms of the nucleon potential inside a given nucleus, determining the interaction between neutrons and protons. Although general nuclei were often taken into consideration, particular care was given by Majorana to light nuclei (deuteron, α particle, etc.). As will be clear from what is published in this volume, the studies performed by Majorana were, at the same time, preliminary studies and generalizations of what had been reported by him in his well-known publication (8), thus revealing a very rich and personal way of thinking. Notice also that, before having understood and thought of all that led him to the paper mentioned, (8), Majorana had seriously attempted to construct a relativistic ﬁeld theory for nuclei as composed of scalar particles (see Sect. 7.6), arriving at a characteristic description of the transitions between diﬀerent nuclei. Other topics in nuclear physics were broached by Majorana (and were presented in the Volumetti too): we shall only mention, here, the study of the energy loss of β particles when passing through a medium, when he deduced the Thomson formula by classical arguments. Such work too might a priori be of interest for a correct historical reconstruction, when confronted with the very important theory on nuclear β decay elaborated by Fermi in 1934. The largest part of the Quaderni is devoted, however, to atomic physics (see Chap. 3), in agreement with the circumstance that it was the main research topic tackled by the Fermi group in Rome in 1928–

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1933. Indeed, also the articles published by Majorana in those years deal with such a subject; and echoes of those publications can be found, of course, in the present Quaderni, showing that, especially in the case of article (5) on the incomplete P triplets, some interesting material did not appear in the published papers (see Sect. 3.18). Several expressions for the wavefunctions and the diﬀerent energy levels of two-electron atoms (and, in particular, of helium) were discovered by Majorana, mainly in the framework of a variational method aimed at solving the relevant Schr¨ odinger equation. Numerical values for the corresponding energy terms were normally summarized by Majorana in large tables, reproduced in this book. Some approximate expressions were also obtained by him for three-electron atoms (and, in particular, for lithium), and for alkali metals; including the eﬀect of polarization forces in hydrogen-like atoms. In the present Quaderni, the problem of the hyperﬁne structure of the energy spectra of complex atoms was moreover investigated in some detail, revealing the careful attention paid by Majorana to the existing literature. The generalization, for a non-Coulombian atomic ﬁeld, of the Land`e formula for the hyperﬁne splitting was also performed by Majorana, together with a relativistic formula for the Rydberg corrections of the hyperﬁne structures. Such a detailed study developed by Majorana constituted the basis of what was discussed by Fermi and Segr`e in a well-known 1933 paper of theirs on this topic, as acknowledged by those authors themselves. A small part of the Quaderni was devoted to various problems of molecular physics (see Sect. 4.3). Majorana studied in some detail, for example, the helium molecule, and then considered the general theory of the vibrational modes in molecules, with particular reference to the molecule of acetylene, C2 H2 (which possesses peculiar geometric properties). Rather important are some other pages (see Sects. 5.3, 5.4, 5.5), where the author considered the problem of ferromagnetism in the framework of Heisenberg’s model with exchange interactions. However, Majorana’s approach in this study was, as always, original, since it followed neither Heisenberg’s nor the subsequent van Vleck formulation in terms of a spin Hamiltonian. By using statistical arguments, instead, Majorana evaluated the magnetization (with respect to the saturation value) of the ferromagnetic system when an external magnetic ﬁeld acts on it, and the phenomenon of spontaneous magnetization. Several examples of ferromagnetic materials, with diﬀerent geometries, were analysed by him as well.

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A number of other interesting questions, even dealing with topics that Majorana had encountered during his academic studies at Rome University (see Chaps. 8, 9), can be found in these Quaderni. This is the case, for example, of the electromagnetic and electrostatic mass of the electron (a problem that was considered by Fermi in one of his 1924 known papers), or of his studies on tensor calculus, following his teacher Levi-Civita. We cannot discuss them here, however, our aim being that of drawing the attention of the reader to a few speciﬁc points only. The discovery of the large number of exceedingly interesting and important studies that were undertaken by Majorana, and written by him in these Quaderni, is left to the reader’s patience.

About the format of this volume As is clear from what we have discussed already, Majorana used to put on paper the results of his studies in diﬀerent ways, depending on his opinion about the value of the results themselves. The method used by Majorana for composing his written notes was sometimes the following. When he was investigating a certain subject, he reported his results only in a Quaderno. Subsequently, if, after further research on the topic considered, he reached a simpler and conclusive (in his opinion) result, he reported the ﬁnal details also in a Volumetto. Therefore, in his preliminary notes we ﬁnd basically mere calculations, and only in some rare cases can an elaborated text, clearly explaining the calculations, be found in the Quaderni. In other words, a clear exposition of many particular topics can be found only in the Volumetti. The 18 Quaderni deposited at the Domus Galilaeana are booklets of approximately of 15 cm × 21 cm, endowed with a black cover and a red external boundary, as was common in Italy before the Second World War. Each booklet is composed of about 200 pages, giving a total of about 2,800 pages. Rarely, some pages were torn oﬀ (by Majorana himself), while blank pages in each Quaderno are often present. In a few booklets, extra pages written by the author were put in. An original numbering style of the pages is present only in Quaderno 1 (in the centre at the top of each page). However, all the Quaderni have nonoriginal numbering (written in red ink) at the top-left corner of their odd pages. Blank pages too were always numbered. Interestingly enough, even though original numbering by Majorana in general is not present, nevertheless sometimes in a Quaderno there appears an original reference to some pages of that same booklet. Some other strange crossreferences, not easily understandable to us, appear (see below) in several

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booklets. Some of them refer, probably, to pages of the Volumetti, but we have been unable to interpret the remaining ones. As was evident also from a previous catalogue of the unpublished manuscripts, prepared long ago by Baldo, Mignani and Recami [14], often the material regarding the same subject was not written in the Quaderni in a sequential, logical order: in some cases, it even appeared in the reverse order. The major part of the Quaderni contains calculations without explanations, even though, in few cases, an elaborated text is fortunately present. At variance with what is found for the Volumetti, in the Quaderni no date appears, except for Quaderni 16 (“1929–1930”), 17 (“started on 20 June 1932”) and, probably, 7 (“about year 1928”). Therefore, the actual dates of composition of the manuscripts may be inferred only from a detailed comparison of the topics studied therein with what is present in the Volumetti and in the published literature, including Majorana’s published papers. Some additional information comes from some crossreferences explicitly penned by the author himself, referring either to his Quaderni or to his Volumetti. In a few cases, references to some of the existing literature are explicitly introduced by Majorana. Since no consequential or time order is present in the present Quaderni, in this book we have grouped the material by subject, and grouped the topics into four (large) parts. To identify the correspondence between what is reproduced by us in a given section and the material present in the original manuscripts, we have added a “code” to each section (or, in some cases, subsection). For instance, the code Q11p138 means that section contains material present in Quaderno 11, starting from page 138. Of course, we have also reported, in a second index (to be found at the end of this Preface, after the Bibliography), the complete list of the subjects present in the 18 Quaderni. If a particular subject is reproduced also in the present volume, this is indicated by the mere presence of the corresponding “code”. We have made a major eﬀort in carefully checking and typing all equations and tables, and, even more, in writing down a brief presentation of the argument exploited in each subsection. In addition, we have inserted among Majorana’s calculations a minimum number of words, when he had left his formalism without any text, trying to facilitate the reading of Majorana’s research notebooks, but limiting as much as possible the insertion of any personal comments of ours. Our hope is to have rendered the intellectual treasures, contained in the Quaderni, accessible for the ﬁrst time to the widest audience. With such an aim,

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we have had frequent recourse to more modern notations for the mathematical symbols. For example, the Laplacian operator has been written ∇2 by us, instead of Δ2 ; the gradient has been denoted by ∇ , instead of grad; and the vector product is represented by ×, instead of ∧; and so on. Analogously, we have treated the scalar product between vectors. In some cases, when the corresponding vectorial quantities were operators, we have retained the original Majorana notation, (a, b), which is still used in many mathematical books. The ﬁgures appearing in the Quaderni have been reproduced anew, without the use of photographic or scanning devices, but they are otherwise true in form to the original drawings. The same holds for tables; several tables had gaps, since in those cases Majorana for some reason did not perform the corresponding calculations. Other minor corrections performed by us, mainly related to typos in the original manuscripts, have been explicitly pointed out in suitable footnotes. More precisely, all changes with respect to the original, introduced by us in the present English version, have been pointed out by means of footnotes. Many additional footnotes have been introduced, whenever the interpretation of some procedures, or the meaning of particular parts, required some more words of presentation. Footnotes which are not present in the original manuscript are denoted by the symbol @. Moreover, all the additions we have made ourselves in the present volume are written, as a rule, in italics, while the original text written by Majorana always appears in Roman characters. At the end of this Preface, we attach a short Bibliography. Far from being exhaustive, it provides just some references about the topics touched upon in this Preface.

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Acknowledgements This work was partially supported by grants from MIUR-University of Bergamo and MIUR-University of Perugia. For their kind helpfulness, we are indebted to C. Segnini, the former curator of the Domus Galileana at Pisa, as well as to the previous curators and directors. Thanks are moreover due to A. Drago, A. De Gregorio, E. Giannetto, E. Majorana Jr. and F. Majorana for valuable cooperation over the years. The realization of this book has been possible thanks to a valuable technical contribution by G. Celentano, which is gratefully acknowledged here. The Editors

Bibliography

Biographical papers, written by witnesses who knew Ettore Majorana, are the following: 1. Amaldi, E.: La Vita e l’Opera di Ettore Majorana. Accademia dei Lincei, Rome (1966); Amaldi, E.: Ettore Majorana: man and scientist. In: Zichichi, A. (ed.) Strong and Weak Interactions. Academic, New York (1966); Amaldi, E.: Ettore Majorana, a cinquant’anni dalla sua scomparsa. Nuovo Saggiatore 4, 13–26 (1988); Amaldi, E.: From the discovery of the neutron to the discovery of nuclear ﬁssion. Phys. Rep. 111, 1–322 (1984) 2. Pontecorvo, B.: Fermi e la Fisica Moderna. Riuniti, Rome (1972); Pontecorvo, B.: Proceedings of the International Conference on the History of Particle Physics, Paris, July 1982. Journal de Physique 43, 221–236 (1982) 3. Segr`e, E.: Enrico Fermi, Physicist. University of Chicago Press, Chicago (1970); Segr`e, E.: A Mind Always in Motion. University of California Press, Berkeley (1993) Accurate biographical information, completed by the reproduction of many documents, is to be found in the following book (where almost all the relevant documents existing by 2002—discovered or collected by that author—appeared for the ﬁrst time): 4. Recami, E.: Il Caso Majorana: Epistolario, Documenti, Testimonianze, 2nd edn. Mondadori, Milan (1991); Recami, E.: Il Caso Majorana: Epistolario, Documenti, Testimonianze, 4th edn., pp. 1–273. Di Renzo, Rome (2002) See also: 5. Recami, E.: Ricordo di Ettore Majorana a sessant’anni dalla sua scomparsa: l’opera scientiﬁca edita e inedita. Quad. Stor. Fis. Soc. Ital. Fis. 5, 19–68 (1999)

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6. Cordella, F., De Gregorio, A., Sebastiani, F.: Enrico Fermi. Gli Anni Italiani. Riuniti, Rome (2001) 7. Esposito S.: Fleeting genius. Phys. World 19, 34–36 (2006); Recami, E.: Majorana: his scientiﬁc and human personality. In: Proceedings of the International Conference on Ettore Majorana’s legacy and the physics of the XXI century, PoS(EMC2006)016. SISSA, Trieste (2006) 8. Dragoni, G. (ed.): Ettore e Quirino Majorana tra Fisica Teorica e Sperimentale. CNR, Rome, (in press) Scientiﬁc published articles by Majorana have been discussed and/or translated into English in the following papers: 9. Majorana, E.: On nuclear theory. Z. Phys. 82, 137–145 (1933); English translation in Brink, D.M.: Nuclear Forces, part 2. Pergamon, Oxford (1965) 10. Majorana, E.: Relativistic theory of particles with arbitrary intrinsic angular momentum. Nuovo Cimento 9, 335–344 (1932); English translation in Orzalesi, C.A.: Technical report no. 792. Department of Physics and Astrophysics, University of Maryland, College Park (1968) 11. Majorana, E.: Symmetrical theory of the electron and the positron. Nuovo Cimento 14, 171–184 (1937); English translation in Sinclair, D.A.: Technical translation no. TT-542, National Research Council of Canada (1975) 12. Majorana, E.: A symmetric theory of electrons and positrons. Nuovo Cimento 14, 171–184 (1937); English translation in Maiani, L.: Soryushiron Kenkyu 63, 149–162 (1981) 13. Bassani, G.F. (ed.): Ettore Majorana—Scientiﬁc Papers. Societ` a Italiana di Fisica, Bologna/Springer, Berlin (2006) A preliminary catalogue of the unpublished papers by Majorana ﬁrst appeared [5] as well as in: 14. Baldo, M., Mignani, R., Recami E.: Catalogo dei manoscritti scientiﬁci inediti di E. Majorana. In: Preziosi, B. (ed.) Ettore Majorana—Lezioni all’Universit` a di Napoli. Bibliopolis, Naples (1987)

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The English translation of the Volumetti appeared as: 15. Esposito, S. Majorana, E., Jr., van der Merwe, A., Recami, E. (eds.): Ettore Majorana—Notes on Theoretical Physics. Kluwer, Dordrecht (2003) The original Italian version, was published in: 16. Esposito, S., Recami, E. (eds.): Ettore Majorana—Appunti Inediti di Fisica Teorica. Zanichelli, Bologna (2006) The anastatic reproduction of the original notes for the lectures delivered by Majorana at the University of Naples (during the ﬁrst months of 1938) is in: 17. Preziosi, B. (ed.): Ettore Majorana—Lezioni all’Universit` a di Napoli. Bibliopolis, Naples (1987) The complete set of the lecture notes (including the so-called Moreno document) was published in: 18. Esposito, S. (ed.): Ettore Majorana—Lezioni di Fisica Teorica. Bibliopolis, Naples (2006) See also: 19. Drago, A., Esposito, S.: Ettore Majorana’s course on theoretical physics: a recent discovery. Phys. Perspect. 9, 329–345 (2007) An English translation of (only) his notes for his inaugural lecture appeared as: 20. Preziosi, B., Recami, E.: Comment on the preliminary notes of E. Majorana’s inaugural lecture. In: Bassani, G.F. (ed.) Ettore Majorana—Scientiﬁc Papers, pp. 263–282. Societ` a Italiana di Fisica, Bologna/Springer, Berlin (2006) Other previously unknown scientiﬁc manuscripts by Majorana have been revaluated (and/or published with comments) in the following articles: 21. Mignani, R., Baldo, M., Recami, E.: About a Dirac-like equation for the photon, according to Ettore Majorana. Lett. Nuovo Cimento 11, 568–572 (1974); Giannetto, E.: A Majorana–Oppenheimer formulation of quantum electrodynamics. Lett. Nuovo Cimento 44, 140–144 & 145–148 (1985); Giannetto, E.: Su alcuni manoscritti inediti di E. Majorana. In: Bevilacqua, F. (ed.) Atti del IX Congresso Nazionale di Storia della Fisica, p. 173, Milan (1988);

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Esposito, S.: Covariant Majorana formulation of electrodynamics. Found. Phys. 28, 231–244 (1998) 22. Esposito, S.: Majorana solution of the Thomas–Fermi equation. Am. J. Phys. 70, 852–856 (2002); Esposito, S.: Majorana transformation for diﬀerential equations. Int. J. Theor. Phys. 41, 2417–2426 (2002); Esposito, S.: Fermi, Majorana and the statistical model of atoms. Found. Phys. 34, 1431–1450 (2004) 23. Majorana, E.: Ricerca di un’espressione generale delle correzioni di Rydberg, valevole per atomi neutri o ionizzati positivamente. Nuovo Cimento 6, 14–16 (1929). The corresponding original material is contained in [15, 16], while a comment is in Esposito, S.: Again on Majorana and the Thomas–Fermi model: a comment about physics/0511222. arXiv:physics/0512259 24. Esposito, S.: A peculiar lecture by Ettore Majorana. Eur. J. Phys. 27, 1147–1156 (2006); Esposito, S.: Majorana and the path-integral approach to quantum mechanics. Ann. Fond. Louis De Broglie 31, 1–19 (2006) 25. Esposito, S.: Hole theory and quantum electrodynamics in an unknown manuscript in French by Ettore Majorana. Found. Phys. 37, 956–976 & 1049–1068 (2007) 26. Esposito S.: An unknown story: Majorana and the Pauli–Weisskopf scalar electrodynamics. Ann. Phys. (Leipzig) 16, 824–841 (2007). 27. Esposito, S.: A theory of ferromagnetism by Ettore Majorana. Annals of Physics (2008), doi: 10.1016/j.aop.2008.07.005 Some scientiﬁc papers elaborating on several intuitions by Majorana are the following: 28. Fradkin, D.: Comments on a paper by Majorana concerning elementary particles. Am. J. Phys. 34, 314–318 (1966) 29. Penrose, R.: Newton, quantum theory and reality. In: Hawking, S.W., Israel, W. (eds.) 300 Years of Gravitation. Cambridge University Press, Cambridge (1987); Zimba, J., Penrose, R.: Stud. Hist. Philos. Sci. 24, 697–720 (1993); Penrose, R.: Ombre della Mente, pp. 338–343, 371–375. Rizzoli, Milan (1996) 30. Leonardi C., Lillo, F., Vaglica, A., Vetri, G.: Majorana and Fano alternatives to the Hilbert space. In: Bonifacio, R. (ed.) Mysteries,

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Puzzles, and Paradoxes in Quantum Mechanics, p. 312. AIP, Woodbury (1999); Leonardi C., Lillo, F., Vaglica, A., Vetri, G.: Quantum visibility, phase-diﬀerence operators, and the Majorana sphere. Preprint. Physics Deparment, University of Palermo (1998); Lillo, F.: Aspetti fondamentali nell’interferometria a uno e due fotoni. Ph.D. thesis, Department of Physics, University of Palermo (1998) 31. Casalbuoni, R.: Majorana and the inﬁnite component wave equations. arXiv:hep-th/0610252 Further scientiﬁc papers can be found in: 32. Licata, I. (ed.): Majorana legacy in contemporary physics. Electronic J. Theor. Phys. 3 issue 10 (2006); Dvoeglazov, V. (ed.): Ann. Fond. Louis De Broglie 31 issues 2–3 (2006) Further historical studies on Majorana’s work may be found in the following recent papers: 33. De Gregorio, A.: Il ‘protone neutro’, ovvero della laboriosa esclusione degli elettroni dal nucleo. arXiv:physics/0603261 34. De Gregorio, A., Esposito, S.: Teaching theoretical physics: the cases of Enrico Fermi and Ettore Majorana. Am. J. Phys. 75, 781–790 (2007) 35. Di Grezia, E., Esposito, S.: Majorana and the quasi-stationary states in nuclear physics. Found. Phys. 38, 228–240 (2008) 36. Drago A., S. Esposito, S.: Following Weyl on quantum mechanics: the contribution of Ettore Majorana. Found. Phys. 34, 871–887 (2004) 37. Esposito, S.: Ettore Majorana and his heritage seventy years later. Ann. Phys. (Leipzig) 17, 302–318 (2008)

TABLE OF CONTENTS OF THE COMPLETE SET OF MAJORANA’S QUADERNI (ca. 1927-1933)

Quaderno 11 Quasi coulombian scattering of particles [6.6] . . . . . . . . . . . . . . . . . . . . . . . . 1 Coulomb scattering: another regularization method [6.7] . . . . . . . . . . . . 8 Coulomb scattering [6.5] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Lorentz group and relativistic equations of motion . . . . . . . . . . . . . . . . . 14 Algebra of the Dirac spinors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Lorentz group and spinor algebra; relativistic equations, non-relativistic limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 Hydrogen atom (relativistic treatment) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .42 Quantization rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 Hydrogen atom (relativistic treatment) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .51 Relativistic spherical waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 Basic lagrangian and hamiltonian formalism for the electromagnetic ﬁeld [2.1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 Electromagnetic ﬁeld: plane wave operators [2.3] . . . . . . . . . . . . . . . . . . . 68 25 blank pages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 Electron theory (two free electrons; starting of the study of two interacting electrons) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Scattering from a potential: the Dirac method [6.3] . . . . . . . . . . . . . . . 106 Scattering from a potential: the Born method [6.4] . . . . . . . . . . . . . . . . 109 Plane waves in parabolic coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 Oscillation frequencies of ammonia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 Oriented atoms passing through a point with vanishing magnetic ﬁeld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 Quantization of the Dirac ﬁeld [1.3] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 Dirac theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 Dirac theory (Weyl equation) for a two-component neutrino . . . . . . . 150 Rigid body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 Internal orbitals of calcium (Coulomb potential plus a screened term); 1s terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 1 The number at the end of any dotted line denotes the page number of the given Quaderno where the topic was ﬁrst covered, while the number embraced in square brackets gives the section number of the present volume where Majorana’s calculations are now presented.

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Representation of the rotation group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 Theory of unstable states (time-energy uncertainty relation) . . . . . . 186 End of Quaderno 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

Quaderno 2 Classical electromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Problem of diatomic molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 General relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 Klein-Gordon theory: quantum dynamics of electrons interacting with an electromagnetic ﬁeld (continuation of p.102-112) [2.8] . . . . . . . . . . . 37 Dirac theory: vibrating string [1.1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 Dirac theory: semiclassical theory for the electron [1.2] . . . . . . . . . . . . . 39 Dirac theory (calculations) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 Problem of deformable charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 Dirac theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 Klein-Gordon theory: relativistic equation for a free particle or a particle in an electromagnetic ﬁeld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 Scalar ﬁeld theory for nuclei? [7.6] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 Electric capacity of the rotation ellipsoid . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 Potential experienced by an electric charge [2] . . . . . . . . . . . . . . . . . . . . 101 Klein-Gordon theory: quantum dynamics of electrons interacting with an electromagnetic ﬁeld [2.7] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Dirac spinors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 Diatomic molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 Atomic eigenfunctions [3.10] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 Interacting Dirac ﬁelds [1.4] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .137 Dirac theory: symmetrization [1.5] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 Dirac theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 Perturbative calculations (transition probability) . . . . . . . . . . . . . . . . . . 157 Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 Hydrogen atom in an electric ﬁeld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 Maxwell equations and Lorentz transformations . . . . . . . . . . . . . . . . . . . 182 Dirac spinors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 Isomorphism between the Lorentz group and the unimodular group in two dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 End of Quaderno 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

Enclosures Analogy between the electromagnetic ﬁeld and the Dirac ﬁeld (4 pages) [2.2] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101/1÷101/4

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Quaderno 3 Dirac theory generalized to higher spins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Maxwell equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Maxwell equations in the Dirac-like form . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Table of contents of several topics? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 Two-electron scattering [6.8] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 Electron in an electromagnetic ﬁeld (Hamiltonian) . . . . . . . . . . . . . . . . . 31 The operator 1 − ∇2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 Coulomb scattering (transformation of a diﬀerential equation) [6] . . .35 Hydrogen atom (relativistic treatment) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .36 Coulomb scattering? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 Compton eﬀect [6.9] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 19 blank pages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Quantization of the electromagnetic ﬁeld [2.4] . . . . . . . . . . . . . . . . . . . . . . 61 Quantization of the electromagnetic ﬁeld (including the matter ﬁelds) [2.6] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 Spinor representation of the Lorentz group . . . . . . . . . . . . . . . . . . . . . . . . . 71 20 blank pages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 Atom in a time-dependent electromagnetic ﬁeld . . . . . . . . . . . . . . . . . . . . 95 Electrostatic problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 Starting of the study of the Auger eﬀect . . . . . . . . . . . . . . . . . . . . . . . . . . 100 Calculations about the continuum spectrum of a system . . . . . . . . . . 101 Group of permutations (Young tableaux) . . . . . . . . . . . . . . . . . . . . . . . . . 102 Quasi-stationary states [6.10] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 5 blank pages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 Darboux formulae, Bernoulli polynomials, diﬀerential equations . . . 113 Gamma function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .119 Riemann ζ function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 Hydrogen atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 5 blank pages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Calculations (continuation from p.180-187) . . . . . . . . . . . . . . . . . . . . . . . .144 Quantization of the electromagnetic ﬁeld (angular momentum) [2.5] 155 Magnetic charges [2.15] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 Pointing vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 Calculations (Dirac equation) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 1 blank page follow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 Reduction of a three-fermion system to a two-particle one (H2+ molecule?) [4.3] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

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Calculations (Dirac equation; continuation from p.170-173) . . . . . . . 180 Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 End of Quaderno 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

Enclosures Dirac equation generalized to higher spins (15 pages) . . . A/1-1÷A/4-3 Dirac equation (angular momentum) (4 pages) . . . . . . . . . . B/2-1÷B/2-4 Dirac equation for a ﬁeld interacting with an electromagnetic ﬁeld (4 pages) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C/1-1÷C/1-4 Dirac equation for a ﬁeld interacting with an electromagnetic ﬁeld (4 pages) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C/11-1÷C/11-4 Field quantization of the Dirac equation (1 page) . . . . . . . . . . . .Z/1÷Z/2

Quaderno 4 Spectroscopic (numerical and theoretical) calculations (lithium?) . . . . 1 Calculations (Group theory; Lorentz group) . . . . . . . . . . . . . . . . . . . . . . . . 22 Oscillator; (D’Alembert) wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Quantum mechanics; Fourier transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Group theory; Euler’s functions; Euler relation for a geometric solid; permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 Blackbody . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 Group theory; spherical functions; group of rotations . . . . . . . . . . . . . . . 48 Angular momentum matrices; rigid rotator . . . . . . . . . . . . . . . . . . . . . . . . . 55 Second order diﬀerential equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 Rigid rotator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 Time-dependent perturbation theory (applications) . . . . . . . . . . . . . . . . 65 Statistical thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 Evaluation of an integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 Statistical thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 Hydrogen molecular ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 Calculations (theoretical) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 Standard thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 1 blank page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 Stock exchange list . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 3 blank pages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 (Generalized) Dirac equation “et similia”; 12-component spinors . . . 87 3 blank pages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 Plane-wave expansion (Spherical coordinates); Schr¨ odinger equation (for hydrogen) and the Laplace transform; Legendre polynomials . . . . . . . 98 Spatial rotations in 4 dimensions (spherical coordinates; generators) 108

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16 blank pages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 Variational principle in the Minkowski space-time . . . . . . . . . . . . . . . . . 137 1 blank page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 Variational principle and Hamilton equations . . . . . . . . . . . . . . . . . . . . . 139 Hyperﬁne structure: relativistic Rydberg corrections [3.19] . . . . . . . . 143 Dirac equation: non-relativistic decomposition, electromagnetic interaction of a two charged particle system, radial equations [3.20] . . . . . . 149 Dirac equation for spin-1/2 particles (4-component spinors) [1.7.1] 154 Dirac equation for spin-7/2 particles (16-component spinors) [1.7.2] 155 Dirac equation for spin-1 particles (6-component spinors) [1.7.3] . . . 157 Dirac equation for 5-component spinors [1.7.4] . . . . . . . . . . . . . . . . . . . . 160 Hyperﬁne structures and magnetic moments: formulae and tables [3.21] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 Hyperﬁne structures and magnetic moments: calculations [3.22] . . . 169 Dirac equation (generalized) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 Representations of the Lorentz group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 End of Quaderno 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

Enclosures Calculations for atomic eigenfunctions (3 pages) . . . . . . . . . . . 74/1÷74/3 Calculations for atomic eigenfunctions (3 pages) . . . . . . . . . 106/1÷106/3 Relativistic motion of a particle; hypergeometric functions (2 pages) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139/1÷139/2

Quaderno 5 Dirac equation for electrons and positrons . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Schr¨ odinger equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Field quantization of the Schr¨ odinger equation (Jordan-Klein theory) 8 Field quantization (Jordan-Klein theory) . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Creation and annihilation operators (Jordan-Klein theory) . . . . . . . . . 14 Planar motion of a point in a central ﬁeld (canonical transformations) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Dirac equation (non-relativistic limit) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Maxwell equations (variational principle) . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Phase space; classical and quantum “product” . . . . . . . . . . . . . . . . . . . . . 31 Complex spectra and hyperﬁne structures [3.14] . . . . . . . . . . . . . . . . . . . . 51 Wave equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 Phase space (continuation from p.45-50) . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 Relativistic dynamics of particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 Retarded ﬁelds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .76

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Legendre polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 Intensity of the spectral lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 Maxwell equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 Atomic spectral terms (angular momentum operators) . . . . . . . . . . . . 102 Phase space (continuation from p.71-73) . . . . . . . . . . . . . . . . . . . . . . . . . . 109 Maxwell equations (variational principle) . . . . . . . . . . . . . . . . . . . . . . . . . 117 Phase space (continuation from p.109-116) . . . . . . . . . . . . . . . . . . . . . . . . 119 6 (almost) blank pages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 Table of integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 Calculations about complex spectra [3.15] . . . . . . . . . . . . . . . . . . . . . . . . . 131 10 blank pages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 Calculations (angular momentum) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 Wavefunctions for the helium atom [3.3] . . . . . . . . . . . . . . . . . . . . . . . . . . 156 Maxwell equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 Wavefunctions for the helium atom (continuation from p.156-163) [3.3] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 Legendre polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 Spherical functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 11 blank pages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 Spherical functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 Integrals; Fourier transform for the Coulomb potential . . . . . . . . . . . . 194 End of Quaderno 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

Quaderno 6 Helium molecular ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Dirac equation (representations of the spin operator) . . . . . . . . . . . . . . . . 6 Ferromagnetism (Slater determinants) [5.5] . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Scattering from a potential well [6.1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Simple perturbation method for the Schr¨ odinger equation [6.2] . . . . . 24 Atomic energy tables [3.12] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Anomalous terms of He . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 Vibration modes in molecules [4.2] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Acetylene molecule [4.2.1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Vibration modes in molecules [4.2] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 H2 molecule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 H2 O molecule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Scattering from a potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 ? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 Numerical tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .101

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Calculations and tables (about helium and hydrogen) . . . . . . . . . . . . . 107 Table of contents of several topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 End of Quaderno 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

Quaderno 7 (dated about 1928) Legendre polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Energy levels for two-electron atoms [3.6] . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Theory of incomplete P triplets (spin-orbit couplings and energy levels) [3.18.1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 Molecular calculations (for the diatomic molecule and further generalization?); Slater determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 Two-electron atoms (3d 3d 1D terms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 Two-electron atoms (calculations) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 Theory of incomplete P triplets (energy levels for M g and Zn) [3.18.2] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 Theory of incomplete P triplets (calculations) . . . . . . . . . . . . . . . . . . . . . 92 Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 Theory of incomplete P triplets (energy levels for Zn, Cd and Hg) [3.18.3] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Calculations (quasi-stationary states, applied to the theory of incomplete P triplets?) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 Perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 Resonance between a p ( = 1) electron and an electron of azimuthal quantum number [3.16] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 Calculations on some applications of the Thomas-Fermi model . . . . 123 ? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 Dirac equation (applications) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 Wave ﬁelds (variational principle) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 2P spectroscopic terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 Scattering from a potential (Dirac and Pauli equation) . . . . . . . . . . . . 181 End of Quaderno 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

Quaderno 8 Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 blank pages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Ferromagnetism [5.3] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14 Calculations on three coupled oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2 blank pages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

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Linear equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Ferromagnetism: applications [5.4] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 Diﬀerential equations; oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 Wave Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 Legendre polynomials (multiplication rules) . . . . . . . . . . . . . . . . . . . . . . . 133 Diﬀerential equations; oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 Geometric and wave optics; diﬀerential equations . . . . . . . . . . . . . . . . . 144 Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 End of Quaderno 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

Quaderno 9 Doppler eﬀect; diﬀraction and interference; mirrors . . . . . . . . . . . . . . . . . . 1 Determination of the electron charge and the Townsend eﬀect; methods by Townsend, Zaliny, Thomson, Wilson, Millikan, Rutherford & Challook [8.4] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Electrometers, electrostatic machines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .39 Experiments by Persico, Rolland, Wood; oscillographs (cathode rays) 41 Thomson’s method for the determination of e/m [8.2] . . . . . . . . . . . . . . 44 Wilson’s chamber; Townsend eﬀect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 Electromagnetic and electrostatic mass of the electron [8.5] . . . . . . . . . 48 Wien’s method for the determination of e/m (positive charges) [8.3] 48 Dampses and Aston experiments; calculations . . . . . . . . . . . . . . . . . . . . . . 50 Isotopes, mass spectrographs, Edison eﬀect . . . . . . . . . . . . . . . . . . . . . . . . .52 Oscillographs; Richardson, photoelectric eﬀects; Langmuir experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Fermat principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 Classical oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 Mirror, lenses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 Integrals; numerical tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 Numerical calculations; Clairaut’s problem . . . . . . . . . . . . . . . . . . . . . . . . 120 Determination of a function from its moments . . . . . . . . . . . . . . . . . . . . 140 Wave Mechanics (Schr¨ odinger); angular momentum; spin . . . . . . . . . .151 π/2 Evaluation of the integral 0 sin kx/ sin x dx . . . . . . . . . . . . . . . . . . . . 164 Characters of Dj ; anomalous Zeeman eﬀect . . . . . . . . . . . . . . . . . . . . . . . 173 Harmonic oscillators; Born and Heisenberg matrices . . . . . . . . . . . . . . . 188 End of Quaderno 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

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Quaderno 10 (Master thesis, chapter I) Spontaneous ionization . . . . . . . . . . . . . . . . . . . 1 (Master thesis, chapter II) Fundamental law for the radioactive phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3 blank pages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (Master thesis, chapter III) Scattering of an α particle . . . . . . . . . . . . . 30 4 blank pages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (Master thesis, chapter IV) Gamow and Houtermans calculations . . 44 3 blank pages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 (Master thesis) Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 1 blank page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 ∞ Evaluation of a sin x/x dx; solutions of integral equations; ∇2 u + k 2 u = 0; ∇2 ϕ = z; retarded potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 Forced oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 Interference; mirrors and Fresnel biprism; Fizeau dispersion; retarded potentials and oscillators; geometric optics and interference . . . . . . . . 98 Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 End of Quaderno 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

Quaderno 11 Representations of groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 6 blank pages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Helium atom (average energy with the variational method; asymmetric potential barrier; potential of the internal masses; eigenfunctions of oneand two-electron atom; limit Stark eﬀect) . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Hartree method for two-electron atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 7 blank pages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 Green functions (applications); integral logarithm function . . . . . . . . . 72 Helium atom (variational method) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 Linear partial diﬀerential equations (complete systems) [9.1] . . . . . . . .87 Absolute diﬀerential calculus (covariant and contravariant vectors) [9.2] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Absolute diﬀerential calculus (equations of parallelism, Christoﬀel’s symbols, permutability, line elements, Euclidean manifolds, angular metric, coordinate lines, geodesic lines) [9.3] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 Absolute diﬀerential calculus (geodesic curvature, parallel displacement, autoparallelism of geodesics, associated vectors, indeﬁnite metric) [9.4] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 Absolute diﬀerential calculus (geodesic coordinates, divergence of a vector and of a tensor, transformation laws, ε systems, vector product, ﬁeld

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extension, curl of a vector, geodesic manifolds) [9.5] . . . . . . . . . . . . . . . 119 Absolute diﬀerential calculus (cyclic displacement, Riemann’s symbols, Bianchi identity and Ricci lemma, tangent geodesic coordinates) [9.6] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 Dirac equation in presence of an electromagnetic ﬁeld . . . . . . . . . . . . . 160 Group theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 Eigenvalue problem (p + ax)ψ = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 Scattering from a potential (partial waves) . . . . . . . . . . . . . . . . . . . . . . . . 180 End of Quaderno 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

Quaderno 12 Dipoles (?); oscillators (?); Bernoulli polynomials . . . . . . . . . . . . . . . . . . . . 1 Poisson brackets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Dirac equation; elementary physical quantities . . . . . . . . . . . . . . . . . . . . . 32 Calculations on applications of the Thomas-Fermi model . . . . . . . . . . . 45 Mean values of rn between concentric spherical surfaces . . . . . . . . . . . . 48 Theoretical calculations on the Townsend experiment . . . . . . . . . . . . . . 51 Dirac equation (spinning electron in a central ﬁeld) . . . . . . . . . . . . . . . . 53 Surface waves in a liquid [8.1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 Ground state energy of a two-electron atom [3.1] . . . . . . . . . . . . . . . . . . . 58 Integral representation of the Bessel functions . . . . . . . . . . . . . . . . . . . . . . 70 Radiation theory (matter-radiation interaction) . . . . . . . . . . . . . . . . . . . . 76 Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 Radiation theory (“dispersive” motion of an electron) . . . . . . . . . . . . . . 82 Variational principle; Hamilton formalism . . . . . . . . . . . . . . . . . . . . . . . . . . 88 Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Legendre spherical functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 Vector spaces; dual spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 Mendeleev’s table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 Unitary geometry and hermitian forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 Inﬁnite-dimensional vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 Evaluation of some integrals (for the helium atom) . . . . . . . . . . . . . . . . 145 1 blank page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 Evaluation of some integrals (for the helium atom) . . . . . . . . . . . . . . . . 154 Dirac equation (non-relativistic limit) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 Diamagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

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Evaluation of some integrals (for the helium atom) . . . . . . . . . . . . . . . . 157 End of Quaderno 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

Quaderno 13 Numerical calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Variational principle; Lagrange and Hamilton formalism . . . . . . . . . . . . . 2 Dirac equation for free or interacting (with the electromagnetic ﬁeld) particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 End of Quaderno 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Quaderno 14 Absolute diﬀerential calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 End of Quaderno 14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Quaderno 15 Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Scattering from a potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Dirac equation (spinning electron; Lorentz group; Maxwell equations) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Inﬁnite-component Dirac equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 End of Quaderno 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

Quaderno 16 (dated 1929-30) Helium molecule [4.1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 3 blank pages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Helium molecule [4.1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Perturbations, resonances (group theory) . . . . . . . . . . . . . . . . . . . . . . . . . . .31 Polarization forces in alkali [3.13] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 1 blank page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Calculations (group theory) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 Helium molecule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 Helium molecule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3 blank pages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 Eigenfunctions for the lithium atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

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Symmetric group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 Thomson formula for β particles in a medium [7.4] . . . . . . . . . . . . . . . . . 83 Calculations (group theory; atomic eigenfunctions) . . . . . . . . . . . . . . . . . 84 Ground state of the lithium atom (electrostatic potential) [3.8.1] . . . 98 Self-consistent ﬁeld in two-electron atoms [3.4] . . . . . . . . . . . . . . . . . . . . 100 Numerical calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Ground state of the lithium atom [3.8.2] . . . . . . . . . . . . . . . . . . . . . . . . . . 112 Numerical calculations and tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 Helium atom; two-electron atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 Asymptotic expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 Ground state of three-electron atoms [3.7] . . . . . . . . . . . . . . . . . . . . . . . . .157 2s terms for two-electron atoms [3.5] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 Asymptotic behavior for the s terms in alkali [3.9] . . . . . . . . . . . . . . . . 158 Calculations (group theory) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 Numerical calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 4 blank pages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 Eigenvalue equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 End of Quaderno 16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

Quaderno 17 (dated 20 June 1932) Proton-neutron scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Radioactivity [7.2] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Nuclear potential (mean nucleon potential) [7.3.1] . . . . . . . . . . . . . . . . . . . 6 Nuclear potential (interaction potential between nucleons) [7.3.2] . . . . 9 Nuclear potential (nucleon density) [7.3.3] . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Nuclear potential (nucleon interaction) [7.3.4] . . . . . . . . . . . . . . . . . . . . . . 14 Nuclear potential (nucleon interaction) [7.3.5] . . . . . . . . . . . . . . . . . . . . . . 20 Nuclear potential (simple nuclei) [7.3.6] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Nuclear potential (simple nuclei) [7.3.7] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Magnetic moment and diamagnetic susceptibility for a one-electron atom (relativistic calculation) [3.17] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 General transformations for matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 Symmetrical theory of the electron and positron . . . . . . . . . . . . . . . . . . . 40 General transformations for matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Dirac equation (real components); A + λA = p . . . . . . . . . . . . . . . . . . . 45 Maxwell equations in the Dirac-like form; spinor transformations (continuation from p.159-160) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 Numerical calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 Symmetrical theory of the electron and positron (continuation from p.4042) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

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Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 Maxwell equations in the Dirac-like form; spinor transformations . . . 83 1 blank page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 ? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 Dirac equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 Calculations (perturbation theory) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 3 blank pages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 Degenerate gas of spinless electrons [5.1] . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 Calculations (spherical harmonics; recursive relations) . . . . . . . . . . . . . . 98 Phase space; classical and quantum “product” . . . . . . . . . . . . . . . . . . . . 104 2 blank pages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 Wave equation for the neutron [7.1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 Quantized radiation ﬁeld [2.9] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .129 Free electron scattering [2.12] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 Wave equation of light quanta [2.10] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 Bound electron scattering [2.13] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 Wave equation of light quanta (continuation from p.142) [2.11] . . . . 151 Wavefunctions of a two-electron atom [3.2] . . . . . . . . . . . . . . . . . . . . . . . . 152 Maxwell equations in the Dirac-like form; spinor transformations . . 156 Atomic eigenfunctions (lithium) [3.11] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 Classical theory of multipole radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 Calculations (quantum mechanics) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 Atomic eigenfunctions (hydrogen) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 Calculations (quantum mechanics) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 Formulae (relativistic quantum mechanics) . . . . . . . . . . . . . . . . . . . . . . . . 183 End of Quaderno 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

Quaderno 18 Maxwell electrodynamics (variational principle) . . . . . . . . . . . . . . . . . . . . . 1 Bessel functions; generalized Green functions; Hamilton equations . . . 8 Scattering from a potential (Green functions) . . . . . . . . . . . . . . . . . . . . . . 18 Scattering from a potential (α particles); Ritz method . . . . . . . . . . . . . .27 Calculations (quantum ﬁeld theory) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 Cubic symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 Thermodynamics; van der Waals equation . . . . . . . . . . . . . . . . . . . . . . . . . 59 Calculations (quantum mechanics; perturbation theory) . . . . . . . . . . . . 66 “Double” (second) quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 Calculations (permutations; Young tableaux) . . . . . . . . . . . . . . . . . . . . . . .74 Atomic calculations (helium?); Dirac matrices; van der Waals curves 89 Numerical calculations (helium? hydrogen?) . . . . . . . . . . . . . . . . . . . . . . 106

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E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

Diﬀerential equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 Pauli paramagnetism [5.2] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 Helium (anomalous terms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 End of Quaderno 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

PART I

1 DIRAC THEORY

1.1.

VIBRATING STRING

Starting from the problem of the vibrating string (which is studied in the framework of the canonical formalism), Majorana obtained a (classical) Dirac-like equation for a two-component ﬁeld u = (u1 , u2 ), where Pauli matrices σ appear. 2 2 ∂q ∂q 1 − dτ = 0, − δ 2 ∂t ∂x q¨ = 1 H= 2

∂2q , ∂x2

p=

2

p +

∂q ∂x

∂q , ∂t 2

dx,

(q1 , p1 ) (q2 , p2 ) (q3 , p3 ) . . . , 1 2 2 H= (λ qλ + p2λ ). 2 λ

1 ∂2 = 2 − ∇2 = c ∂t

1∂ ∂ ∂ ∂ + σx + σy + σr c ∂t ∂x ∂y ∂z ∂ ∂ ∂ 1∂ − σx − σy − σz , × c ∂t ∂x ∂y ∂z

∂ ∂ ∂ 1 ∂ − σx σy σz c ∂t ∂x ∂y ∂z

u = 0,

u = (u1 , u2 ),

∂ ∂ ∂ ∂u = c σx + σy + σz ∂t ∂x ∂y ∂z

3

u,

4

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

0 1 σx = 1 0

,

0 −i , σy = i 0

1 0 σz = 0 −1

1 ∂u1 ∂ ∂ = −i u2 + c ∂t ∂x ∂y ∂ ∂ 1 ∂u2 = +i u1 − c ∂t ∂x ∂y

∂ 1∂ − c ∂t ∂z ∂ 1∂ + c ∂t ∂z

u1 =

u2 = x0 x1 x2 x3

∂ ∂ −i ∂x ∂y ∂ ∂ +i ∂x ∂y

,

∂ u1 , ∂z ∂ u2 , ∂z u2 , u1 .

= ict, = x, = y, = z,

∂ ∂ ∂ ∂ u1 = u2 , +i − i ∂x0 ∂x3 ∂x1 ∂x2 ∂ ∂ ∂ ∂ u2 = u1 . −i + i ∂x0 ∂x3 ∂x1 ∂x2

1.2.

1.2.1

A SEMICLASSICAL THEORY FOR THE ELECTRON

Relativistic Dynamics

In the following, the relativistic equations of motion for an electron in a force ﬁeld F are considered in a non-usual way, by separating the radial F r and the transverse component F t (with respect to the particle velocity βc) of the force. Expressions for the time derivative of the charge density ρ and current density i, which satisfy the continuity equation, are obtained.

5

DIRAC THEORY

charge + e mass m ρ,

ix = ρβx ,

iy = ρβy ,

iz = ρβz ;

βx = vx /c, βy = vy /c, βz = vz /c; β = βx2 + βy2 + βz2 = v/c. d dt d dt d dt

mv

x = eFx , 1 − β2 mvy

= eFy , 1 − β2 mv

z = eFz . 1 − β2 k=

e . mc

β 1 d

= F, 2 dt 1 − β k ˙ β β˙ (β · β)β 1 d

=

+ =

2 )3/2 2 dt 1 − β 2 (1 − β 1−β 1 − β2

β · β˙ β β˙ + 1 − β2

1 1 1 ˙

(β · β)β = F. β˙ + 3/2 2 2 k (1 − β ) 1−β 1 1 ˙ (β · β), F ·β = k (1 − β 2 )3/2 1 1 F ×β =

β˙ × β; 2 k 1−β 1 β r = (1 − β 2 )3/2 F r , k

1 βt = 1 − β 2 F t ; k

,

6

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

β˙ = β˙ r + β˙ t , F = F r + F t.

βy βx , (Fx βx + Fy βy + Fz βz ) 2 , 2 β β βz (Fx βx + Fy βy + Fz βz ) 2 , β βy βx F t = Fx − (Fx βx + Fy βy + Fz βz ) 2 , Fy − (Fx βx + Fy βy + Fz βz ) 2 , β β βr Fz − (Fx βx + Fy βy + Fz βz ) 2 . β

Fr =

(Fx βx + Fy βy + Fz βz )

d 1 − β2 [Fx − (Fx βx + Fy βy + Fz βz )βz ] = βx , β˙ x = k dt

d 1 − β2 ˙ [Fx − (Fx βx + Fy βy + Fz βz )βz ] = βy , βy = dt

k 2 d 1 − β [Fx − (Fx βy + Fy βy + Fz βz )βz ] = βz . β˙ r = k dt ∂ix ∂iy ∂iz + + = 0; ∂x ∂y ∂z ∂ρ ∂ρ ∂ρ ∂ρ dρ ; = + c βx + βy + βz dt ∂t ∂x ∂y ∂z dρ ∂ρ ∂ρ ∂ρ ∂ix ∂iy ∂iz = c βx + βy + βz − − − ; dt ∂x ∂y ∂z ∂x ∂y ∂z ∂iy dix ∂ix ∂iz ∂ix = − c βx + βy + βz ; ∂t dt ∂x ∂y ∂z ∂ρ +c ∂t

dix dt

d dβx dρ (ρβx ) = βx +ρ dt dt dt ∂iz ∂ρ ∂ρ ∂ρ ∂ix ∂iy + βy + βz − − − = βx · c βx ∂x ∂y ∂z ∂x ∂y ∂z

1 − β2 [Fx − (Fx βx + Fy βy + Fz βz )βx ] . +ρ k =

7

DIRAC THEORY

1.2.2

Field Equations

The author began now to study the ﬁeld equations for an electron in an electromagnetic potential (ϕ, C) by following two diﬀerent approaches. In the ﬁrst part, he “tries” with a semiclassical Hamilton-Jacobi equation corresponding to the relativistic expression for the energy-momentum relation, by imposing the constraint of a positive value for the energy. From appropriate correspondence relations, he then deduced a KleinGordon equation for the ﬁeld ψ and, on introducing the Pauli matrices, the Dirac equations for the electron 4-component wavefunction. Some (mathematical) consequences of the formalism adopted (mainly related to the charge-current density) were also analyzed. In the second part, Majorana focused his attention on the standard formalism for the Dirac equation, again discussing in detail the expressions for the Dirac charge-current density (ρ, i) and some peculiar constraints on Lorentz-invariant ﬁeld quantities. He introduced and studied the consequences of several ansatz leading to Dirac-like equations for the electron. 2 2 1 ∂S e ∂S e − − + ϕ + + Cx + m2 c2 = 0; c ∂t c ∂x c x

−

1 ∂S e + ϕ>0. c ∂t c

ψ = A e2πiS/h ,

∂ψ ∂x ∂2ϕ ∂x2

1 ∂A 2πi ∂S ∂A 2πi ∂S 2πiS/h = = +A e + ψ ∂x h ∂x A ∂x h ∂x 2 ∂A 2πi ∂S 4π 2 ∂S 2πi ∂S ∂ A e2πiS/h = + 2 − A + A ∂x2 ∂x h ∂x h ∂x2 h2 ∂x2 1 ∂2A 2 2π ∂S 2πi ∂ 2 S 4π 2 ∂S = + − 2 ψ + A ∂x2 A h ∂x h ∂x2 h ∂x2

Versuchsweise: 1@

A = |ψ|.

1

This German word means “tentatively”, and refers to the successive assumptions. Note, however, that in the original paper the cited word is written as “versucherweiser”.

8

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

⎧ ∂S h 1 ∂ψ ⎪ ⎪ = ; ⎪ ⎪ 2πi ψ ∂x ⎨ ∂x

∂S h 1 ∂ψ = ; ∂t 2πi ψ ∂t

⎪ ⎪ ∂S h 1 ∂ψ ⎪ ⎪ =− ; ⎩ ∂x 2πi ψ ∂x

∂S h 1 ∂ψ =− . ∂t 2πi ψ ∂t

2 2 h ∂ e e 1 h ∂ + ϕ + Cx ψ +m2 c2 ψ 2 = 0. (B) + − − c 2πi ∂t c 2πi ∂x c x Approximate condition: 1 h ∂ e 1 h ∂ + ϕ ψ+ψ + eϕ ψ > 0. ψ − c 2πi ∂t c c 2πi ∂t In exact form: 2 1 ∂S e 2 ∂S e + ϕ + + Cx + m2 c2 = 0, − − c ∂t c ∂x c x |ψ| = 1;

(C)

ψ = e2πiS/h , 2πi ∂S ∂ψ = ψ. ∂x h ∂x (A) ≡ (B) + (C). 2π S, h 2π ∂S 2π ∂ψ0 = cos S, ∂x h ∂x h ψ0 = sin

2π S; h ∂ψ1 2π ∂S 2π =− sin S; ∂x h ∂x h ψ1 = cos

∂S h ∂ψ0 = ψ1 , 2π ∂x ∂x ∂S h ∂ψ1 = − ψ0 , 2π ∂x ∂x

(A)

9

DIRAC THEORY

1 h ∂ψ0 1 h ∂ψ1 ∂S = =− . ∂x ψ1 2π ∂x ψ0 2π ∂x ——————– 1 h ∂ψ1 e 1 h ∂ϕ0 e δ − ϕψ1 + ϕψ0 c 2π ∂t c c 2π ∂t c h ∂ψ0 e h ∂ψ1 e 2 2 + + Cx ψ1 − Cx ψ0 + m c ψ0 ψ1 dτ = 0 2π ∂x c 2π ∂x c x

(dτ = dV dt).

2

e h ∂ 1 h ∂ψ0 e 1 h ∂ϕ1 e − ϕψ1 + ϕ + ϕψ0 2π ∂t c 2π ∂t c c c 2π ∂t c h ∂ h ∂ψ0 e e 2 ∂ψ1 e − + Cx ϕ1 − Cx − Cx ψ0 2π ∂x 2π ∂x c c 2π ∂x c x +m2 c2 ψ0 = 0. e e h 1 h ∂ + ϕ + ρ3 σ · ∇ + C + ρ1 mc ψ = 0, − c 2πi ∂t c 2πi c 0 1 0 −i 1 0 , σy = σx = i 0 , σz = 0 −1 ; 1 0 A = (ψ1 , ψ2 ),

B = (ψ3 , ψ4 ).

e e 1 h ∂ h − + ϕ+σ· ∇ + C A + mcB = 0, c 2πi ∂t c 2πi c 1 h ∂ e e h − + ϕ−σ· ∇ + C B + mcA = 0. c 2πi ∂t c 2πi c ˜ + BB ˜ = ψ 1 ψ1 + ψ 2 ψ2 + ψ 3 ψ3 + ψ 4 ψ4 , ρ = AA ˜ x A + Bσ ˜ x B = −ψ 1 ψ2 − ψ 2 ψ1 + ψ 3 ψ4 + ψ 4 ψ3 , ix = Aσ ˜ y A + Bσ ˜ y B = i(ψ 1 ψ2 − ψ 2 ψ1 − ψ 3 ψ4 + ψ 4 ψ3 ), iy = Aσ ˜ r A + Bσ ˜ r B = −ψ 1 ψ1 + ψ 2 ψ2 + ψ 3 ψ3 − ψ 4 ψ4 . iz = Aσ 2@

Note that, more appropriately, it should be written d4 τ = d3 V dt, since dτ denotes the 4-dimensional volume element, while drmV is the 3-dimensional space volume element.

10

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

ψ1 , ψ2 ∼ −ψ 4 , +ψ 3 , ψ3 , ψ4 ∼ ψ 2 , −ψ 1 . Versuchsweise:

ψ3 = k ψ 2 , ψ4 = −k ψ 1 ; ψ1 = −(1/k) ψ 4 , ψ2 = (1/k) ψ 3 ;

k = k(x, y, r, t), ψ 1 ψ3 + ψ 2 ψ4 = 0.

h e ∂ ∂ 1 h ∂ + ϕ ψ1 + −i + − c 2πi ∂t c 2πi ∂x ∂y e h ∂ + Cz ψ1 + mc ψ3 = 0, + 2πi ∂z c h e ∂ ∂ 1 h ∂ + ϕ ψ2 + +i + − c 2πi ∂t c 2πi ∂x ∂y h ∂ e + Cz ψ2 + mc ψ4 = 0, − 2πi ∂z c e ∂ ∂ 1 h ∂ h + ϕ ψ3 − −i + − c 2πi ∂t c 2πi ∂x ∂y h ∂ e + Cz ψ3 + mc ψ1 = 0, − 2πi ∂z c e h ∂ ∂ 1 h ∂ + ϕ ψ4 − +i + − c 2πi ∂t c 2πi ∂x ∂y e h ∂ + Cz ψ4 + mc ψ2 = 0. + 2πi ∂z c

——————– k = k(x, y, r, t)

e (Cx − iCy ) ψ2 c

e (Cx + iCy ) ψ1 c

e (Cx − iCy ) ψ4 c

e (Cx + iCy ) ψ3 c

11

DIRAC THEORY

1 h ∂ e ∂ ∂ h + − + ϕ ψ1 + −i c 2πi ∂t c 2πi ∂x ∂y e h ∂ + + Cz ψ1 + kmc ψ2 = 0, 2πi ∂z c e ∂ ∂ 1 h ∂ h + − + ϕ ψ2 + +i c 2πi ∂t c 2πi ∂x ∂y e h ∂ − + Cz ψ2 − kmc ψ1 = 0, 2πi ∂z c

e (Cx − iCy ) ψ2 c

e (Cx + iCy ) ψ1 c

e e ∂ ∂ 1 h ∂ h + (Cx − iCy ) (−kψ 1 ) − + ϕ (kψ 2 ) − −i c 2πi ∂t c 2πi ∂x ∂y c e h ∂ − + Cz (kψ 2 ) + mc ψ1 = 0, 2πi ∂z c e e ∂ ∂ 1 h ∂ h + (Cx + iCy ) (kψ 2 ) − + ϕ (−kψ 1 ) − +i c 2πi ∂t c 2πi ∂x ∂y c e h ∂ + + Cz (−kψ 1 ) + mc ψ2 = 0. 2πi ∂z c

——————– without ﬁeld3 :

k = ±1;

h 1 h ∂ − ψ1 + c 2πi ∂t 2πi h 1 h ∂ − ψ2 + c 2πi ∂t 2πi h 1 h ∂ − ψ2 + c 2πi ∂t 2πi h 1 h ∂ + ψ − c 2πi ∂t 1 2πi

ψ3 = ψ 2 ;

∂ ∂ −i ∂x ∂y ∂ ∂ +i ∂x ∂y ∂ ∂ −i ∂x ∂y ∂ ∂ +i ∂x ∂y

ψ4 = −ψ 1 ;

ϕ, C = 0

ψ2 +

h ∂ ψ1 + mc ψ 2 = 0, 2πi ∂r

ψ1 −

h ∂ ψ2 − mc ψ 1 = 0, 2πi ∂r

ψ1 −

h ∂ ψ + mc ψ1 = 0, 2πi ∂r 2

ψ2 −

h ∂ ψ + mc ψ2 = 0. 2πi ∂r 1

For real u1 , u2 , u3 , u4 : This interesting side note is present in the original manuscript: we can use ±m in place of k = ±1: k = 1 corresponds to m and k = −1 corresponds to −m.

3@

12

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

k=1:

k = −1 :

ρ ix iy iz

1 h ∂ u1 − c 2π ∂t 1 h ∂ u2 − c 2π ∂t 1 h ∂ u3 − c 2π ∂t 1 h ∂ u4 − c 2π ∂t

ψ1 =

u1 + iu2 √ , 2

ψ2 =

u3 + iu4 √ , 2

ψ3 =

u3 − iu4 √ , 2

ψ4 =

−u1 + iu2 √ ; 2

ψ1 =

u1 + iu2 √ , 2

ψ2 =

u3 + iu4 √ , 2

ψ3 =

−u3 + iu4 √ , 2

= = = =

ψ4 =

u1 − iu2 √ . 2

u21 + u22 + u23 + u24 , − (2u1 u3 + 2u2 u4 ) , − (2u1 u4 − 2u2 u3 ) , − u21 + u22 − u23 − u24 .

h ∂ u3 − 2π ∂x h ∂ u4 + 2π ∂x h ∂ u1 + 2π ∂x h ∂ u2 − 2π ∂x

h ∂ u4 − 2π ∂y h ∂ u3 − 2π ∂y h ∂ u2 + 2π ∂y h ∂ u1 + 2π ∂y

h ∂ u1 − mc u4 2π ∂z h ∂ u2 − mc u3 2π ∂z h ∂ u3 + mc u2 2π ∂z h ∂ u4 + mc u1 2π ∂z

= 0, = 0, = 0, = 0.

h ∂ ∂ ∂ 1 h ∂ u= γ1 + γ2 + γ3 + δ mc u. c 2π ∂t 2π ∂x ∂y ∂r 0 0 1 0 0 0 0 1 0 0 0 1 , γ2 = 0 0 −1 0 , γ1 = 1 0 0 0 0 −1 0 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 1 0 0 1 0 1 0 , δ = γ3 = . 0 0 −1 0 0 −1 0 0 0 0 0 −1 −1 0 0 0 γ1 = ρ1 ,

γ2 = −σy ρ2 ,

γ3 = ρ3 ,

δ = −iσx ρ2 .

13

DIRAC THEORY

For u = u(r, t): ∂ h 1∂ − u1 2π c ∂t ∂z ∂ h 1∂ − u2 2π c ∂t ∂z ∂ h 1∂ + u3 2π c ∂t ∂z ∂ h 1∂ u4 + 2π c ∂t ∂z u1 = λ 1 R e u2 = λ 2 R e u3 = λ 3 R e u1 = λ 4 R e −i −i

a c a

c a −i c a −i c a2 c2 λ4 λ1

+ b λ1 + b λ2 − b λ3 − b λ4

= mcu4 , = mcu3 , = −mcu2 , = −mcu1 ;

2πi (−at+bz) h

,

2πi (−at+bz) h

,

2πi (−at+bz) h

,

2πi (−at+bz) h

.

= mc λ4 , = mc λ3 , = −mc λ2 , = −mc λ1 ;

= m2 c2 + b2 , = −

λ i a 3 . +b = mc c λ2

——————– ρ = u† L0 u, L0 =

1 0 0 0

0 1 0 0

ix = u† L1 u, 0 0 1 0

0 0 0 1

,

iy = u† L2 u,

L1 = −

0 0 1 1

0 0 0 0

1 0 0 0

iz = u† L3 u; 0 1 0 0

= −γ1 ,

14

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

L2 = −

0 0 0 1 0 0 −1 0 = −γ2 , 0 −1 0 0 1 0 0 0

L3 = −

1 0 0 0

0 0 0 1 0 0 0 −1 0 0 0 −1

= −γ3 .

ρ2 = (u21 + u22 + u23 + u24 )2 = u41 + u42 + u43 + u44 + 2u21 u22 + 2u21 u23 + 2u21 u24 + 2u22 u23 +2u22 u24 + 2u23 u24 , i2x = 4(u1 u3 + u2 u4 )2 = 4u21 u23 + 4u22 u24 + 8u1 u2 u3 u4 , i2y = 4(u1 u4 − u2 u3 )2 = 4u21 u24 + 4u22 u23 − 8u1 u2 u3 u4 , i2z = (u21 + u22 − u23 − u24 )2 , = u41 + u42 + u43 + u24 + 2u21 u22 − 2u21 u23 − 2u21 u24 − 2u22 u23 −2u22 u24 + 2u23 u24 ; ρ2 − i2x − i2y − i2z = 0. ——————– 2

2

2

2

(ψ 1 ψ1 + ψ 2 ψ2 + ψ 3 ψ3 + ψ 4 ψ4 )2 = ψ 1 ψ12 + ψ 2 ψ22 + ψ 3 ψ32 + ψ 4 ψ42 + 2ψ 1 ψ 2 ψ1 ψ2 + 2ψ 1 ψ 3 ψ1 ψ3 + 2ψ 1 ψ 4 ψ1 ψ4 + 2ψ 2 ψ 3 ψ2 ψ3 + ψ 2 ψ 4 ψ2 ψ4 + 2ψ 3 ψ 4 ψ3 ψ4 , 2

2

2

2

(−ψ 1 ψ2 − ψ 2 ψ1 + ψ 3 ψ4 + ψ 4 ψ3 )2 = ψ 1 ψ22 + ψ 2 ψ12 + ψ 3 ψ42 + ψ 4 ψ32 + 2ψ 1 ψ 2 ψ1 ψ2 − 2ψ 1 ψ 3 ψ2 ψ4 − 2ψ 1 ψ 4 ψ2 ψ3 − 2ψ 2 ψ 3 ψ1 ψ4 − 2ψ 2 ψ 4 ψ1 ψ3 + 2ψ 3 ψ 4 ψ3 ψ4 , 2

2

2

2

−(ψ 1 ψ2 − ψ 2 ψ1 − ψ 3 ψ4 + ψ 4 ψ3 )2 = −ψ 1 ψ22 − ψ 2 ψ12 − ψ 3 ψ42 − ψ 4 ψ32 + 2ψ 1 ψ 2 ψ1 ψ2 + 2ψ 1 ψ 3 ψ2 ψ4 − 2ψ 1 ψ 4 ψ2 ψ3 − 2ψ 2 ψ 3 ψ1 ψ4 + 2ψ 2 ψ 4 ψ1 ψ3 + 2ψ 3 ψ 4 ψ3 ψ4 , 2

2

2

2

(−ψ 1 ψ1 + ψ 2 ψ2 + ψ 3 ψ3 − ψ 4 ψ4 )2 = ψ 1 ψ12 + ψ 2 ψ22 + ψ 3 ψ32 + ψ 4 ψ42 2

2

− 2ψ 1 ψ 2 ψ1 ψ2 − 2ψ 1 ψ 3 ψ1 ψ3 + 2ψ 1 ψ 4 ψ1 ψ4 + 2ψ 2 ψ 3 ψ2 ψ3 − 2ψ 2 ψ 4 ψ2 ψ4 − 2ψ 3 ψ 4 ψ3 ψ4 .

ρ2 − i2z = 4ψ 1 ψ 2 ψ1 ψ2 + 4ψ 1 ψ 3 ψ1 ψ3 + 4ψ 2 ψ 4 ψ2 ψ4 + 4ψ 3 ψ 4 ψ3 ψ4 , i2x + i2y = 4ψ 1 ψ 2 ψ1 ψ2 − 4ψ 1 ψ 4 ψ2 ψ3 − 4ψ 2 ψ 3 ψ1 ψ4 + 4ψ 3 ψ 4 ψ3 ψ4 .

15

DIRAC THEORY

ρ2 − i2x − i2y − i2r = 4ψ 1 ψ 3 ψ1 ψ3 + 4ψ 2 ψ 4 ψ2 ψ4 + 4ψ 1 ψ 4 ψ2 ψ3 + 4ψ 2 ψ 3 ψ1 ψ4 = 4(ψ 1 ψ3 + ψ 2 ψ4 )(ψ1 ψ 3 + ψ2 ψ 4 ) = QQ; Q = 2(ψ 1 ψ3 + ψ 2 ψ4 ),

Q = (ψ1 ψ 3 + ψ2 ψ 4 ).

——————– e e W + ϕ + ρ3 σx px + Cx + ρ1 mc ψ = 0. c c c x e e W σx px + Cx + ρ1 mc ψ dτ = 0; + ϕ + ρ3 δ ψ˜ c c c x

dτ = dV dt. ψ1 ψ 3 + ψ2 ψ 4 − ψ 1 ψ3 − ψ 4 ψ2 = 0. e e W σx px + Cx + ρ1 mc ψ + ϕ + ρ3 δ ψ˜ c c c x + λ i(ψ 1 ψ3 + ψ 2 ψ4 − ψ1 ψ 3 − ψ2 ψ 4 ) dτ = 0. δ

0 0 0 0 −i 0 0 −i

i 0 0 0

0 i 0 0

= −ρ2 .

e e W + ϕ + ρ3 σx px + Cx + ρ1 mc − λρ2 ψ dτ = 0. ψ˜ c c c x ⎧ ⎪ e e W ⎪ ⎪ + ϕ + ρ3 σx px + Cx + ρ1 mc ϕ = λ ρ2 ψ, ⎨ c c c x ⎪ ⎪ ⎪ ⎩ ˜ ψρ2 ψ = 0. ρ3 σx = αx ,

ρ3 σy = αy ,

ρ3 σz = αz ,

ρ1 = α4 ,

ρ2 = α5 ;

16

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

αi αk + αk αi = 2δik ; α = (αx , αy , αz ).

e e W + ϕ + α · p + C + α4 mc ψ = α5 λψ, c c c −

˜ 5 ψ = 0. ψα

e W e ψ = ϕ + α · p + C + α4 mc − α5 λ ψ, c c c

e W e ˜ ˜ ˜ ˜ 4 α5 mc ψ − λψψ. ˜ ψαx α5 px + Cx ψ − ψα −ψα5 ψ = ϕ ψα5 ψ − c c c x A = (ψ1 , ψ2 ),

B(ψ3 , ψ4 ).

e e W + ϕ + σ · p + C A + mc B = −λ iB, c c c

˜ − AB ˜ = 0. BA e e W + ϕ − σ · p + C B + mc A = λ iB. c c c e ∂ ∂ h 1 h ∂ + ϕ ψ1 + −i + − c 2πi ∂t c 2πi ∂x ∂y e h ∂ + Cz ψ1 + mc ψ3 = −λ iψ3 , + 2πi ∂z c h e ∂ ∂ 1 h ∂ + ϕ ψ2 + +i + − c 2πi ∂t c 2πi ∂x ∂y h ∂ e + Cz ψ2 + mc ψ4 = −λ iψ4 , − 2πi ∂z c e ∂ ∂ h 1 h ∂ + ϕ ψ3 − −i + − c 2πi ∂t c 2πi ∂x ∂y h ∂ e + Cz ψ3 + mc ψ1 = λ iψ1 , − 2πi ∂z c e h ∂ ∂ 1 h ∂ + ϕ ψ4 − +i + − c 2πi ∂t c 2πi ∂x ∂y h ∂ e + + Cz ψ4 + mc ψ2 = λ iψ2 . 2πi ∂z c

e (Cx − iCy ) ψ2 c e (Cx + iCy ) ψ1 c e (Cx − iCy ) ψ4 c e (Cx − iCy ) ψ3 c

17

DIRAC THEORY

ψ 1 ψ3 + ψ 2 ψ4 − ψ 3 ψ1 − ψ 4 ψ2 = 0. h e ∂ ∂ 1 h ∂ + ϕ ψ1 − +i − c 2πi ∂t c 2πi ∂x ∂y e h ∂ − Cz ψ 1 + mc ψ 3 = λ iψ 3 , − 2πi ∂z c e ∂ ∂ h 1 h ∂ + ϕ ψ2 − −i − c 2πi ∂t c 2πi ∂x ∂y h ∂ e − Cz ψ 2 + mc ψ 4 = λ iψ 4 , + 2πi ∂z c

e (Cx + iCy ) ψ 2 c e (Cx − iCy ) ψ 1 c

e h ∂ ∂ e 1 h ∂ + ϕ ψ3 + +i − (Cx + iCy ) ψ 4 c 2πi ∂t c 2πi ∂x ∂y c h ∂ e − Cz ψ 3 + mc ψ 1 = −λ iψ 1 , + 2πi ∂z c e h ∂ ∂ e 1 h ∂ + ϕ ψ4 + −i − (Cx − iCy ) ψ 3 c 2πi ∂t c 2πi ∂x ∂y c h ∂ e − Cz ψ 4 + mc ψ 2 = −λ iψ 2 . − 2πi ∂z c

1 h ∂ (ψ ψ3 + ψ 2 ψ4 − ψ 3 ψ1 − ψ 4 ψ2 ) c 2πi ∂t 1 e ∂ ∂ e h + (Cx − iCy ) ψ4 −i = ψ 1 ϕ ψ3 − ψ 1 c 2πi ∂x ∂y c h ∂ e + Cz ψ3 + mc ψ 1 ψ1 − λ iψ 1 ψ1 −ψ 1 2πi ∂z c ∂ ∂ e e h +i + (Cx + iCy ) ψ3 +ψ 2 ϕ ψ4 − ψ 2 c 2πi ∂x ∂y c h ∂ e +ψ 2 + Cz ψ4 + mc ψ 2 ψ2 − λ iψ 2 ψ2 2πi ∂z c ∂ ∂ e e h −i + (Cx − iCy ) ψ2 −ψ 3 ϕ ψ1 − ψ 3 c 2πi ∂x ∂y c e h ∂ −ψ 3 + Cz ψ1 − mc ψ 3 ψ3 − λ iψ 3 ψ3 2πi ∂z c ∂ e ∂ e h +i −ψ 4 ϕ ψ2 − ψ 4 + (Cx − iCy ) ψ1 c 2πi ∂x ∂y c

18

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

e h ∂ −ψ 4 + Cz ψ2 − mc ψ 4 ψ4 − λ iψ 4 ψ4 2πi ∂z c + complex conjugate terms.

——————– δ

e e W σx p + Cx + (cos λ ρ1 + sin λ ρ2 ) mc ψ = 0. + ϕ + ρ3 ψ˜ c c c x

1 h ∂ e h ∂ ∂ − + ϕ ψ1 + −i + c 2πi ∂t c 2πi ∂x ∂y e h ∂ + Cz ψ1 + e−iλ mc ψ3 = 0, + 2πi ∂z c e h ∂ ∂ 1 h ∂ + ϕ ψ2 + +i + − c 2πi ∂t c 2πi ∂x ∂y e h ∂ + Cz ψ2 + e−iλ mc ψ4 = 0, − 2πi ∂z c h e ∂ ∂ 1 h ∂ + ϕ ψ3 − −i + − c 2πi ∂t c 2πi ∂x ∂y e h ∂ + Cz ψ3 + eiλ mc ψ1 = 0, − 2πi ∂z c h e ∂ ∂ 1 h ∂ + ϕ ψ4 − +i + − c 2πi ∂t c 2πi ∂x ∂y e h ∂ + Cz ψ4 + eiλ mc ψ2 = 0. + 2πi ∂z c

e (Cx − iCy ) ψ2 c

e (Cx + iCy ) ψ1 c

e (Cx − iCy ) ψ4 c

e (Cx + iCy ) ψ3 c

˜ sin λ ρ1 + cos λ ρ2 )ψ = 0. ψ(− ρ1 =

0 0 1 0

0 0 0 1

1 0 0 0

0 1 0 0

,

ρ2 =

0 0 i 0

0 −i 0 0 0 −i , 0 0 0 i 0 0

19

DIRAC THEORY

0 0 cos λ ρ1 + sin λ ρ2 = iλ e 0 0 0 − sin λ ρ1 + cos λ ρ2 = iλ ie 0

0 e−iλ 0 0 0 e−iλ 0 0 0 eiλ 0 0

,

0 −ie−iλ 0 0 0 −ie−iλ 0 0 0 0 0 ieiλ

.

˜ sin λ ρ1 + cos λ ρ2 )ψ ψ(− = (1/i) e−iλ ψ 1 ψ3 + e−iλ ψ 2 ψ4 − eiλ ψ 3 ψ1 − eiλ ψ 4 ψ2 = 0. e−iλ (ψ 1 ψ3 + ψ 2 ψ4 ) − eiλ (ψ 3 ψ1 + ψ 4 ψ2 ) = 0. 1 h ∂ −iλ e ψ 1 ψ3 + e−iλ ψ 2 ψ4 − eiλ ψ 3 ψ1 − eiλ ψ 4 ψ2 c 2πi ∂t =−

D =

+

+

+

∂λ 1 h −iλ (e ψ 1 ψ3 + e−iλ ψ 2 ψ4 + eiλ ψ 3 ψ2 + eiλ ψ 4 ψ2 ) + D + D, c 2π ∂t ∂ ∂ e e h −i + (Cx − iCy ) ψ4 ψ 1 ϕ ψ3 − ψ 1 e c 2πi ∂x ∂y c e h ∂ + Cz ψ3 + eiλ mc ψ 1 ψ1 −ψ 1 2πi ∂z c e ∂ ∂ e h −iλ e +i + (Cx + iCy ) ψ3 ψ 2 ϕ ψ4 − ψ 2 c 2πi ∂x ∂y c e h ∂ + Cz ψ4 + eiλ mc ψ 2 ψ2 +ψ 2 2πi ∂z c ∂ ∂ e h e +iλ −i + (Cx − iCy ) ψ2 ψ 3 ϕ ψ1 + ψ 3 e c 2πi ∂x ∂y c e h ∂ + Cz ψ1 + e−iλ mc ψ 3 ψ3 +ψ 3 2πi ∂z c ∂ ∂ e e h +iλ +i + (Cx + iCy ) ψ1 ψ 4 ϕ ψ2 + ψ 4 e c 2πi ∂x ∂y c e h ∂ −ψ 4 + Cz ψ2 + e−iλ mc ψ 4 ψ4 2πi ∂z c −iλ

20

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

e = − Cx e−iλ (+ψ 1 ψ4 + ψ 2 ψ3 ) + eiλ (ψ 3 ψ2 + ψ 4 ψ1 ) c e −iλ iλ i e C (ψ 1 ψ4 − ψ 2 ψ3 ) − i e (ψ 4 ψ1 − ψ 3 ψ2 ) + y c e −iλ iλ − Cz e (ψ 1 ψ3 − ψ 2 ψ4 ) + e (ψ 3 ψ1 − ψ 4 ψ2 ) c +mc ψ 1 ψ1 + ψ 2 ψ2 − ψ 3 ψ3 − ψ 4 ψ4 ∂ ∂ ∂ ∂ h −iλ ψ 1 ψ4 + ψ 2 ψ3 e + ψ 3 ψ2 + ψ 4 ψ1 eiλ − 2πi ∂x ∂x ∂x ∂x h ∂ ∂ ∂ ∂ −iλ + ψ 1 ψ4 − ψ 2 ψ 3 e + ψ 3 ψ2 − ψ 4 ψ1 eiλ 2πi ∂y ∂x ∂x ∂x ∂ ∂ ∂ ∂ h −iλ − ψ 1 ψ1 − ψ 2 ψ4 e + ψ 3 ψ1 − ψ 4 ψ2 eiλ . 2πi ∂z ∂z ∂z ∂z [4 ] −iλ 0 0 0 e −iλ 0 0 0 e , β = −iλ 0 0 0 e 0 0 0 e−iλ 0 0 0 −ie−iλ −iλ 0 0 0 −ie . γ = iλ ie 0 0 0 0 eiλ 0 0 ψ1 ψ † = |ψ1 , ψ2 , ψ3 , ψ4 ), ψ2 , ψ = ψ3 ψ˜ = |ψ 1 ψ 2 ψ 3 ψ4 ). ψ4 β = β(λ), β = cos λ ρ1 + sin λ ρ2 , βγ = γβ = 0,

γ = γ(λ); γ = − sin λ ρ1 + cos λ ρ2 ; β 2 = γ 2 = 1.

˜ ψγψ = 0. 4@

Note that some things in the last three square brackets (the x, y, z-derivatives and the indices 1, 2, 3, 4 of the ψ components) should be slightly corrected. However, at variance with what is usually done by us, we choose to leave unchanged the expressions appearing in the original manuscript.

21

DIRAC THEORY

0=−

1 h ˜ ∂λ e˜ ˜ ψβψ − 2 ψβσ · Cψ − ψβσ · pψ + ψ † βσ · pψ. c 2π ∂t c ——————–

˜ ψβψ = e−iλ (ψ 1 ψ3 + ψ 2 ψ4 ) + eiλ (ψ 3 ψ1 + ψ 4 ψ2 ) = 2e−iλ (ψ 1 ψ3 + ψ 2 ψ4 ). 1 h ∂ −iλ (e ψ 1 ψ3 + e−iλ ψ 2 ψ4 + eiλ ψ 3 ψ1 + eiλ ψ 4 ψ2 ) c 2π ∂t ∂λ 1 h −i e−iλ ψ 1 ψ3 − i e−iλ ψ 2 ψ4 + i eiλ ψ 3 ψ1 + i eiλ ψ 4 ψ1 = c 2π ∂t + L + L, ∂ h e ∂ e ψ 2 ϕ ψ3 − ψ 1 L = ie −i + (Cx − iCy ) ψ4 − . . . c 2π ∂x ∂y c + i e−iλ . . . −iλ

+iλ

...

+ ie

+iλ

+ ie

...

e = i ϕ e−iλ (ψ 1 ψ3 + ψ 2 ψ4 ) + eiλ (ψ 3 ψ1 + ψ 4 ψ2 ) c e −iλ + i Cx e (ψ 1 ψ4 + ψ 2 ψ3 ) − eiλ (ψ 3 ψ2 + ψ 4 ψ1 ) c e + i Cy . . . c e ± i Cz . . . c ∂ ∂ ∂ ∂ h −iλ − ψ 3 ψ2 + ψ 4 ψ1 eiλ (ψ 1 ψ4 + ψ 2 ψ3 )e − 2π ∂x ∂x ∂x ∂x h ... − 2π h − ... . 2π

22

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

e ˜ 1 h ∂ ˜ ˜ σ · p ψ − ψ x γ σ · p ψ. (ψβψ) = 2 ψγ σ · C ψ + ψγ c 2π ∂t c ——————– e−iλ (ψ 1 ψ3 + ψ 2 ψ4 ) − eiλ (ψ 3 ψ1 + ψ 4 ψ2 ) = 0. eiλ =

ψ 1 ψ3 + ψ 2 ψ4 ; ψ 3 ψ1 + ψ 4 ψ2

e−iλ (ψ 1 ψ3 + ψ 2 ψ4 ) > 0;

|ψ 1 ψ3 + ψ 2 ψ| > 0, provided that not all ψi be zero (ψ1 = ψ2 = ψ3 = ψ4 = 0) at the same time.

1.3.

QUANTIZATION OF THE DIRAC FIELD

The canonical quantization of a Dirac ﬁeld ψ is here considered (starting from a Lagrangian density L), by introducing the ﬁeld variables P, P conjugate to ψ, ψ. After imposing the commutation rules, the Hamiltonian H was deduced, and an expression for the energy W was obtained in terms of the annihilation and creation operators a, b. The quantities ni are number operators. W A = V A − c σ · p B − mc2 A, W B = V B − c σ · p A − mc2 B.

W0 B0 A0

p2 2 + mc B0 , = V + 2m σ · p B0 . = − 2mc

W =−

L =

1 2m

h ∂ 2πi ∂t

px =

h ∂ 2πi ∂x

e e W W + ϕ ψ + ϕ ψ − c c c c e e −px + Ax ψ px + Ax ψ + m2 c2 ψψ . + c c x

23

DIRAC THEORY

ψ, ψ,

W e P = − + ϕ ψ; c c e W + ϕ ψ. P = c c

ψ(q) ψ(q ) − ψ(q ) ψ(q) = 0, ψ(q) ψ(q ) − ψ(q ) ψ(q) = 0, ψ(q) ψ(q ) − ψ(q ) ψ(q) = 0,

P (q) P (q ) − P (q ) P (q) = 0, P (q) P (q ) − P (q) P (q) = 0, P (q) P (q ) − P (q ) P (q) = 0.

ψ(q) P (q ) − P (q ) ψ(q) = δ(q − q ) 2mc, ψ(q) P (q ) − P (q ) ψ(q) = 0, ψ(q) P (q ) − P (q ) ψ(q) = 0, ψ(q) P (q ) − P (q ) ψ(q ) = −δ(q − q ) 2mc. W e W W e W 1 − + ϕ ψ ψ+ + ϕ ψ ψ −L H = 2m c c c c c c e e 1 P (P − ϕ ψ + P P − ϕ ψ − P P = 2m c c e e + −px + Ax ψ px + Ax ψ + m2 c2 ψψ c c x e 1 P P − ϕ (P ψ + P ψ) = 2m c e e + −px + Ax ψ px + Ax ψ + m2 c2 ψψ . c c x a, a;

b, b.

ab − ba = 2mc, ab − ba = −2mc.

n =

1 √ 2 mc

n =

1 √ 2 mc

1

4 b + m2 c2 + p2 a ,

1

b − 4 m2 c2 + p2 a .

4 m2 c2 + p2

4 m2 c2 + p2

24

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

1 1

bb + m2 c2 + p2 aa , 2mc m2 c2 + p2 1 ab + ab ; 2mc

1 + n1 + n2 = n1 − n2 =

n1 =

1 4mc

n2 =

1 4mc

=

1

b − 4 m2 c2 + p2 a

4 m2 c2 + p2

1 4 b − m2 c2 + p2 a . ×

4 m2 c2 + p2

ψ=

W

b + 4 m2 c2 + p2 a

4 m2 c2 + p2

1 b + 4 m2 c2 + p2 a , ×

4 m2 c2 + p2

ψ=

1 2m

1

e − c i,k

e + c

ai fi ,

P =

ai f i ,

P =

bi bi +

i

+

c2

i

f i (q) fk (q) ϕ(q) dq · (bi ak + bk ai ) f i (q) fk (q) (pi + pk ) · A dq · ai ak

i,k

f i (q) fk (q)A2 dq

⎫ ⎬ ⎭

.

m2 c2 (ui − v i ), m2 c2 + p2i $ 2 2 2 4 m c + pi = mc (ui + vi ); m2 c2

ai = bi

bi fi .

(m2 c2 + p2i ) ai ai

i,k

e2

bi f i ;

4

25

DIRAC THEORY

m2 c2 + p2i (ui uk − vi v k − ui v k + vi uk ), m2 c2 + p2k mc (ui uk + vi v k − ui v k − vi uk ). 4 (m2 c2 + p2i )(m2 c2 + p2k )

bi ak = mc ai ak =

1.4.

4

INTERACTING DIRAC FIELDS

In the following pages, the author again studied the problem of the electromagnetic interaction of a Dirac ﬁeld ψ; the electromagnetic scalar and vector potentials are denoted with ϕ and C, respectively. After some explicit passages on the (interacting) Dirac equation (see Sect. 1.4.1), Majorana considered in some detail also the Maxwell equations for the electromagnetic ﬁeld (see Sect. 1.4.2). The starting point are the ﬁeld equations deduced from a variational principle, and the role of the gauge constraints is particularly pointed out. The superposition of Dirac and Maxwell ﬁelds was, then, studied using again a canonical formalism (see Sect. 1.4.3); choosing appropriate state variables and conjugate momenta, the quantization of both the Dirac and the Maxwell ﬁeld was carried out. An expression for the Hamiltonian of the interacting system was deduced and, ﬁnally, normal mode decomposition was as well introduced (see Sect. 1.4.3.1). This part ends with some explicit matrix expressions for the Dirac operators in particular representations (see Sect. 1.4.3.2).

1.4.1

Dirac Equation e e e W + ϕ + αx px + Cx + αy py + Cy c c c c e +αz pz + Cz + βmc ψ = 0; c

αx = ρ1 σx , 1 − ρ = ψψ, e

αy = ρ1 σy ,

1 − ix = −ψαx ψ, e

αz = ρ1 σz , 1 − iy = ψαy ψ, e

β = ρ3 ; 1 − iz = ψαz ψ; e

26

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

0 1 ρ1 = 1 0 0 1 σx = 1 0 αx = αz =

P0 =

W e + ϕ, c c

0 0 0 1

,

0 −i , ρ2 = i 0

, 0 0 1 0

0 0 0 0 1 0 0 −1

0 −i , σy = i 0 0 1 0 0

1 0 0 0

,

1 0 0 −1 , 0 0 0 0

e Px = px + Cx , c

F = (Px , Py , Pz ),

αy = β =

1 0 ρ3 = 0 −1

,

1 0 σz = 0 1

,

0 0 0 −i 0 0 i 0 , 0 −i 0 0 i 0 0 0 1 0 0 0

0 0 0 1 0 0 0 −1 0 0 0 −1

e Py = py + Cy , c

.

e Pz = pz + Cz . c

α = (αx , αy , αz ).

[P0 + α · F + βmc] ψ = 0.

(P0 + mc)ψ1 + (Px − iPy )ψ4 + Pz ψ3 = 0, (P0 + mc)ψ2 + (Px + iPy )ψ3 − Pz ψ4 = 0, (P0 − mc)ψ3 + (Px − iPy )ψ2 + Pz ψ1 = 0, (P0 − mc)ψ4 + (Px + iPy )ψ1 − Pz ψ2 = 0.

W + mc ψ1 + (px − ipy )ψ4 + pz ψ3 c e + [ϕ ψ1 + (Cx − iCy )ψ4 + Cz ψ3 ] = 0, c

27

DIRAC THEORY

W + mc ψ2 + (px + ipy )ψ3 − pz ψ4 c e + [ϕ ψ2 + (Cx + iCy )ψ3 − Cz ψ4 ] = 0, c W − mc ψ3 + (px − ipy )ψ2 + pz ψ1 c e + [ϕ ψ3 + (Cx − iCy )ψ2 + Cz ψ1 ] = 0, c W − mc ψ4 + (px + ipy )ψ1 − pz ψ2 c e + [ϕ ψ4 + (Cx + iCy )ψ1 − Cz ψ3 ] = 0; c W + mc ψ 1 − (px + ipy )ψ 4 − pz ψ 3 − c e + [ϕ ψ 1 + (Cx + iCy )ψ 4 + Cz ψ 3 ] = 0, c W − + mc ψ 2 − (px − ipy )ψ 3 + pz ψ 4 c e + [ϕ ψ 2 + (Cx − iCy )ψ 3 − Cz ψ 4 ] = 0, c W − − mc ψ 3 − (px + ipy )ψ 2 − pz ψ 1 c e + [ϕ ψ 3 + (Cx + iCy )ψ 2 + Cz ψ 1 ] = 0, c W − mc ψ 4 − (px − ipy )ψ 1 + pz ψ 2 − c e + [ϕ ψ 4 + (Cx − iCy )ψ 1 − Cz ψ 2 ] = 0. c u0 = ψ 1 ψ 1 + ψ 2 ψ 2 + ψ 3 ψ 3 + ψ 4 ψ 4 , ux = −(ψ 1 ψ4 + ψ 2 ψ3 + ψ 3 ψ2 + ψ 4 ψ1 ), uy = i(ψ 1 ψ4 − ψ 2 ψ3 + ψ 3 ψ2 − ψ 4 ψ1 ), uz = −(ψ 1 ψ3 − ψ 2 ψ4 + ψ 3 ψ2 − ψ 4 ψ2 ).

1.4.2

Maxwell Equations x0 = ict, S0 = iρ,

x1 = x,

S1 = ρ

vx , c

x2 = y, S2 = ρ

vy , c

x3 = z; S3 = ρ

vz ; c

28

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

φ0 = iϕ,

φ1 = Cx , Fik =

φ2 = Cy ,

φ3 = Cz ;

∂φk ∂φi − . ∂xi ∂xk

F01 = iEx , F02 = iEy , F03 = iEz ,

F23 = Hx , F31 = Hy , F12 = Hz .

The Maxwell equations are: ∂Fik k

∂xk

= 4πSi ,

I

∂Fkl ∂Fli ∂Fik + + = 0. ∂xl ∂xi ∂xk

I

4πSi =

∂Fik k

=

∂xk

=

II

∂ ∂φk ∂ 2 − φi ∂xi ∂xk ∂xk k

k

∂ ∇ · φ − ∇2 φi , ∂xi

4πS = ∇ × ∇ · φ − ∇2 φ. Additional constraint: ∇ · φ = 0; ∇2 φ + 4πS = 0. Variational approach: 2 Fik dτ δ i
∂φk 2

∂φk ∂φi = δ − dτ ∂xi ∂xi ∂xk ∂ 2 ∇ · φ δφk ∇ φk − = −2 ∂xk k ∂ 2 = 2 ∇ · φ − ∇ φk δφk ; ∂xk k

29

DIRAC THEORY

S · φ dτ =

δ

Sk δφk ;

k

1 2 1 2 Fik dτ = − ∇ φk Sk + δ −S · φ + 8π 4π i
δ

k

1 2 +S · φ − Fik dτ 8π

1 ∂ − ∇ · φ δφk . 4π ∂xk

= 0, (A)

i
4πS + ∇2 φ − ∇ (∇ · φ) = 0.

I

The Maxwell equations are obtained from: 1 ∂φk 2 δ dτ = 0; +S · φ − 8π ∂xi ⎫ ∇2 φ + 4πS = 0, ⎬ ∇ · φ = 0.

1.4.3

⎭

I

Maxwell-Dirac Theory

e e W + ϕ + α · p + C + βmc = M ; c c c M ψ = 0.

The Dirac equation is obtained from: δ

ψM ψ dτ = 0;

˜ ψ + ψM ˜ δψ = 2 Re (δ ψ)M ˜ ψ = 0, (δ ψ)M M ψ = 0.

30

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

In

δ

1 2 ˜ ψM ψ − Fik dτ = 0, 8π i
the Dirac equation Mψ=0 is obtained from a variation of the variables ψ, while the Maxwell equations −4πS − ∇2 φ + ∇ (∇ · φ) = 0 come from a variation of φ. Eichinvarianz:5 ϕ = 0. State variables: ψ1 ,

ψ2 ,

Cx ,

ψ3 ,

Cy ,

ψ4 ;

Cz ;

Conjugate momenta: −

h ψ , 2πi 1

Px = −

−

Ex , 4πc

E=

h ψ , 2πi 2

−

Py = − 1 ∂C , c ∂t

h ψ , 2πi 3

Ey , 4πc

−

h ψ ; 2πi 4

Pz = −

Ez . 4πc

H = ∇ × C;

ϕ = 0, ∇ · C = 0. δ

e ψ˜ +W + c α · p + C + βmc2 ψ c 2 1 ∂C 1 (∇ × C)2 − 2 dτ = 0. − 8π c ∂t

5 @ This German word means “gauge invariance”; the author uses this property in order to set the potential ϕ to zero.

31

DIRAC THEORY

Pi (q)Ck (q ) − Ck (q )Pi (q) =

h δ(q − q ), 2πi

ψi (q)ψ k (q ) + ψ k (q )ψi (q) = δ(q − q ). C = ABA, Air Brs Ask = Brs Air Aks , Cik = = Brs Akr Ais = Bsr Air Aks = B rs Air Ars ;

Cki

∂Ci = −c Ei = 4πc2 Pi = Cik . ∂t 1 e 2 2 2 2 ˜ |∇ × C| + 2πc |F | dτ. H= −ψ c α · p + C + βmc ψ + c 8π

1.4.3.1

Normal mode decomposition. ψ= ar ψr , ψ = ar ψr ; ar as + as ar = δrs . C=

q ν uν ,

P =

pν q ν − q ν pν =

pν u ν ;

h . 2π

˜i ak − a ˜i ak ak = ak a ˜i ak + a ˜i ak ak = δik ak , ak a ˜i bk − a ˜i bk ak = ak a ˜i bk − a ˜i ak bk = (ak a ˜− a ˜i ak )bk , ak a ak ˜bi ak − ˜bi ak ak = ˜bi (ak ak − ak ak ). ∂Ci 2 ∂Ci ∂Ck − |∇ × C|2 = . ∂xk ∂xk ∂xi i,k

∂ck = 0, ∂xk ∂ 2 Ck ∂ ∂ck ∂Ci ∂Ck dτ = − Ci = − Ci , ∂xk ∂xi ∂xi ∂xk ∂xi ∂xk Ci

32

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

∂Ci ∂Ck ∂ ∂Ck dτ = − = − C · ∇ (∇ · C) = 0. Ci ∂xk ∂xi ∂xi ∂xk i,k

i,k

|∇ × C|2 dτ = −

C∇2 Cdτ =

|F |2 dτ =

qν2

4π 2 v 2 , c2

p2ν .

π e qν uν (q) + βmc2 + 2 qv 2 v 2 + 2πc2 p2ν . H = − cα · p + c 2c v2 2 πv 2 2 2 2 2 2 q + 2πc pν = 2πc pν + 2 qν 2c2 ν 4c νi νi 2 pν + 2 q ν . = 2πc pν − 2 qν 2c 2c $ cν = c

2π hν

νi pν − 2 q ν , 2c

c˜ν cν =

$ cν = c

2π hν

νi pν = 2 q ν ; 2c

Wν 1 Wν 1 − , cν c˜ν = + , hν 2 hν 2 cν c˜ν − c˜ν cν = 1

1 . Wν = hν c˜ν cν + 2

1.4.3.2

Particular representations of Dirac operators. 0 1 1 0 0 0 , ε = ρ = 0 0 , ε 1 0 . 0 −1 ε2 = 0,

ε2 = 0,

ρ2 = 1;

ερ + ρε = 0, ερ + ρε = 0, εε + εε = 1; 0 0 1 0 . , εε = εε = 0 1 0 0 ar as + as ar = δrs ,

ar as + as ar = 0,

ar as + as ar = 0.

33

DIRAC THEORY

For s > r: ar ar as as ar as as ar ar as + as ar ar as + as ar ar as as ar ar ar ar ar ar ar + ar ar

ρ1 ρ2 · · · ρr−1 εr , ρ1 ρ2 · · · ρr−1 εr , ρ1 ρ2 · · · ρr−1 ρr · · · ρs−1 εs , ρ1 ρ2 · · · ρr−1 ρr · · · ρs−1 εs , −ρr ρr+1 · · · ρs−1 εr εs , ρr ρr+1 · · · ρs−1 εr εs , 0, 0, −ρr · · · ρs−1 εr εs , ρr · · · ρs−1 εs εr , εr εr , εr εr , 1.

= = = = = = = = = = = = =

c˜ c − c˜c = 1, c˜ c = r. cr−1,r cr,r−1 crs crs (c˜ c)rs

√ = r, √ = r, √ = δr+1,s s, √ = δr−1,s r; = crt cts = tδr+1,t δt−1,s = tδrs = (r + 1)δrs , t

(˜ cc)rs =

crt cts =

√ √ r sδr−1,t δt+1,s = rδrs .

t

c˜ c − c˜c = 1. c =

0 0 0 0 0

√

1 √0 0 0 0 2 √0 0 3 √0 0 0 4 0 0 0 0 0 0 0

, ...

c =

0 0 0 √0 1 √0 0 0 0 2 √0 0 0 0 3 √0 4 0 0 0

0 0 0 0 0

; ...

34

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

c˜c =

0 0 0 0 0

0 1 0 0 0

0 0 2 0 0

0 0 0 3 0

0 0 0 0 4

, ...

c˜ c =

1 0 0 0 0

0 2 0 0 0

0 0 3 0 0

0 0 0 4 0

0 0 0 0 5

. ...

——————– 0 1 ε = 0 0

,

0 0 ε = 1 0

,

a1 = ε1 , a2 = ρ1 ε2 , 0 0 1 0 0 0 0 1 a1 = 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 a2 = 0 0 0 −1 0 0 0 0

, ,

1 0 ρ = 0 −1

,

0 0 εε = 0 1

a1 = ε1 , a2 = ρ1 ε2 . a1 = a2 =

0 0 1 0

0 0 0 1

0 0 0 0

0 1 0 0

0 0 0 0 0 0 0 −1

0 0 0 0

, 0 0 0 0

——————– 0 0 0 , a = 1 0 0 , √ 0 2 0 1 0 0 0 0 1 0 , aa = 0 2 0 ; 0 0 0 0 2 √ 0 0 0 2 0 , a2 = √0 0 0 ; 2 0 0 0 1 0 0 aa + aa = 0 3 0 . 0 0 2

0 1 a = 0 0 0 0 0 aa = 0 0 0 0 2 a = 0 0 0 0

√0 2 0

.

.

35

DIRAC THEORY

1.5.

SYMMETRIZATION

Inserted in the discussion of the Maxwell-Dirac theory (see Sect. 1.4.3), we ﬁnd a page where the (anti-)symmetrization of Dirac ﬁelds, describing spin-1/2 particles, was considered. ψ = ar ψr , ϕ = ϕ(nr ), with nr = 0, 1. % (1) nr = 1; ns is diﬀerent from zero: ϕ = ϕ(s) = cs ; ϕ∼ cs ψs (q). (2)

%

nr = 2; ns , nt are diﬀerent from zero (s < t): ϕ = ϕ(s, t) = cst ; ϕ∼

(3)

%

s
cst

ψs (q1 )ψt (q2 ) − ψt (q2 )ψs (q1 ) √ . 2

nr = n; ni1 , ni2 , . . . , nin are diﬀerent from zero (ii < i2 < i3 < . . . < in ): ϕ = ϕ(i1 , i2 , . . . in ); 1 (−1)p Pq ψi1 (q1 )ψi2 (q2 ) · · · ψin (qn ). ϕ∼ √ n! p

1.6.

PRELIMINARIES FOR A DIRAC EQUATION IN REAL TERMS

What is reported in the following appears to be a preliminary study for Majorana’s article on a Symmetrical theory of electrons and positrons [Nuovo Cim. 14 (1937) 171], where he put forth the known Majorana representation for spin-1/2 ﬁelds. The Dirac equation and its consequences were considered using slightly diﬀerent formalisms (diﬀerent decompositions of the wave function ψ). An expression was obtained for the total angular momentum carried by the ﬁeld ψ, starting from the Hamiltonian. In some places, the interaction with the electromagnetic potential

36

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

(ϕ, A) was included as well in a somewhat interesting fashion. Note, however, that real ﬁelds (that is: directly related to the Majorana representation mentioned above) were considered only in very few points in the following pages.

1.6.1

First Formalism αx = ρ1 σx ,

αy = ρ3 ,

αz = ρ1 σz ,

β = −ρ1 σy . Without ﬁeld (That is, without interaction with the electromagnetic ﬁeld), and for U = U , we have: W + (α, p) + βmc U = 0. c For ψ = U + iV : e W + (α, p) + βmc U + i [ϕ + (α, A)] V = 0, c c W e + (α, p) + βmc V − i [ϕ + (α, A)] U = 0. c c

β = −iβ;

2πmc ; μ= h

2πe ε= hc

=

1 137e

.

1∂ − (α, ∇ ) + β μ U + ε [ϕ + (α, A)] V = 0, c ∂t 1∂ − (α, ∇ ) + β μ V − ε [ϕ + (α, A)] U = 0. c ∂t

1∂ 1 − (α, ∇ ) + β μ U + εV ∗ [ϕ + (α, A)] V∗ c ∂t 2 1 + εU ∗ [ϕ + (α, A)]U dq dt = 0. 2

δ

——————– ψ = U + iV,

ψ˜ = U ∗ − iV ∗ .

37

DIRAC THEORY

1∂ − (α, ∇ ) + β μ U + ε[ϕ + (α, A)]V = 0, c ∂t 1∂ − (α, ∇ ) + β μ V − ε[ϕ + (α, A)]U = 0. c ∂t

[6 ] δ

1∂ U − (α, ∇ ) + β μ U c ∂t 1∂ +V∗ − (α, ∇ ) + β μ V c ∂t ∗ + εU [ϕ + (α, A)]V − εV ∗ [ϕ + (α, A)]U } dq dt = 0.

hc i 2π

∗

[7 ] 6@ 7@

In the original manuscript, the author neglect to equate the following expression to zero. Here, the following insert appears in the original manuscript, reporting what follows: Z X X Bik qi qk )dt = 0. iδ ( Aik qi q˙k +

Aik = Aik (t) = Aki (t), Bik = Bik (t) = −Bki (t). A = A, B = B. P By taking the variation with respect to the conjugate variables qk and i Aik qi : X X (Aik q˙k + Bik qk ) − (Aik q˙k + Bik qk )δqi = 0. δqi k

k

X [Aik q˙k + Bik qk ]) = 0. (δqi , k

X (Aik q˙k + Bik qk ) = 0. k

H = −i

X

Bik qi qk .

ik

q˙k

= =

X

Aik q˙k

=

−

2π X Aik Brs (qk qr qs − qr qs qr ) h krs

=

−

2π X Aik Brs [(qk qr + qr qk )qs − qr (qk qr + qs qk )]. h krs

k

qr

2ai (qk H − Hqk ) h X 2π − Brs (qk qr qs − qr qs qk ). h rs −

X

! Aik qk

+

k

[The footnote continues on the next page]

X k

! Aik q − k

qr = +

h δir . 4π

38

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

1 Ui (q)Uk (q ) + Uk (q )Ui (q) = δik δ(q − q ), 2 Ui (q)Vk (q ) + Vk (q )Ui (q) = 0, 1 Vi (q)Vk (q ) + Vk (q )Vi (q) = δik δ(q − q ). 2

1.6.2

Second Formalism

W + ρ1 (σ, p) + ρ3 mc ψ = 0. c

A = (ψ1 , ψ2 ), B = (ψ3 , ψ4 ): W + mc A + (σ, p)B = 0, c W − mc B + (σ, p)A = 0. c −1 W + mc (σ, p)B, A = − c −1 W − mc (σ, p)A. B = − c

ε=

m2 c2 + p2 . W = ±ε. c

7

X

Aik q˙k

=

k

=

1X 1X Bis qs + Bri qr 2 s 2 r X − Bik qk . −

k

qr qs + qs qr

=

=

+

+

h X −1 Asi δir 4π i

h −1 A . 4π rs

39

DIRAC THEORY

W = ε: c

1)

A = −(ε + mc)−1 (σ, p)B, A˜ = −[(ε + mc)−1 pB] , σ .

˜ AA =

[(ε + mc)−1 pB] , [(ε + mc)−1 pB]

+i[(ε + mc)−1 px B][(ε + mc)−1 py σz B] −i[(ε + mc)−1 py B][(ε + mc)−1 px σz B] +i[(ε + mc)−1 py B](ε + mc)−1 pz σx B −i[(ε + mc)−1 pz B](ε + mc)−1 py σx B +i[(ε + mc)−1 pz B](ε + mc)−1 px σy B −i[(ε + mc)−1 px B](ε + mc)−1 pz σy B.

˜ dq = AA

˜ + mc)−2 p2 B dq = B(ε

˜ + BB) ˜ dq = (AA

2)

W = ε: c

˜ B

˜ + mc)−1 (ε − mc)B dq, B(ε 2ε B dq. ε + mc

B = (ε + mc)−1 (σ, p)A,

˜ dq = BB

˜ + mc)−1 (ε − mc)A dq, A(ε

˜ + BB) ˜ dq = (AA

A˜

2ε A dq. ε + mc

——————– $

ε + mc (σ, p) A −

B, 2ε 2ε(ε + mc) $ ε + mc (σ, p) A + B. B=

2ε 2ε(ε + mc) A=

40

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

˜ + BB) ˜ dq = (AA

˜ B) dq. (A˜ A + B

$

ε + mc (σ, p) B, A+

2ε 2ε(ε + mc) $ (σ, p) ε + mc A+ B. B =

2ε 2ε(ε + mc)

A =

1.6.3

Angular Momentum

ψ = (A, B), ψ = (A , B ). H = −cρ1 (σ, p) − ρ3 mc2 − eϕ − ρ1 (σ, eU ). ˜ ψHψ dq =

˜ ˜ ˜ −cA(σ, p)B − cB(σ, p)A − mc2 AA

˜ − eAϕA ˜ ˜ +mc2 BB − eBϕB & ˜ ˜ −eA(σ, U )B − eB(σ, U )A dq ˜ 1 ψ dq. ˜ 0 ψ dq + ψH = ψH H = H0 + H1 , H0 = −cρ1 (σ, p) − ρ3 mc2 , ˜ 0 ψ dq = ψH

H1 = −eϕ − ρ1 (σ, eU ).

˜ ˜ −eA(σ, p)B − cB(σ, p)A & ˜ + mc2 BB ˜ ˜ εB − A˜ εA ) dq. −mc2 AA dq = c (B

Nx =

1 2

H0 H0 H0 h x + x =x − ρ1 σx , c c c 4πi xε − εx = −

h px . 2π ε

41

DIRAC THEORY

˜ x ψ dq = ψN

=

ψ˜ Nx ψ dq

˜ xεB − A˜ xεA) dq (B

px ˜ px B + A˜ σx B + B ˜ σx A A˜ A − B ε ε px (σ, p) px (σ, p) ˜ ˜ B−B A dq −A ε(ε + mc) ε(ε + mc) (σ, p) (ε − mc)(2ε + mc) h (ε − mc)mcpx ˜ A + − σx B + 3 2πi 4ε 2ε 4ε3 m2 c2 px mcσx (σ, p) (ε − mc)(2ε + mc)mcpx + ∓ A dq + 4ε3 2ε(ε + mc) 4ε3 (ε + mc) ε − mc (2ε + mc)px (ε − mc)(σ, p) h mcpx (σ, p) A˜ + σx + − 3 2πi 4ε 2ε 4ε3 (ε + mc) m2 c2 px (σ, p) mcσx (2ε + mc)px (σ, p)mc + − − 4ε3 (ε + mc) 2ε 4ε3 (ε + mc) h ˜ {. . .} A dq + h ˜ {. . .} B dq B B + 2πi 2πi h mcpx + εσx (σ, p) ˜ ˜ ˜ −A A = (B xεB − A xεA) dq + 2πi 2ε(ε + mc) mcpx + εσx (σ, p) ˜ +B B dq. 2ε(ε + mc) h − 4πi

Nx

h mcpx + εσx (σ, p) = −ρ3 xε + 4πi ε(ε + mc) h py σz − pz σy h px + . = −ρ3 xε + 4πi ε 4π ε + mc ——————– H0 = −ρ3 ε. c

42

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

˜ dq = ψxψ

˜ xB) dq (A˜ xA + B

σx (σ, p) h mcpx A˜ − + 2πi 4ε3 2ε(ε + mc) (ε − mc)(2ε + mc)px A dq − 4ε3 (ε + mc) σx mc(σ, p)px h A˜ + + 2πi 4ε3 (ε + mc) 2ε (2ε + mc)(σ, p)px − B dq 4ε3 (ε + mc) h ˜[B [. . .] A dq + h ˜ [. . .] dq B + 2πi 2πi ˜ xB ) dq = (A˜ xA + B +

i(py σz − pz σy ) A˜ A dq 2ε(ε + mc) (σ, p)px h σx ˜ A B dq − + 2πi 2ε 2ε2 (ε + mc) σx (σ, p)px h ˜ A dq + B − + 2 2πi 2ε 2ε (ε + mc) h ˜ i(py σz − pz σy ) B dq. + B 2πi 2ε(ε + mc) h + 2πi

h h py σz − pz σy + ρ2 x =x+ 2π 2ε(ε + mc) 2π

Nx

=

1 2

σx (σ, p)px − 2 2ε 2ε (ε + mc)

H0 h px H0 x + x = −ρ3 xε − ρ3 c c 4πi ε

h py σz − pz σy ρ3 2π 2(ε + mc) h py σz − pz σy h px + . = −ρ3 xε + 4πi ε 4π ε + mc −

.

43

DIRAC THEORY

Nx Ny − Ny Nx =

=

= =

h h2 εσz (xpy − ypx ) + 2 2πi 4π i ε + mc +

h2 i(py pz σy + p2z σz + pz px σx ) 8π 2 (ε + mc)2

+

h2 −p2y σz + py pz σy + px pz σx − p2x σz 8π 2 i (ε + mc)2

h h2 (xpy − ypx ) + 2 σz 2πi 8π i 2 h (σ, p)pz (σ, p)pz + 2 − 8π i (ε + mc)2 (ε + mc) h h2 (xpy − ypx ) + 2 σz 2πi 8π i h h xpy − ypx + σx . 2πi 4π

[8 ] 8@

Here, the following insert appears in the original manuscript, reporting what follows:

For a relativistic Hamiltonian system described by the variables q, p, t, W : Z=0 (for example: Z = −W + H(p, q, t)). dqi : dpi : dt : dW =

∂Z ∂Z ∂Z ∂Z : . : − : − ∂pi ∂qi ∂W ∂t

For the states: S = S(p, q, W, t), ZS = 0. X ∂S ∂Z X ∂S ∂Z ∂S ∂Z ∂S ∂Z + = 0, − − ∂q ∂p ∂p ∂q ∂t ∂W ∂W ∂t i i i i i i [S, Z] = 0. For example: S = S0 (p, q, t)δ(−W + H), H = H(p, q, t); X ∂S ∂H X ∂S0 ∂H ∂S = 0. − + ∂q ∂p ∂p ∂q ∂t i i i i i i

44

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

1.6.4

Plane-Wave Expansion

For the Dirac ﬁeld: H = H0 + H1 ,

H = H0 + H1 ;

H0 = −cρ1 (σ, p) − ρ3 mc2 , H0 = −ρ3 cε, ε=

H1 = −eϕ − eρ1 (σ, U ); ε=

m2 c2 + p2 .

m2 c2 + h2 γ 2 .

ψ = (A, B), ψ = (A , B ):

A(q) = B(q) = A (q) = B (q) =

2πi(γ,q)

dγ,

a(γ) =

b(γ) e2πi(γ,q) dγ,

b(γ) =

a (γ) e2πi(γ,q) dγ,

a (γ) =

b(γ) e2πi(γ,q) dγ,

b (γ) =

a(γ) e

A(q) e−2πi(γ,q) dq; B(q) e−2πi(γ,q) dq; A (q) e−2πi(γ,q) dq; B (q) e−2πi(γ,q) dq.

$

h(σ, γ) ε + mc a (γ) −

b (γ), 2ε 2ε(ε + mc) $ h(σ, γ) ε + mc b(γ) =

a (γ) + b (γ); 2ε 2ε(ε + mc)

a(γ) =

$

ε + mc h(σ, γ) b(γ), a(γ) +

2ε 2ε(ε + mc) $ h(σ, γ) ε + mc a(γ) + b(γ). b (γ) = −

2ε 2ε(ε + mc)

a (γ) =

χ(γ) = (a, b), χ (γ) = (a , b ): ε + mc ihρ2 (σ, γ) χ (γ), −

χ(γ) = 2ε 2ε(ε + mc) $ ihρ ε + mc (σ, γ) 2 +

χ(γ). χ (γ) = 2ε 2ε(ε + mc) $

45

DIRAC THEORY

ε=

1.6.5

m2 c2 + h2 γ 2 ,

ε =

m2 c2 + h2 γ 2 .

Real Fields

Dirac equation with real ﬁelds:

ψ=

W + ρ1 (σ, p) + ρ3 mc ψ = 0. c

1 − iρ2 σy √ ψ, 2

ψ =

1 + iρ2 σy √ ψ. 2

W 1 (1 + iρ2 σy ) + ρ1 (σ, p) + ρ3 mc (1 − iρ3 σy )ψ 0 = 2 c W + ρ1 σx px + ρ3 py + ρ1 σz − ρ1 σy ψ = 0. = c

1.6.6

Interaction With An Electromagnetic Field hc ∗ 1 ∂ δ − (α, ∇ ) + β μ U i U 2π c ∂t hc ∗ 1 ∂ +i V − (α, ∇ ) + β μ V 2π c ∂t +ieU ∗ [ϕ + (α, A)]V − ieV ∗ [ϕ + (α, A)]U 2 1 1 1 2 2 ϕ˙ + ∇ · A + (E − H ) − dq dt = 0. 8π 8π c 1 ϕ˙ + ∇ · A = 0 c 1 2 ˙ ∇ ϕ + ∇ · A + 4πρ = 0 . c

46

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

1∂ 2πe − (α, ∇ ) + β U + [ϕ + (α, A)]V = 0, c ∂t hc 2πe 1∂ − (α, ∇ ) + β V − [ϕ + (α, A)]U = 0. c ∂t hc

1 ∂2 − ∇2 c2 ∂t2 1 ∂ − ∇2 c2 ∂t2

ϕ + 4πei(U ∗ V − V ∗ U ) = 0, A − 4πei(U ∗ αV − V ∗ αV ) = 0.

ρ = −ei(U ∗ V − V ∗ U ) = −e I = ei(U ∗ αV − V ∗ αU ) = e

˜ − ψ∗ψ ψψ , 2 ˜ ψαψ − ψ ∗ αψ 2

(ψ = U + iV ). 1 P0 = − 4πc

1 ϕ˙ + ∇ · A , c

1 Ex , 4πc 1 Ey , Py = − 4πc 1 Ez . Pz = − 4πc Px = −

1 ϕ˙ + ∇ · A = 0 : c ˙ + 4πρ = 0 : ∇2 ϕ + ∇ · A

P0 = 0; ρ = −c ∇ · F

(F = (Px , Py , Pz )). H=

1 2 2 2 2 ˜ |∇ × A| dq. ψ −c(α, p) − βmc ψ − (A, I) + 2πc P + 8π

47

DIRAC THEORY

1.7.

DIRAC-LIKE EQUATIONS FOR PARTICLES WITH SPIN HIGHER THAN 1/2

By starting from the known Dirac equation for a 4-component spinor, the author then wrote down the corresponding equations for 16-component, 6-component and 5-component spinors. Explicit expressions for the Dirac matrices for the cases considered were given, thus producing for the ﬁrst time Dirac-like equations for particle with spin higher than 1/2. In the following we report what found in the Quaderno 4 in the same order as the material appears there; it seems evident, in fact, that the author has obtained the reported results just in this order, i.e., not in the more obvious way from 4-component case to 5-component, to 6-component, to 16-component case.

1.7.1

Spin-1/2 Particles (4-Component Spinors)

e px + A x → px , c

e W + A0 c c

→ p0 ,

e py + A y → py , c

p0 ψ1 + px ψ4 − ipy ψ4 + pz ψ3 + mc p0 ψ2 + px ψ3 + ipy ψ3 − pz ψ4 + mc p0 ψ3 + px ψ2 − ipy ψ2 + pz ψ1 − mc p0 ψ4 + px ψ1 + ipy ψ1 + pz ψ2 − mc ψ1 ψ1 p0 + mc

ψ1 ψ2 ψ3 ψ4

e pz + A z → pz . c = = = =

0, 0, 0, 0.

ψ2

ψ3

ψ4

0

pz

px − ipy −pz

ψ2

0

p0 + mc px + ipy

ψ3

pz

px − ipy

p0 − mc

0

−pz

0

p0 − mc

ψ4 px + ipy

48

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

1.7.2

Spin-7/2 Particles (16-Component Spinors)

[See the matrix on page 49.]9 Let us set M = 2m, P0 = p0 + p0 , Q0 = p0 − p0 , and so on: [See the matrix on page 50.]

[See the matrix on page 51.] [See the matrix on page 52.]10

1.7.3

Spin-1 Particles (6-Component Spinors) e 1 e e W + A0 + mc ψ1 + px + Cx + i py + Cy ψ2 c c 2 c c 1 e 1 e − pz + Cz ψ3 − pz + Cz ψ4 2 c 2 c 1 e e px + Cx − i py + Cy ψ5 = 0, − 2 c c e e W 1 e px + Cx − i py + Cy ψ1 + + A0 ψ2 2 c c c c 1 e e px + Cx − i py + Cy ψ6 = 0, − 2 c c

9 In

the following matrices, for obvious editorial reasons, we have introduced the shortened ± ± ± ± notations: p± 00 = p0 ± mc, p00 = p0 ± mc, pxy = px ± ipy , pxy = px ± ipy , p0z = p0 ± pz , ± ± ± ± ± p0z = p0 ± pz ; P00 = P0 ± M c, Pxy = Px ± iPy , Qxy = Qx ± iQy , P0z = P0 ± Pz , Q± 0z = Q0 ± Qz . 10 @ Note that such a matrix was left incomplete by the author.

11

21

31

41

12

22

32

42

13

23

33

43

14

24

34

44

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

0

+ p00 − p00

p+ xy

pz

p+ xy

pz

0

−pz

p+ xy

0

p− xy

pz

p+ xy

pz

0

0

−pz

p+ xy

0

pz

0

+ p00 + p00

p+ xy

pz

0

+ p00 − p00

p− xy

41

31

21

11

+ p00 + p00

4

3

2

1

−pz

p− xy

−pz

p− xy

−pz

p+ xy

−pz

p− xy

0

0

+ p00 − p00

−pz

p+ xy

+ p00 − p00

p+ xy

p− xy

−pz

pz

p− xy

0

42

8

0

0

32

7

+ p00 + p00

0

22

6

pz

0

+ p00 + p00

0

12

5

0

p+ xy

pz

0

− p00 + p00

p+ xy

pz

13

9

0

−pz

0

0

0

− p00 − p00

0

− p00 − p00

p+ xy

p− xy

−pz

pz

p− xy

p+ xy

pz

43

12

0

p+ xy

pz

33

11

− p00 + p00

p+ xy

pz

23

10

p+ xy

pz

0

− p00 + p00

0

−pz

p− xy

14

13

−pz

0

0

− p00 − p00

− p00 − p00

−pz

p+ xy p− xy

pz

p− xy

0

−pz

p− xy

44

16

0

0

−pz

p− xy

34

15

− p00 + p00

0

−pz

p− xy

24

14

DIRAC THEORY

49

0

0

0

Qz √ 2 Q+ xy √ 2

Q+ xy √ 2

Q− xy √ 2 Qz −√ 2

0

0

0

0

0

0

0

0

Qz √ 2

0

Q− xy √ 2

Qz √ 2

0

0

0

Pz √ 2

0

− Pxy √ 2

0

0

0

0

0

− Pxy √ 2 Pz −√ 2

0

+ Pxy √ 2

0

0

0

0

0

Pz √ 2

0

0

0

0

Pz √ 2 + Pxy √ 2

0

0

− P00

0

0

21 + 12 √ 2 31 + 13 √ 2 41 + 14 √ 2 32 + 23 √ 2 42 + 24 √ 2 43 + 34 √ 2 21 − 12 √ 2 31 − 13 √ 2 41 − 14 √ 2 32 − 23 √ 2 42 − 24 √ 2 43 − 34 √ 2

0

0

− P00

0

0

33

44

0

0

0

+ P00

0

0

2

2 Q+ xy

2 Qz

−

2 Qz

Q− xy

0

0

2

2 + Pxy

2 Pz

−

2 Pz

− Pxy

+ P00

0

0

22

0

0

0

21 + 12 √ 2

+ P00

44

11

33

22

11

2

Q+ xy

0

0

0

0

2

− 2

Qz

0

0

0

0

2

2 Qz

2 Q+ xy −

Pz

0

0

P0

0

+ Pxy

0

0

0

P0

2

+ Pxy

0

0 + Pxy √ 2 Pz − 2

0

0

− Pxy √ 2

Pz √ 2

Pz √ 2

41 + 14 √ 2

31 + 13 √ 2

−

−

−

2

Qz

0

0

0

0

2

2 Qz

Pz

0

P0

0

0

2

Pz

0

+ Pxy √ 2 − Pxy √ 2

0

32 + 23 √ 2

−

−

2

Q− xy

0

0

0

0

2

2 Q− xy

− Pxy

P0

0

0

0

2

Pz √ 2 − Pxy

0

Pz √ 2

0

42 + 24 √ 2

2 Pz

−

2 Qz

0

2

2 Q+ xy

2 Qz

−

−

Q− xy

0

− P00

2

2 + Pxy

−

2 Pz

− Pxy

0

0

0

0

0

43 + 34 √ 2

0

2

2 Q+ xy

2 Qz

−

2 Pz

0

2

2 + Pxy

−

−

2 Pz

+ P00 − Pxy

−

−

−

2 Qz

Q− xy

0

0

0

0

0

21 − 12 √ 2

Q+ xy

−

2

+ Pxy

0

0

0

P0

2

2 + Pxy

−

0

0

0

0

2

Q+ xy

0

Qz √ 2

0

Qz √ 2

31 − 13 √ 2

0

2

2 Pz

Qz

2

Pz

0

0

P0

−

−

0

0

0

0

Q+ xy √ 2 Qz − 2

0

0

Q− xy √ 2

41 − 14 √ 2

0

0

2

2 Pz

2

Pz

0

P0

−

Qz

0

0

0

0

2

Qz

0

Q+ xy √ 2 Q− xy √ 2

0

32 − 23 √ 2

P0

0

0

0

2

2 + Pxy

Q− xy

2

− Pxy

−

−

0

0

0

0

2

Qz −√ 2 Q+ xy

0

Qz −√ 2

0

42 − 24 √ 2

2 Pz

2 Pz

− Pxy

0

0

2

2 Q+ xy

2 Qz

2 Qz

− P00

2

2 + Pxy

−

−

−

−

Q− xy

0

0

0

0

0

43 + 34 √ 2

50 E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

+ p0z

p+ xy

0

31

41

12

44

34

24

14

43

33

23

13

42

32

p+ xy

+ p0z

0

21

22

0

0 0

11

p+ xy

+ p0z

0

0 0 p− xy − p0z

21

11

p+ xy

+ p0z

0

p+ xy

+ p0z

0

0 0

0

0 0

0

+ p0z

−p− xy

− p0z

−p+ xy

41

31

− p0z − p0z − p0z

p− xy

0

− p0z

p+ xy

p− xy

0 0

p− xy

−p+ xy

0 0 p− xy

0

− p0z

0

32

0

0

22

+ p0z

0 0

0

12

− p0z

p− xy

0 0

0

+ p0z

−p− xy

0

42

0

p+ xy

+ p0z

0

0 0

−p+ xy

− p0z

13

0

− p0z

p− xy

0 0

0

−p+ xy

− p0z

23

0

0

0 0

−p+ xy

− p0z

−p+ xy

− p0z

33

0

0 0

0

+ p0z

−p− xy

−p+ xy

− p0z

43

p+ xy

+ p0z

0

0 0

0

+ p0z

−p− xy

14

0 0 p− xy − p0z

0

0

+ p0z

−p− xy

24

0

0 0

−p+ xy

− p0z

0

+ p0z

−p− xy

34

0 0

0

+ p0z

−p− xy

0

+ p0z

−p− xy

44

DIRAC THEORY

51

0

0

33

44

21 − 12 √ 2 31 − 13 √ 2 41 − 14 √ 2 32 − 23 √ 2 42 − 24 √ 2 43 − 34 √ 2

0

0

0

+ Q0z √ 2 Q+ xy √ 2

0

0

0

0

+ P0z √ 2 + Pxy √ 2

0

0

22

21 + 12 √ 2 31 + 13 √ 2 41 + 14 √ 2 32 + 23 √ 2 42 + 24 √ 2 43 + 34 √ 2

0

11

11

0

Q+ xy − √ 2

Q− xy √ 2 + Q0z √ 2

0

0

0

− Q − √0z 2

0

0

0

0

0

0

0

0

+ Pxy √ 2

0

− P0z √ 2

0

0

0

0

0

33

− Pxy √ 2 − P0z √ 2

0

0

0

0

0

0

22

0

+ Q − √0z 2

0

0

0

0

0

0

44

0

2

2 Q+ xy

2 − Q0z 2 + Q0z

Q− xy

0

0

0

0

21 + 12 √ 2

2

Q+ xy

0

0

0

0

2

Q+ xy

+ Q − 0z 2

0

0

0

0

+ Q0z 2

0

−

0

0

− Pxy √ 2

− P0z √ 2

+ P0z √ 2

41 + 14 √ 2

31 + 13 √ 2

0

0

0

0

− Q0z 2

− Q0z 2

−

−

2

Q− xy

0

0

0

0

2

Q− xy

2

2 + Q − 0z 2 + Q − 0z 2 Q+ xy

Q− xy

0

0

0

+ P0z √ 2

+ Pxy √ 2 − Pxy √ 2 0

0

−

43 + 34 √ 2

0

42 + 24 √ 2

0

32 + 23 √ 2

0

2 − P0z 2 + P − 0z 2 + Pxy − 2

− Pxy

0

0

0

0

21 − 12 √ 2

−

−

2

+ Pxy

0

0

0

0

2

+ Pxy

+ Q − √0z 2

0

− Q0z √ 2

31 − 13 √ 2

+ P0z 2

0

0

0

0

+ P0z 2

0

0

Q− xy − √ 2

41 − 14 √ 2

0

0

0

0

− P0z 2

− P − 0z 2

−

Q+ xy − √ 2 Q− xy √ 2

0

32 − 23 √ 2

2

− Pxy

0

0

0

0

2

− Pxy

0

+ Q0z √ 2

0

42 − 24 √ 2

0

2 − P0z 2 + P − 0z 2 + Pxy − 2

− Pxy

0

0

0

0

43 + 34 √ 2

52 E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

53

DIRAC THEORY

1 e W 1 e e − pz + Cz ψ1 + + A0 ψ3 + pz + Cz ψ6 = 0, 2 c c c 2 c 1 e e W 1 e − pz + Cz ψ1 + + A0 ψ4 + pz + Cz ψ6 = 0, 2 c c c 2 c e e e W 1 + A0 ψ5 − px + Cx + i py + Cy ψ1 + 2 c c c c 1 e e + px + Cx + i py + Cy ψ6 = 0, 2 c c −

e e 1 e 1 px + Cx + i py + Cy ψ2 + pz + Cz ψ3 2 c c 2 c 1 e 1 e e pz + Cz ψ4 + px + Cx − i py + Cy ψ5 + 2 c c c 2 W e + A0 − mc = 0. + c c ——————–

In ﬁrst approximation, for Cx = Cy = Cz = 0: ψ1 = 0,

ψ2 =

ψ4 = − −

px − ipy ψ6 , 2mc

pz ψ6 ; 2mc

ψ5 = −

ψ3 = −

pz ψ6 , 2mc

px + ipy ψ6 ; 2mc

p2x + p2y + p2z W e + + A0 − mc ψ6 = 0, 2mc c c p2z + p2y + p2z . W = mc − eA0 + 2m ——————–

e e e W + A0 + αx px + Cx + αy py + Cy c c c c e +αz pz + Cz + βmc = 0; c

54

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

αx =

0

1 2

0

0

−

1 2

0

0

0

0

−

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1 2

0

0

1 2

0

,

1 2

−

0

0

−

1 2

−

0

αz =

0 0

1 2 0 0

−

0

1 2

0

1 2

0

1 2

0

0

1 − 2

0

0

0

0

1 2

1 2

0

0

0

0

1 2

0

0

0

0

0

0

0

0

1 2

1 2

0

0

−

αy = ,

i 2

0

0

i 2

0

0

0

0

0

i 2

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

i 2

i 2

0

0

−

0 −

−

β=

i 2

i 2

−

0

i 2

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

−1

0

.

——————–

px − ipy 2

W + mc c

px + ipy 2

px − ipz 2

W c

0

0

0

−

pz 2

−

pz 2

−

−

pz 2

0

W c

0

0

−

pz 2

0

0

W c

0

0

0

0

W c

pz 2

pz 2

px − ipy 2

px + ipy 2 0

−

px + ipy 2

px − ipy − 2 pz 2 = 0. pz 2 px + ipy 2 W − mc c 0

,

55

DIRAC THEORY

W 0 2 c W 0 c 0 0 0 0 0 0 W px + ipy − mc − c 2

0

0

0

0

0

0

W c

0

0

0

W c

0

0

0

W c

px − ipy − 2 pz 2 = 0, pz 2 px + ipy 2 W − mc c W − mc c

p− xy 2 W6 W5 W4 W4 W2 2 2 2 6 − 2 5 mc − 4 (p2x + p2y + p2z ) − 4 − 2W m + m c = 0, c c c c c2 pz 2

pz 2

2 W2 2 2 2 2 − m c − p + p + p = 0. x y z c2

1.7.4

5-Component Spinors

e W 1 e e + A0 + mc ψ1 + px + Cx + i py + Cy ψ2 c c 2 c c 1 e 1 e e px + Cx − i py + Cy ψ4 = 0, − √ pz + Cz ψ3 − c 2 c c 2 e e W e 1 px + Cx − i py + Cy ψ1 + + A0 ψ2 2 c c c c 1 e e − px + Cx − i py + Cy ψ5 = 0, 2 c c e e W 1 e 1 + A0 ψ3 + √ pz + Cz ψ5 = 0, − √ pz + Cz ψ1 + c c c c 2 2 e e W e 1 + A0 ψ4 − px + Cx + i py + Cy ψ1 + 2 c c c c 1 e e + px + Cx + i py + Cy ψ5 = 0, 2 c c

56

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

1 px + 2 1 + px + 2 −

e Cx + i py + c e Cx − i py + c

e 1 e Cy ψ2 + √ pz + Cz ψ3 c c 2 W e e Cy ψ4 + + A0 − mc ψ5 = 0. c c c

e e e W + A0 + αx px + Cx + αy py + Cy c c c c e +αz pz + Cz + βmc = 0, c αx =

0

1 2

0

−

1 2

0

0

0

−

0

0

0

0

0

0

0

0

1 2

0

1 2

0

,

1 −√ 2

0

0

0

0

0

0

0

0

0

1 √ 2

0

0

0

0

0

1 √ 2

0

0

−

1 2

0

−

0 0 1 αz = − √ 2 0 0

1 2

0

1 2

0 1 2

αy = ,

0 −

i 2

0 −

i 2

0

i 2

0

0

0

0

i 2

0

0

0

0

0

0

0

i 2

0

−

−

0

β=

i 2

i 2

i 2

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

−1

0

.

,

2 QUANTUM ELECTRODYNAMICS

2.1.

BASIC LAGRANGIAN AND HAMILTONIAN FORMALISM FOR THE ELECTROMAGNETIC FIELD

The author studied the dynamics of the electromagnetic ﬁeld in a lagrangian framework; the Lagrangian density L was deduced from a least action principle and, following a canonical formalism, the Hamiltonian density H was then obtained. δ L ds dt = 0, 1 ϕ˙ + ∇ · A = 0, c L =

1 8π

1 2 1 ϕ˙ + |∇ ϕ|2 + 2 (A˙ 2x + A˙ 2y + A˙ 2z ) c2 c 2 2 2 − |∇ Ax | − |∇ Ay | − |∇ Az | .

−

ϕ, Ax , Ay , Az ,

1 ϕ, ˙ 4πc2 1 ˙ Ax , Px = 4πc2 1 ˙ Ay , Py = 4πc2 1 ˙ Az , Pz = 4πc2

P0 = −

ϕ = 0, A = 0.

57

58

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

1 ˙ = −∇ ϕ − A. c H = ∇ × A. E

H = P0 ϕ˙ + Px A˙ x + Py A˙ y + Pz A˙ z − L 1 1 1 = − 2 ϕ˙ 2 − |∇ ϕ|2 + 2 (A˙ 2x + A˙ 2y + A˙ 2z ) + |∇ Ax |2 8π c c + |∇ Ay |2 + |∇ Az |2 = 2πc2 (−P02 + Px2 + Py2 + Pz2 ) 1 −|∇ ϕ|2 + |∇ Ax |2 + |∇ Ay |2 + |∇ Az |2 , + 8π 4πcP0 = ∇ · A, 1 ϕ˙ + ∇ · A = 0, c 1 ˙ = 0. ∇2 ϕ + ∇ · A c H ds =

=

1 −(∇ · A)2 − |∇ ϕ|2 + 2 (A˙ 2x + A˙ 2y + A˙ 2z ) c + |∇ A2x | + |∇ Ay |2 + |∇ Az |2 ds 1 1 −(∇ · A)2 + ϕ ∇2 ϕ + 2 (A˙ 2x + A˙ 2y + A˙ 2z ) 8π c − A · ∇2 A ds. 1 8π

1 ˙ = −∇ ϕ − A, c 2 1 ˙2 2 2 2 2 ˙ ˙ ˙ |∇ ϕ| + (∇ ϕ) · A + 2 (Ax + Ay + Az ) ds E ds = c c 2 1 2 2 2 2 ˙ + (A˙ + A˙ + A˙ ) ds = −ϕ ∇ ϕ − ϕ ∇ · A y z c c2 x 1 = ϕ ∇2 ϕ + 2 (A˙ 2x + A˙ 2y + A˙ 2z ) ds, c E

59

QUANTUM ELECTRODYNAMICS

H = ∇ × A, |∇ × A|2 ds = A · ∇ × ∇ × A ds H2 ds = = A · ∇ (∇ · A) − A · ∇2 A ds = −(∇ · A)2 − A · ∇2 A ds,

[1 ] H ds =

2.2.

1 8π

(E 2 + H2 ) ds.

ANALOGY BETWEEN THE ELECTROMAGNETIC FIELD AND THE DIRAC FIELD

In the following pages, the author explored the possibility of describing the electromagnetic ﬁeld in full analogy with what usually done for a Dirac ﬁeld. In a three-dimensional formalism, he then introduced a wavefunction ψ in terms of the electric and magnetic ﬁelds E, H (and, more speciﬁcally, in terms of quantities E ± iH), and its dynamics (for free ﬁelds) was developed in close analogy with the Dirac procedure for spin-1/2 ﬁelds. Commutation (rather than anticommutation) rules for Dirac-like matrices were adopted, and energy eigenvalues and eigenvectors were calculated. For further details, see R. Mignani, M. Baldo and E. Recami, Lett. Nuovo Cim. 11 (1974) 568; E. Giannetto, Atti del IX Congresso Nazionale di Storia della Fisica, edited by F. Bevilacqua (Milan, 1988) 173; S. Esposito, Found. Phys. 28 (1998) 231.

1@

In the original manuscript, the author pointed out that, from: 1 ϕ˙ + ∇ · A = 0, c

ϕ = 0,

it follows that: ∇2 ϕ +

1 ∇ · A˙ = 0. c

60

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

4πρ − ∇ · E = 0, 4πI +

∇ · H = 0,

1 ∂E = ∇ × H, c ∂t

−

1 ∂H = ∇ × E. c ∂t

ψ1 = E1 − iH1 = Ex − iHx , ψ2 = E2 − iH2 = Ey − iHy , ψ3 = E3 − iH3 = Ez − iHz .

∇ · ψ = ∇ · E − i∇ · H = 4πρ.

(1)

1 ∂E 1 ∂H − − 4πiI c ∂t c ∂t

∂H i ∂E −i − 4πiI, = − c ∂t ∂t

∇ × ψ = ∇ × E − i∇ × H = −

4πI +

1 ∂ψ = +i∇ × ψ. c ∂t

(2)

——————– The Maxwell equations are given by: 1 ∂ψ − i∇ × ψ + 4πI = 0, c ∂t ∇ · ψ − 4πρ = 0.

∂ψ3 ∂ψ2 1 ∂ψ1 −i +i + 4πIx c ∂t ∂y ∂z ∂ψ1 ∂ψ3 1 ∂ψ2 −i +i + 4πIy c ∂t ∂z ∂x 1 ∂ψ3 ∂ψ2 ∂ψ1 −i +i + 4πIz c ∂t ∂x ∂y ∂ψ1 ∂ψ2 ∂ψ3 + + − 4πρ ∂x ∂y ∂z

= 0, = 0, = 0, = 0.

61

QUANTUM ELECTRODYNAMICS

Without charge: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

W ψ1 + ipy ψ3 − ipz ψ2 = 0, c W ψ2 + ipz ψ1 − ipx ψ3 = 0, c

⎪ W ⎪ ⎪ ψ3 + ipx ψ2 − ipy ψ1 = 0, ⎪ ⎪ ⎪ c ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ px ψ1 + py ψ2 + pz ψ3 = 0. [2 ]

W c 0 αx = 0 0 αz =

+ αx px + αy py + αz pz ψ = 0. 0 0 0 −i , +i 0

0 −i 0 +i 0 0 , 0 0 0

0 αy = 0 −i 1 1 = 0 0

(3)

0 +i 0 0 , 0 0 0 0 1 0 . 0 1

[3 ] αx αy − αy αx = −iαz , [αx , αz ]− = +iαy , [αy , αz ]− = iαx . βx = |1 0 0|,

βy = |0 1 0|,

βz = |0 0 1|.

(βx px + βy py + βz pz ) ψ = 0.

(4)

Following the Dirac method, the eigenvalues of the Maxwell equation are obtained from: 2@

The line before the fourth equation means that it is deduced from the previous three equations. 3 @ Note that the signs on the RHS of the following two equations were wrong: correctly, we have αx αy − αy αx = iαz and [αx , αz ]− = −iαy .

62

p=

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

p2x + p2y + p2z . W/c

W/c −ipz ipy ipz W/c −ipx = 0, −ipy ipx W/c 3 W W = 0, − p2 c c ⎧ ⎨ p, W = −p, ⎩ c 0,

ψ1

ψ2

ψ3

p

p2y + p2z −px py − ippz −px pz + ippy

−p

p2y + p2z −px py + ippz −px pz − ippy

0

px

py

pz

——————– For t = 0: ψ1 = a δ(x − x0 )δ (y − y0 )δ (r − r0 ), ψ2 = b δ (x − x0 )δ(y − y0 )δ (z − z0 ), ψ3 = −(a + b) δ (x − x0 )δ (y − y0 )δ(z − z0 ). ∂ψ1 ∂ψ2 ∂ψ3 + + = 0. ∂x ∂y ∂z ψ1 (x, y, z) = ψ2 (x, y, z) =

ψ3 (x, y, z) =

ψ1 =

A(x0 , y0 , z0 ) δ(x − x0 )δ (y − y0 )δ (z − z0 ) dx0 dy0 dz0 , B(x0 , y0 , z0 ) δ (x − x0 )δ(y − y0 )δ (z − z0 ) dx0 dy0 dz0 , −(A + B) δ (x − x0 )δ (y − y0 )δ(z − z0 ) dx0 dy0 dz0 . ∂2A , ∂y∂z

ψ2 =

∂2B , ∂z∂x

ψ3 = −

∂ 2 (A + B) ; ∂x∂y

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QUANTUM ELECTRODYNAMICS

∂3A ∂ψ1 = , ∂x ∂x∂y∂z

∂ψ2 ∂2B = , ∂y ∂x∂y∂z

∂ψ3 ∂ 2 (A + B) =− . ∂z ∂x∂y∂z

——————– ∂A = ψ1 , ∂y∂z ∂A = ψ1 dz + fy , ∂y A = A0 + F1 (x, y) + F2 (x, z); ∂2B = ψ2 , ∂z∂x B = B0 + F3 (x, y) + F4 (y, z). ∂ 2 (A + B) ∂ 2 (A0 + B0 ) =− + F (x, y). ∂x∂y ∂x∂y By substituting the expressions: ψ3 = −

∂2A ∂2B , ψ2 = , ∂y∂z ∂z∂x into the Maxwell equations, we get: ψ1 =

ψ3 =

∂2C , ∂x∂y

∂3C ∂2B 1 ∂3A −i + i = 0, c ∂y∂z∂t ∂x∂ 2 y ∂x∂ 2 z ∂3A ∂3C 1 ∂3B −i + i = 0, c ∂z∂x∂t ∂y∂ 2 z ∂y∂ 2 x ∂3B ∂3A 1 ∂3C −i + i = 0; c ∂x∂y∂t ∂z∂ 2 x ∂z∂ 2 y ∂ 3 (A + B + C) = 0. ∂x∂y∂z A + B + C = 0.

2

∂2 ∂2 ∂ 1 ∂2 ∂ ∂ +i + A+i B = 0, ∂y c ∂z∂t ∂x∂y ∂x ∂ 2 y ∂ 2 z

2

∂2 ∂ ∂ ∂2 ∂ 1 ∂2 −i B−i + A = 0, ∂x c ∂z∂t ∂x∂y ∂y ∂ 2 x ∂ 2 z

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E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

∂ ∂y

1 ∂2 ∂2 −i c ∂x∂t ∂y∂z

A−

∂ ∂x

1 ∂2 ∂2 +i c ∂y∂t ∂x ∂z

B = 0.

——————–

A = −a ei(γ1 x+γ2 y+γ3 z) , B = −b ei(γ1 x+γ2 y+γ3 z) , C = −c ei(γ1 x+γ2 y+γ3 z) ; ψ1 = a γ2 γ3 ei(γ1 x+γ2 y+γ3 z) , ψ2 = b γ3 γ1 ei(γ1 x+γ2 y+γ3 z) , ψ3 = c γ1 γ2 ei(γ1 x+γ2 y+γ3 z) .

2.3.

ELECTROMAGNETIC FIELD: PLANE WAVE OPERATORS

Plane wave expansion of the electromagnetic ﬁeld was considered in a way similar to what is usually done for a Dirac or a Klein-Gordon ﬁeld. In the second part, the author again introduced a sort of photon wave ﬁeld Ψ, in close analogy to the Dirac ﬁeld for a spin-1/2 particle and in a full Lorentz-invariant formalism. The properties of this ﬁeld are deduced from general group-theoretic arguments. ϕ, Ax , Ay , Az , P0 , Px ,

1 ϕ, ˙ 4πc2 1 ˙ Ax , Px = 4πc2 1 ˙ Ay , Py = 4πc2 1 ˙ Az , Pz = − 4πc2

P0 = −

ϕ˙ = 4πc2 P0 ; A˙ x = 4πc2 Px ; A˙ y = 4πc2 Py ; A˙ z = 4πc2 Pz ;

1 P˙0 = − ∇2 ϕ; 4π 1 2 ˙ ∇ Ax ; −Ax , Px = 4π 1 2 ∇ Ay ; Py , −Ay , P˙y = 4π 1 2 ∇ Az . Pz , −Az P˙z = 4π −ϕ,

65

QUANTUM ELECTRODYNAMICS

[4 ] U0 (γ) = Ux (γ) = Uy (γ) =

e−2πi(γ1 x+γ2 y+γ3 z) ϕ(x, y, z) dx dy dz,

e−2πiγ ·q Ax (q) dq. e−2πiγ ·q Ay (q) dq,

e−2πiγ ·q Az (q) dq.

Uz (γ) =

L(q) dq =

M (γ) dγ,

[5 ] M

=

1 1 − 2 U˙ 0 U˙ 0 + 4π 2 γ 2 U 0 U0 + 2 (U˙ x U˙ x + U˙ y U˙ y + U˙ z U˙ z ) c c 2 2 − 4π γ (U x Ux + U y Uy + U z Uz ) .

1 8π

U0 , Ux , Uy , Uz ,

4@

1 ˙ U 0, 4πc2 1 ˙ U x, Vx = 4πc2 1 ˙ U y, Vy = 4πc2 1 ˙ Vz = U z. 4πc2 V0 = −

U = (Ux , Uy , Uz ),

V = (Vx , Vy , Vz ),

U˙ = (U˙ x , U˙ y , U˙ z ),

V˙ = (V˙ x , V˙ y , V˙ z ),

In the original manuscript, the author considered in what follows the role of the operators √ ∇2 = L2 and L = ∇2 . He denoted with q the vector (x, y, z). 5 @ A bar over a quantity denotes complex conjugation.

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E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

U 0 (γ) = U0 (−γ), U˙ 0 (γ) = U˙ 0 (−γ), U (γ) = U (−γ), U˙ (γ) = U˙ (−γ), V (γ) = V (−γ), V˙ (γ) = V˙ (−γ), V 0 (γ) = V0 (−γ), V˙ 0 (γ) = V˙ 0 (−γ). 1 ¨ U0 + 4π 2 γ 2 U0 = 0, c2 1 ¨ U + 4π 2 γ 2 U = 0, c2 1˙ U0 + 2πi(γ1 Ux + γ2 Uy + γ3 Uz ) = 0, c 1 2πiγ 2 U0 + (γ1 U˙ x + γ2 U˙ y + γ3 U˙ z ) = 0. c [6 ]

i 1 ϕ(q) ˙ dq, 2πγc ϕ(q) + √ ψ0 (γ) = e · √ 2πγc 2c h

i 1 −2πi(γ ·q ˙ ψx (γ) = e Ax (q) dq, 2πγc Ax (q) + √ · √ 2πγc 2c h

i 1 −2πiγ ·q ψy (γ) = e A˙ y (q) dq, 2πγc Ay (q) + √ · √ 2πγc 2c h

i 1 −2πiγ ·q ˙ Az (q) dq. ψz (γ) = e 2πγc Az (q) + √ · √ 2πγc 2c h

−2πiγ ·q

√ 1 √ [ψ0 (γ) + ψ 0 (−γ)] e2πiγ ·q dγ, ϕ(q) = c h 2πγc √ c h 2πγc [ψ0 (γ) − ψ 0 (−γ)] e2πiγ ·q dγ, ϕ(q) ˙ = i 6@

Probably, the author proceeded in analogy with the Dirac ﬁeld .

QUANTUM ELECTRODYNAMICS

67

√ 1 √ Ax (q) = c h [ψx (γ) + ψ x (−γ)] e2πiγ ·q dγ, 2πγc ..., √ c h ˙ 2πγc [ψx (γ) − ψ x (−γ)] e2πiγ ·q dγ, Ax (q) = i ..., [7 ] ϕ = =

1 ϕ¨ − ∇2 ϕ c√2 h 2πγc ψ˙ 0 (γ) − ψ˙ 0 (−γ) ci

+ 2πγc i ψ0 (γ) + 2πγc i ψ 0 (−γ) e2πiγ ·q dγ.

ψ0 (γ) = −2πγc i ψ0 (γ), ψ˙ 0 (γ) = 2πγc i ψ 0 (γ), ˙ ψ(γ) = −2πγc i ψx (γ), ψ˙ x (γ) = 2πγc i ψ x (γ), ....

1 2 1 ϕ˙ − |∇ ϕ|2 + 2 (A˙ 2x + A˙ 2y + A˙ 2z ) c2 c 2 2 2 + |∇ Ax | + |∇ Ay | + |∇ Az | dq ψ0 (γ)ψ 0 (γ) + ψ 0 (γ)ψ0 (γ) ψx (γ)ψ x (γ) + ψ x (γ)ψx (γ) + = hγc − 2 2 ψy (γ)ψ y (γ) + ψ y (γ)ψy (γ) ψz (γ)ψ z (γ) + ψ z (γ)ψz (γ) + dγ, + 2 2 1 8π

−

W =

hγc −ψ0 (γ)ψ 0 (γ) + ψ x (γ)ψx (γ) + ψ y (γ)ψy (γ) + ψ z (γ)ψz (γ) dγ.

7@

In the original manuscript, the author also cited the following (seeming) identity, whose meaning in this general framework is not clear: 0

= =

ϕ(q) ˙ − ϕ(q) ˙ n o √ Z 1 c h ψ˙ 0 (γ) + ψ˙ 0 (−γ) + 2πγc i ψ0 (q) − 2πγc i ψ 0 (−γ) e2πiγ·q dγ. √ 2πγc

68

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

ψ0 (γ)ψ 0 (γ ) − ψ 0 (γ )ψ0 (γ) = −δ(γ − γ ), ψx (γ)ψ x (γ ) − ψ x (γ )ψx (γ) = +δ(γ − γ ), ....

1 ˙ = ∇2 ϕ + ∇ · A c √ 2πγ −γ[ψ0 (γ) + ψ 0 (−γ)] + γx [ψx (γ) − ψ x (−γ)] = c h 2π c + γy [ψy (γ) − ψ y (−γ)] + γz [ψz (γ) − ψ z (−γ)] e2πiγ ·q dγ, 1 ϕ˙ + ∇ · A c 2π ch γ[ψ0 (γ) − ψ 0 (−γ)] − γx [ψx (γ) − ψ x (−γ)] =√ γc i − γy [ψy (γ) − ψ y (−γ)] − γz [ψz (γ) − ψ z (−γ)] e2πiγ ·q dγ, γψ0 − γx ψx − γy ψy − γz ψz = 0, γψ 0 − γx ψ x − γy ψ y − γz ψ z = 0, ψ0 = ψ0 (γ), ψx = ψx (γ), . . ., ψ 0 = ψ 0 (γ), ψ x = ψ x (γ), . . ..

2.3.1

Dirac Formalism Ψ = (ψ0 , ψx , ψy , ψz ), h ∂ , 2πi ∂t 0 0 Sx = 0 0

H =−

px = 0 0 0 0

0 0 0 1

Sz

h ∂ h ∂ , pz = , 2πi ∂x 2πi ∂y 0 0 0 0 0 0 , Sy = −1 0 0 0 −1 0 0 0 0 0 0 0 −1 0 ; = 0 0 0 1 0 0 1 0

pz = 0 0 0 0

0 1 0 0

h ∂ ; 2πi ∂z

,

69

QUANTUM ELECTRODYNAMICS

Tx =

0 1 0 0

1 0 0 0

0 0 0 0

0 0 0 0 Tz =

Ty =

, 0 0 0 1

0 0 0 0

0 0 0 0

1 0 0 0

1)

Ψ = HΨ = hγc Ψ

2)

Ψ = px Ψ = hγx Ψ

3)

Ψ = py Ψ = hγy Ψ

4)

Ψ = pz Ψ = hγz Ψ ⎧ 0 ⎪ ⎪ ⎪ ⎪ ⎨ 0 ∂ ∂ + γz + Ψ = Sx Ψ = −γy ⎪ ∂γz ∂γy 0 ⎪ ⎪ ⎪ ⎩ 0

5)

⎧ ⎪ ⎪ ⎪ ⎪ ⎨

6)

∂ ∂ + γx + Ψ = Sy Ψ = −γz ⎪ ∂γz ∂γz ⎪ ⎪ ⎪ ⎩ ⎧ ⎪ ⎪ ⎪ ⎪ ⎨

7)

∂ ∂ + γy + Ψ = Sz Ψ = −γx ⎪ ∂γy ∂γx ⎪ ⎪ ⎪ ⎩ ∂ γx + − Ψ = Tx Ψ = −γ ⎪ ∂γx 2γ ⎪ ⎪ ⎪ ⎩ ⎧ ⎪ ⎪ ⎪ ⎪ ⎨

8)

0 0 1 0 .

0 0 0 0

1 0 0 0

0 0 0 0

,

⎫ 0 ⎪ ⎪ ⎪ ⎪ 0 0 0 ⎬ Ψ 0 0 −1 ⎪ ⎪ ⎪ ⎪ ⎭ 0 1 0 0 0

⎫ 0 0 0 ⎪ ⎪ ⎪ ⎪ 0 0 0 1 ⎬ Ψ 0 0 0 0 ⎪ ⎪ ⎪ ⎪ ⎭ 0 −1 0 0 0

⎫ 0 0 ⎪ ⎪ ⎪ ⎪ 0 0 −1 0 ⎬ Ψ 0 1 0 0 ⎪ ⎪ ⎪ ⎪ ⎭ 0 0 0 0 0 0

⎫ 0 0 ⎪ ⎪ ⎪ ⎪ ⎬ 1 −γx /γ 0 0 − 2πi ct γx Ψ ⎪ 0 −γy /γ 0 0 ⎪ ⎪ ⎪ ⎭ 0 −γz /γ 0 0 0

0

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E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

⎧ ⎪ ⎪ ⎪ ⎪ ⎨

9)

γy ∂ Ψ = Ty Ψ = −γ − + ⎪ ∂γy 2γ ⎪ ⎪ ⎪ ⎩ ∂ γz + − Ψ = Tz Ψ = −γ ⎪ ∂γz 2γ ⎪ ⎪ ⎪ ⎩ ⎧ ⎪ ⎪ ⎪ ⎪ ⎨

10)

ψ0 = 0, Ψ = (ψx , ψy , ψz ).

γ = (γx , γy , γz ), γ =

⎫ 0 ⎪ ⎪ ⎪ ⎪ ⎬ 0 0 −γx /γ 0 − 2πi ct γy Ψ ⎪ 1 0 −γy /γ 0 ⎪ ⎪ ⎪ ⎭ 0 0 −γz /γ 0 0 0

0

⎫ ⎪ ⎪ ⎪ ⎪ ⎬ 0 0 0 −γx /γ − 2πi ct γz Ψ ⎪ 0 0 0 −γy /γ ⎪ ⎪ ⎪ ⎭ 1 0 0 −γz /γ 0 0 0

0

γx2 + γy2 + γz2 .

(γ , γx , γy , γz ) = C(γ, γx , γy , γz ), C = cik (i, k = 0, 1, 2, 3) c200 −

3

c20i = 1,

i=1

c00 ci0 − ci0 ck0 −

3

c0k cik = 0,

k=1 3

(i = 1, 2, 3),

cik cki = −∂ik ,

(i, k = 1, 2, 3).

k=10

−2πic(γ −γ)t

Ψ (γ ) = e

γ D Ψ(γ), γ

D = dik (i, k = 1, 2, 3) γx c01 , γ γ = c12 − x c02 , γ γ = c13 − x c03 , γ

γy c01 , γ γy = c22 − c02 , γ γy = c23 − c03 , γ

γz c01 , γ γ = c32 − z c02 , γ γ = c33 − z c03 . γ

d11 = c11 −

d21 = c21 −

d31 = c31 −

d12

d22

d32

d13

d23

d33

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QUANTUM ELECTRODYNAMICS

γx Ψx + γy Ψy + γz Ψz =

γ −2πc(γ −γ)t e (γx Ψx + γy Ψy + γz Ψz ). γ

0 0 0 ∂ ∂ + γz + 0 0 −1 , Sx = −γy ∂γz ∂γy 0 1 0 0 0 1 ∂ ∂ + γx + 0 0 0 , Sy = −γz ∂γx ∂γz −1 0 0 0 −1 0 ∂ ∂ 0 0 , + γy + 1 Sz = −γx ∂γy ∂γx 0 0 0 γx /γ 0 0 ∂ γx − − 2πi c γx t − γy /γ 0 0 , Tx = −γ ∂γx 2γ γ /γ 0 0 z 0 γx /γ 0 γy ∂ − 2πi c γy t − 0 γy /γ 0 , − Ty = −γ ∂γy 2γ 0 γ /γ 0 y 0 0 γx /γ ∂ γz − − 2πi cγz t − 0 0 γy /γ Tz = −γ ∂γz 2γ 0 0 γ /γ z

,

γx ψx + γy ψy + γz ψz = 0.

2.4.

QUANTIZATION OF THE ELECTROMAGNETIC FIELD

In what follows,8 the author considered the quantization of the electromagnetic ﬁeld inside a box, obtaining the usual equations in terms of 8@

In the original manuscript, the title of this section is “Dispersion”.

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E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

oscillators. Particular care was devoted to distinguish the role of the right-handed polarized states from that of the left-handed ones. ∇ · E = ∇ · C = 0. dS = dx dy dz: 1 8π

E 2 − H 2 dS dt = minimum,

ϕ = 0. 1 ∂C , c ∂t

E=− δE = −

H = ∇ × C;

1 ∂ δC, c ∂t

δH = ∇ × δC.

1 ∂H + ∇ × E = 0, c ∂t 1 ∂E = ∇ × H = ∇ × ∇ × C = ∇ (∇ · C) − ∇2 C c ∂t = −∇2 C. Conjugate variables: Cx , −

1 Ex , 4πc H=

1 8π

Cy ,

−

Cz ;

1 Ey , 4πc

−

1 Ez . 4πc

(E 2 + H 2 ) dS.

Let us consider the electromagnetic ﬁeld conﬁned inside a cube with side k, its volume being S = k 3 : γ1 =

n1 , k

γ2 =

n2 , k

γ3 =

dN = 2k 3 dγ1 dγ2 dγ3 . v = cγ.

n3 . k

73

QUANTUM ELECTRODYNAMICS

γ=

A1s A2s A3s A4s

= = = =

v γ12 + γ22 + γ32 = . c

k1 cos 2π(γ1 x + γ2 y + γ3 z) + k2 sin 2π(γ1 x + γ2 y + γ3 z), −k1 sin 2π(γ1 x + γ2 y + γ3 z) + k2 cos 2π(γ1 x + γ2 y + γ3 z), k1 cos 2π(γ1 x + γ2 y + γ3 z) − k2 sin 2π(γ1 x + γ2 y + γ3 z), k1 sin 2π(γ1 x + γ2 y + γ3 z) + k2 cos 2π(γ1 x + γ2 y + γ3 z);

A1s and A2s correspond to right-handed, circularly polarized waves, while A3s and A4s correspond to the left-handed ones. The direction of s = (v1 , v2 , v3 ) is deﬁned by the right-handed direction of k1 , k2 . Note that γ1 , γ2 , γ3 are given apart from a simultaneous change of sign! s −→ −s, k1 , k2 −→ k2 , k1 . A1−s = A2s , A2−s = A1s , A3−s = A4s , A4−s = A3s . |k1 | = 1, |k2 | = 1; S = k 3 .

C= E=

ais Ais , bis Ais .

Notice that, in these sums, the terms corresponding to s and those corresponding to −s give the same contribution: s ≡ −s. The terms with s and −s are counted only once; the sign of s is deﬁned by the right-handed rotation of k1 , k2 !

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E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

ais bis − bis ais =

1 bis = − a˙ is , c

2hc . iS

ais =

1 ˙i b . 4πγ 2 c s

¨bi + 4π 2 γ 2 c2 bi = 0, s s

a ¨is + 4π 2 γ 2 c2 ais = 0, γ 2 c2 = ν 2 .

a˙ is = −cbis , b˙ is = 4π 2 γ 2 c ais .

H=

4π 2 γ 2 ai2 + bi2 s

s,i

a˙ is = −

4πc ∂H , S ∂bis

pis

=

ais

=

bis =

νSπ i a , hc s

qsi

hc i p , νSπ s

bis

H=

1 ν,i

2

1 pis qsi − qsi pis = , i

p˙is = −2πνqsi = −

s

8π

= =

S.

4πc ∂H . S ∂ais

S bi , 4πνhc s 4πνhc i qs . S

i2 (pi2 s + qs )hν.

ais bis − bis ais =

2π ∂H , h ∂qsi

2hc . iS

q˙si = 2πνpis =

2π ∂H . h ∂pis

75

QUANTUM ELECTRODYNAMICS

s → −s → s → −s →

ps − qs2 √ , 2

pR s =

R P−s =

pL s =

p2s − qs √ , 2

R q−s =

p4s − qs3 √ , 2

pL −s =

qs + ps √ , 2

qsR =

qsL =

p3s − qs4 √ , 2

qs2 + ps √ , 2

qs4 + p3s √ , 2

L q−s =

qs3 + p4s √ , 2

1 R R R pR s q s − q s ps = ; i 1 R R R pR −s q−s − q−s p−s = ; i 1 L L L pL s q s − q s ps = ; i 1 L L L pL −s q−s − q−s p−s = . i

From now on, the terms with s are distinct from those with −s ! 1 R R R pR s q s − q s ps = , i

1 L L L pL s q s − q s ps = . i

as =

R pR s − iqs √ 2

bs =

L pL s − iqs √ 2

as =

R pR s + iqs √ 2

bs =

L pL s + iqs √ 2

as as − as as = 1 bs bs − bs bs = 1 1 1 2 D2 as as = (pD s + qs ) − 2 2 as as = ns ,

1 1 bs bs = (pSs 2 + qsS 2 ) − 2 2

(ns = 0, 1, 2, . . .) bs bs = ns

as (ns , ns + 1) =

√

ns + 1 bs (ns , ns + 1) =

as (ns , ns − 1) =

pR s =

√

as + as √ , 2

qsR = i

ns bs (ns , ns−1 ) =

pL s =

bs + bs √ ; 2

as − as bs + b √ , qsL = i √ s . 2 2

ns + 1

ns

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E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

W

11 hνs (pis 2 + qsi 2 ) 2 2 s,i 1 1 2 D2 = hνs (pD hνs (pSs 2 + qsS 2 ) s + qs ) + 2 2 s s = hνs (ns + ns ) (+ an inﬁnite constant). =

s

p1s =

R pR s + q−s √ , 2

qs1 =

qsR − pR √ −s , 2

p2s =

R pR −s + qs √ , 2

qs2 =

R − pR q−s √ s, 2

p3s =

L pL −s + qs √ , 2

qs3 =

L − pL q−s √ s, 2

p4s =

L pL −s + q−s √ , 2

qs4 =

qsL − pL √ −s 2

(in the LHS s and −s are gathered together, while on the RHS they are kept distinct). 1 p1s = [as + ia−s + as − ia−s ], 2

1 qs1 = [ias − a−s − ias − a−s ], 2

1 p2s = [a−s + ias + a−s − ias ], 2

1 qs2 = [ias − as − ias − as ], 2

1 p3s = [b−s + ibs + b−s − ibs ], 2

1 qs3 = [ib−s − bs − ib−s − bs ], 2

1 p4s = [bs + ib−s + bs − ibs ], 2

1 qs4 = [ibs − b−s − ibs − b−s ]. 2

a˙ s = . . . ,

b˙ s = . . . ,

a˙ ∗s = . . . ,

b˙ ∗s = . . . .

In what follows, the orthogonal functions Ais are deﬁned for all the values of s (see page 73); the indices of k1 , k2 are given in such a way that the vectors k1 , k2 , s form a right-handed trihedron. The vectors k1 and k2 transform one into the other by changing s into −s. Each function Ais is counted twice, due to the relations: A1s = A2−s ,

A2s = A1−s ,

A3s = A4−s ,

A4s = A3−s .

77

QUANTUM ELECTRODYNAMICS

C =

c 2

E =

h 1 √ [(as + as )A1s + i(as − as )A2s πS s νs + i(bs − bs )A3s + (bs + bs )A4s ],

πh √ νs [i(as − as )A1s − (as + as )A2s S s − (bs + bs )A3s + i(bs − bs )A4s ].

√ as (ns , ns+1 ) = ns + 1, √ as (ns , ns−1 ) = ns , as as − as as = 1, as as = ns ,

W =

bs (ns , ns+1 ) = ns + 1, bs (ns , ns−1 ) = ns , bs bs − bs bs = 1, bs bs = ns .

1 ∗ ∗ 2 ∗2 ∗ ∗ hνs [ a2s + a∗2 s + as as + as as − as − as + as as + as as 4 s

∗ ∗ 2 ∗2 ∗ ∗ − b2s − b∗2 s + bs bs + bs bs + bs + bs + bs bs + bs bs ] hνs (ns + ns+1 ) = hνs (ns + Ns ) + an inﬁnite constant, = s

s

with: ns = a∗s as , Ns = b∗s bs .

By absorbing the inﬁnite constant into W , we have: WR =

hνs (ns + Ns ).

s

We have used Ns instead of ns : ns corresponds to right-handed polarized waves, while Ns to the left-handed ones.

78

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

2.5.

CONTINUATION I: ANGULAR MOMENTUM

The author continued9 to study the quantization of the electromagnetic ﬁeld, obtaining explicit expressions for the matrix elements of the creation and the annihilation operators (in the number operator representation) and for the angular momentum of the ﬁeld. Transformation properties of the n-photon states ψ were quickly outlined at the end of this Section.

C =

k

E =

√

2hc pk f k , k

2hck qk f k .

k

q˙k = kc pk ,

1 ∂C c ∂t 1 ∂E c ∂t

=

k

=

k

p˙ k = −kc qk .

√ 2h p˙k f k = −E = − 2hck qk f k , ck k

√ 2hk q˙k f k = −∇2 C = k 2hck pk f k . c k

q˙k =

2π ∂W , h ∂pk

p˙k = −

W =

hνk

2π ∂W . h ∂qk

h 1 1 2 (pk + qk2 ) = ck (p2k + qk2 ). 2 2π 2 k

9@

In the original manuscript, the title of this section is “Irradiation”.

79

QUANTUM ELECTRODYNAMICS

2πi (qk W − W qk ), h 2πi (pk W − W qk ); = − h

q˙k = − p˙k

∂W , ∂pk ∂W ; ∂qk

i(qk W − W qk ) = −i(pk W − W pk ) =

−i(qk pk − pk qk ) = 1, +i(pk qk − qk pk ) = 1.

1 pk q k − q k pk = . i

W =

k

p2 + qk2 1 hνk nk + = hνk k . 2 2

pk + iqk pk − iqk 1 2 1 √ (pk + qk2 ) = √ + , 2 2 2 2 ak =

pk − iqk √ , 2

a∗k =

pk + iqk √ . 2

i ak a∗k − a∗k ak = (pk qk − qk pk + pk qk − qk pk ) = 1. 2 a∗k ak = nk , ak a∗k = nk + 1. pk − iqk √ , i pk + iqk √ a∗k = , i

ak =

ak + a∗k √ , 2 ∗ a − ak qk = k √ . i 2 pk =

80

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

ak =

0 1 √0 0 0 0 ... 0 0 2 √0 0 0 ... 3 √0 0 ... 0 0 0 4 √0 . . . 0 0 0 0 5 ... 0 0 0 0 0 ... ... ... ... ... ... ...

∗ ak =

0 0 0 0 0 0 ... 1 √0 0 0 0 0 ... 0 2 √0 0 0 0 ... 0 0 3 √0 0 0 ... 0 0 0 4 √0 0 ... 5 0 ... 0 0 0 0 ... ... ... ... ... ... ... 0 0 0 0 . . . 0 1 0 0 . . . 0 2 0 . . . , a∗k ak = 0 0 0 0 3 . . . ... ... ... ... ... 1 0 0 0 . . . 0 2 0 0 . . . ∗ 0 3 0 . . . ; ak ak = 0 0 0 0 4 . . . ... ... ... ... ... √ 0 2 0 . . . 1/ √ 1/ 2 0 1 . . . pk = , 1 0 . . . 0 ... ... ... ... √ 0√ i/ 2 0 . . . −i/ 2 0 −i . . . . qk = 0 i 0 . . . ... ... ... ...

C = C + S C, pr = pr + ε

Srs ps ,

, ;

E = E + S E. qr = qr + ε

s

s

Srs = −Ssr .

Srs qs .

81

QUANTUM ELECTRODYNAMICS

ψ = ψ(n1 , n2 , . . .), T ψ = ψ + εψ; i ε q = q + (qT − T q), i ε p = p + (pT − T p). i pr T − T p r = i qr T − T q r = i

T =

Srs ps , Srs qs .

Srs pr qs .

rs

T is the angular momentum in units h/2π. Srs (pr qs − ps qr ). T = Srs pr qs = r<s

pr q s − ps q r =

=

T =

1 ∗ ∗ (a a − ar as − a∗r as + ar a∗s 2i r s − a∗s a∗r − as ar + a∗s ar − as a∗r ) 1 (ar a∗s − as a∗r ). i 1 r<s

i

(ar a∗s − as a∗r )Srs .

For n photons: ψ = ψ(n1 , n2 , . . .) δ

ni − n .

For n = 1, ψ = ψ(n1 , n2 , . . .) and all ni but one vanish, and the non-zero number is equal to 1: ψ(1, 0, 0, 0, 0, . . .) = c1 , ψ(0, 1, 0, 0, 0, . . .) = c2 , ψ(0, 0, 1, 0, 0, . . .) = c3 , ... .

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E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

ψ = (c1 , c2 , c3 , . . .).

ψ = T ψ. r<s

Srs (ar a∗s − as a∗r ) =

Srs ar a∗s ,

r,s

1 Srs ar a∗s ψ = (c1 , c2 , . . .). i rs cs =

1 Srs cr = i Ssr cr . i

cr = i

2.6.

Srs cs .

CONTINUATION II: INCLUDING THE MATTER FIELDS

What had been studied in the Sect. 2.4 was tentatively generalized here to the case of an electromagnetic ﬁeld interacting with a charged Dirac ﬁeld ψ. As above, the scalar potential is assumed to be zero, ϕ = 0, and again the box volume is S = k 3 . Dirac equations:

e W + ρ3 σ · p + C + ρ1 mc ψ = 0. c c

p = (px , py , pz ). For plane waves, px , py , pz are constant. ψpr = (ψ1 , ψ2 , ψ3 , ψ4 ) = e(2πi/h)(px x+py y+pz z) ( 1 , 2 , 3 , 4 ), ⎧ ⎨ + m2 c2 + p2 ,

for r = 1, 2,

− m2 c2 + p2 ,

for r = 3, 4.

W = ⎩ c

83

QUANTUM ELECTRODYNAMICS

The spinor factors are given in the following table: 2 3 2S . . . 4 2S . . . 2 2S . . . 1 2S 1 + Wc mp2zc2 + mp2 c2 1

0

z − W/c+p mc

px −ipy mc

z − W/c+p mc

0

1

0

z − W/c+p mc

px −ipy mc

z − W/c+p mc

0

h px = g1 , k

h py = g2 , k

−

px +ipy mc

1 −

px +ipy mc

1

h pz = g3 ; k

g1 , g2 , g3 = 0, ±1, ±2, ±3, . . . .

H = −cρ3 σ · p − ρ1 mc2 +

hνs (ns + Ns ) − eρ3 σ · C

s

= H0 − eρ3 σ · C = H0 + H1 . H1 = −eρ3 σ · C. Quantities ns , Ns are the numbers of the right-handed and left-handed polarized waves, respectively. p, r, ni , Ni |H0 |p , r , ni , Ni = δ(p − p ) δ(r − ri ) δ(n − n ) δ(N − N ) p,r + hνs (ns + Ns ). × Welectr. s

——————– Expression for ρ3 σ 0 1 ρ3 σx = 0 0

on the states ψp1 , ψp2 , ψp3 , ψp4 : 0 −i 1 0 0 0 0 0 0 0 i 0 0 0 , ρ σ = , 3 y 0 0 0 0 −1 0 i 0 0 −i 0 0 −1 0

84

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

ρ3 σz = ψp1 ψp2 ψp3 ψp4

2.7.

1 0 0 0 0 −1 0 0 0 0 −1 0 0 0 0 1

.

= (1, 0, 0, 0) e(2πi/h)(px x+py y+pz z) , = (0, 1, 0, 0) e(2πi/h)(px x+py y+pz z) , = (0, 0, 1, 0) e(2πi/h)(px x+py y+pz z) , = (0, 0, 0, 1) e(2πi/h)(px x+py y+pz r) .

QUANTUM DYNAMICS OF ELECTRONS INTERACTING WITH AN ELECTROMAGNETIC FIELD

The dynamics of a system composed of interacting electrons and photons is considered in the realm of Quantum Field Theory (Klein-Gordon theory). The electrons are described by a ﬁeld ψ (or P , deduced from ψ), while the electromagnetic ﬁeld is described in terms of the potential (ϕ, C). An expression for the quantized Hamiltonian is given, along with the commutation rules for creation/annihilation operators. For a charge −e we have: !

2 e 2 h ∂ e h ∂ + ϕ − + Cx − m2 c2 ψ = 0. − 2πic ∂t c 2πi ∂x c x

P

=

P

=

∂ 2πi h2 + e ϕ ψ, 8π 2 c2 m ∂t h

h2 ∂ 2πi − e ϕ ψ. 8π 2 c2 m ∂t h

2

2 2πi 2πi ∂ ∂ − eϕ − + e Cx + ∂t h ∂x hc x

2

2 2πi 2πi ∂ 1 ∂ + eϕ − − e Cx + c2 ∂t h ∂x hc x 1 c2

∇2 Cx −

! 4π 2 2 2 m c ψ = 0, h2 ! 4π 2 2 2 m c ψ = 0. h2

∂ 2 Cy ∂ ∂Cx ∂ 2 Cx ∂ 2 Cz + − ∇·C = − . ∂x ∂y 2 ∂r2 ∂x∂y ∂x∂z

QUANTUM ELECTRODYNAMICS

2 2πi ∂ − eϕ ∂t h !

2 h2 ∂ 1 2 2πi − 2 + e Cx + mc ψ = 0, 8π m x ∂x hc 2

(1)

2 ∂ 2πi + eϕ ∂t h !

2 h2 ∂ 1 2 2πi − 2 − e Cx + mc ψ = 0. 8π m x ∂x hc 2

(2)

h2 8π 2 mc2

h2 8π 2 mc2

2

∂ 1 h2 ∂ 2πi 2πi − e ϕ P = − mc2 ψ + 2 + e Cx ψ, ∂t h 2 8π m x ∂x hc

85

h2 ∂ 2πi 1 2 2πi ∂ + e ϕ P = − mc ψ + 2 − e Cx ψ, ∂t h 2 8π m x ∂x hc

(3)

(4)

8π 2 mc2 2πi ∂ P, + eϕ ψ = ∂t h h2

(5)

2πi 8π 2 mc2 ∂ P. − eϕ ψ = ∂t h h2

(6)

∂ ∂ he 2πi 2πi ρ= − eϕ ψ − ψ + eϕ ψ , ψ 4πimc2 ∂t hc ∂t hc

he 2πi 2πi ∂ ∂ ψ ix = − + eϕ ψ − ψ − dx ψ , 4πimc ∂x hc ∂x hc ... .

——————– dτ = dV dt. [10 ]

10 @

Notice that, more appropriately, one should write d4 τ = d3 V dt, since dτ denotes the 4-dimensional volume element, while drmV is the 3-dimensional space volume element.

86

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

1 ∂ 2πi 2πi h2 ∂ + eϕ ψ − eϕ ψ δ 8π 2 m c2 ∂t h ∂t h

! ∂ ∂ 2πix 2πi − e Cx ψ · + e Cx ψ − ∂x hc ∂x hc x " # 2 1 1 ∂C 1 2 + ∇ ϕ − |∇ × C|2 dτ = 0. − mc ψψ + 2 8π c ∂t

(7)

From this, the variation with respect to ψ or ψ gives Eq. (1) or (2), respectively. The variation with respect to ϕ yields:

1 ∂ ∂ϕ 1 ∂C + − 4π x ∂x ∂x c ∂t

2πi ∂ 2πi ∂ he − eϕ ψ − ψ + e ϕ ψ = 0, ψ − 4πimc2 ∂t h ∂t h 1 ∇ · E − ρ = 0. 4π

(8)

The variation with respect to Cx instead gives:

1 ∂ ∂Cy ∂Cx 1 ∂ ∂ϕ 1 ∂Cx + − − − 4πc ∂t ∂x c ∂t 4π ∂y ∂x ∂y

he 2πi ∂ ∂Cx ∂Cz ∂ − − + e Cx ψ ψ − ∂z ∂z ∂x 4πimc ∂x hc

2πi ∂ − e Cx ψ = 0, −ψ ∂x hc 1 1 ∂Ex − 4πc ∂t 4π

∂Hy ∂Hz − ∂y ∂r

+ ix = 0,

and similarly for the other components. A =

=

B =

1 1 ∂Cx ∂ϕ 2 + 8π x c ∂t ∂x

1 1 ∂Cx 2 1 ∂ϕ 2 1 ∂Cx ∂ϕ , + + 8π c2 x ∂t 8π ∂x 4πc x ∂t ∂x

1 ∂Cx ∂ϕ 1 ∂Cx 2 , + 4πc2 x ∂t 4πc x ∂t ∂x

(9)

87

QUANTUM ELECTRODYNAMICS

B−A =

1 ∂Cx 2 1 ∂ϕ 2 − . 8πc2 x ∂t 8π x ∂x ——————–

Without matter ﬁelds, the conjugate Hamiltonian variables are: Cx , Cy , Cz , ϕ, [11 ] H=

1 Ex ; 4πc 1 Ey ; − 4πc 1 − Ez ; 4πc 0 −

1 1 2 1 ∂ϕ |∇ × C|2 + E + Ex , 8π 8π 4π x ∂x

∂Hy ∂Hz − , ∂y ∂z ∂ϕ ∂ϕ 1 ∂Cx C˙ x = −cEx − c , Ex = − − , ∂x ∂x c ∂t ϕ˙ = . . . 1 0˙ = 0 = − ∇ · E. 4π

E˙ x = c

In the following we consider a particle with charge −e and assume ϕ = 0. δ Ldτ = 0, with dτ = dV dt. 1 ∂ h2 ∂ ψ ψ 8π 2 m c2 ∂t ∂t

! ∂ ∂ 2πi 2πi − − e Cx ψ + e Cx ψ ∂x hc ∂x hc x ! 1 1 ∂C 2 1 2 dτ = 0. − |∇ × C|2 − mc ψψ + 2 8π c ∂t

δ

11 @

In the following, the author looked for the variable conjugate to ϕ.

(7 )

88

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

ψ, ψ,

Cx , Cy , Cz ,

h2 ∂ ψ; 8π 2 mc2 ∂t h2 ∂ P = 2 ψ; 8π mc ∂t

P =

1 ∂Cx Ex = : 4πc 4πc2 ∂t Ey 1 ∂Cy = ; − 4πc 4πc2 ∂t Ez 1 ∂Cz = . − 4πc 4πc2 ∂t −

8π 2 mc2 1 2 h2 ∂ 2πi − e Cx ψ P P + mc ψψ + 2 H = h2 2 8π m x ∂x hc

∂ 1 2πi 2 2 × + e Cx ψ + (E + H ) dV, ∂x hc 8π 2 2 1 2 h2 8π mc mc ∇ ψ · ∇ ψ+ P P + ψψ + H = h2 2 8π 2 m hc C · (ψ∇ ψ − ψ∇ ψ) + 4πimc 1 c2 2 2 2 (E + H ) dV. |C| ψψ + + 2mc2 8π

2πi e(ψP − ψP ), h

he 2πi 2πC i = − ψ ∇+ eC ψ − ψ ∇ − eC ψ 4πimc hc hc c2 he (ψ∇ ψ − ψ∇ ψ) − ψψ C. = − 4πimc mc2

ρ =

∇ · f k = 0,

f λ = ∇ ϕλ ;

∇2 ϕλ + λ2 ϕλ = 0.

∇2 f λ + λ2 f λ = 0, ∇2 f k + k 2 f k = 0.

89

QUANTUM ELECTRODYNAMICS

f λ · f λ dV = δλλ , f k · f k dV = δkk , f λ · f k dV = 0;

1 1 ϕλ ϕ dV = 2 f λ · f λ dV = 2 δλλ , λ λ uλ uλ dV = δλλ . λϕλ = uλ ;

λ

ψ = P

=

[Aλ (qλ + Qλ ) + iBλ (pλ − Pλ )] λϕλ ,

(Aλ = Bλ )

[Cλ (pλ + Pλ ) + iDλ (qλ − Qλ )] λϕλ ;

(Cλ = Dλ )

P P dV ψψ dV

% $ Cλ2 (pλ + Pλ )2 + Dλ2 (qλ − Qλ )2 , % $ = A2λ (qλ + Qλ )2 + Bλ2 (pλ − Pλ )2 .

=

2 1 8π 2 mc2 2 2 2 h m c +λ P P dV + ψψ dV h2 2m 4π 2 % 8π 2 mc2 $ 2 2 2 2 (p + P ) + D (q − Q ) C = λ λ λ λ λ λ h2 λ

$ % 2 1 2 2 2 2 2 h 2 2 (q + Q ) + B (p − P ) m c +λ + A λ λ λ λ λ λ 2m 4π 2 λ 1 1 2 1 2 1 2 h2 2 pλ + qλ + Pλ + Qλ c m2 c2 + λ2 2 , = 2 2 2 2 4π λ

2 1 h2 2 2 2 h 2 2 c2 + λ2 m c +λ c = m , B λ 4π 2 2 4π 2

2 8π 2 mc2 2 1 1 h2 2 2 2 h 2 2 c2 + λ2 D + c + λ = , m c m A λ λ h2 2m 4π 2 2 4π 2 1 8π 2 mc2 2 Cλ + h2 2m

90

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

2 1 8π 2 mc2 2 2 2 2 h Bλ2 , m C = c + λ λ h2 2m 4π 2

2 1 8π 2 mc2 2 2 2 2 h Dλ = m c +λ A2λ , h2 2m 4π 2 mc

A2λ = Bλ2 =

h2 2 m c +λ 4π 2 2 2

Cλ2

1 ψ=√ 2 λ

P =

h √ 4π 2

=

Dλ2

&

h2 = 32π 2 mc

mc m2 c2

&

λ

+

λ2 h2 /4π 2

,

2

m2 c2 + λ2

h2 . 4π 2

' ( qλ + qλ + i pλ − pλ uλ ,

( m2 c2 + λ2 h2 /4π 2 ' pλ + pλ + i qλ − qλ uλ . mc

4/i = 2(pλ qλ − qλ pλ ) + 2(pλ qλ − qλ pλ ) ± 2i(qλ qλ − qλ qλ ) ∓2i(pλ pλ − pλ pλ ), 0 = (pλ qλ − qλ pλ ) − (pλ qλ − qλ pλ ) + (pλ qλ − qλ pλ ) − (pλ qλ − qλ pλ ), 0 = (pλ qλ − qλ pλ ) − (pλ qλ − qλ pλ ) − (pλ qλ − qλ pλ ) − (pλ qλ − qλ pλ ), 0 = (pλ qλ − qλ pλ ) + (pλ qλ − qλ pλ ) ± (pλ pλ − pλ pλ ) ± (qλ qλ − qλ qλ ).

pλ qλ − qλ pλ = 1/i, pλ qλ − qλ pλ = 0, pλ pλ − pλ pλ = 0,

pλ qλ − qλ pλ = 1/i, pλ qλ − qλ pλ = 0, qλ qλ − qλ qλ = 0.

——————– −Ze =

2πi e ρ dV = h

(ψP − ψ P ) dV,

91

QUANTUM ELECTRODYNAMICS

Z = = = =

2πi − (ψP − ψP ) dV h

1 1 1 1 p2λ + qλ2 − pλ2 − qλ2 2 2 2 2 λ

1 2 1 2 1 1 2 1 2 1 − p + q − p + qλ − 2 λ 2 λ 2 2 λ 2 2 λ (Nλ − Nλ ) = Zλ . λ

λ

H = H M + HR , 0 + H1 , HM = H M M

where HM and HR account for the matter and radiation ﬁeld contribu0 is the free particle Hamiltion to the Hamiltonian, respectively. HM 1 tonian, while HM describes the particle interaction and that between particles and light quanta. 1 1 1 Nλ = pλ2 + qλ2 − , 2 2 2 1 1 1 Nλ = pλ2 + qλ2 − , 2 2 2 Zλ = Nλ − Nλ .

0 HM

1 1 1 h2 = + + qλ2 pλ2 + qλ2 c m2 c2 + λ2 2 2 2 2 2 4π λ h2 = (Nλ + Nλ )c m2 c2 + λ2 2 + zero point energy. 4π 1

p2λ

λ

——————– [12 ] 12 @ In the original manuscript, some expressions were written in terms of ν instead of k, but the warning “use k instead of ν” appears. We have therefore chosen to use the symbol k throughout.

92

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

C =

Ak Qk f k +

k

−E =

E 2 dV

=

Bλ Pλ f λ ,

λ

Ck Pk f k −

k

(∇ × f λ = 0).

Dλ Qλ f λ

λ

Ck2 Pk2 +

k

Dλ2 Q2λ .

λ

∂Cy ∂Cz − , ∂y ∂z

∂Cz 2 ∂CH 2 ∂Cx ∂Cy , = + −2 ∂y ∂r ∂y ∂x ∂Cx ∂Cy = = |∇ Cx |2 − A2k N 2 Q2k . ∂y ∂x x xy

Hx = Hx2 H2

k

H 2 dV = . . . .

Pk Qk − Qk Pk = 1/i, Pλ Qλ − Qλ Pλ = 1/i.

Ck2 8π A2k 8π Dλ2 8π

= = =

Ck = Ak = Dλ = Bλ =

1 hck , 2 2π 1 hck , 2 2π 1 ; 2 √ 2hck, 2hc , k √ √ 4π = 2 π, hc √ . π

93

QUANTUM ELECTRODYNAMICS

1 1 Nk = (Pk2 + Q2k ) − . 2 2

C =

k

−E =

hc 2hc √ Pλ f λ , Qk f k + k π λ

√

2hckPk f k −

√

k

4πQλ f λ .

λ

νk = c

k . 2π

1 ck 2 1 2 h (Qk + Pk2 ) + Qλ 2 2π 2 k λ 1 1 2 = Qλ (Pk2 + Q2k ) hνk + 2 2 k λ 1 = N hνk + Q2λ + rest energy. 2

HR =

k

λ

——————– [13 ] 1 ∇ψ = √ 2 λ

&

∇ uλ = ∇ λϕλ = λf λ , ' ( mc qλ + qλ + i(pλ − pλ ) λf λ , m2 c2 + λ2 h2 /4π 2

1 4 2 2 2 2 2 (m c + λ h /4π ) (m2 c2 + λ h2 /4π 2 ) λλ ' ( × (pλ − pλ )(qλ + qλ ) − (pλ − pλ )(qλ + qλ ) λ uλ f λ .

ψ∇ ψ − ψ∇ ψ = −imc

∇ · ϕλ f λ = f λ · f λ − λ 2ϕλ ϕλ . 13 @ In the original manuscript, the expression ∇ u = ∇ λu = λf was written down, λ λ λ which is evidently incorrect.

94

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

ψ =

P

=

1 √ 2 λ

&

' ( mc qλ + qλ + i(pλ − pλ ) uλ , 2 2 2 2 2 m c + h λ /4π & ( m2 c2 + h2 λ2 /4π 2 ' h √ pλ + pλ + i(qλ − qλ ) uλ . mc 4π 2 λ

[14 ] 1 aλ = √ (qλ + ipλ ), 2

1 bλ = √ (qλ + ipλ ), 2

1 aλ = √ (qλ − ipλ ), 2

1 bλ = √ (qλ − ipλ ). 2

[aλ , aμ ] − [bλ , bμ ] − [aλ , bμ ] − [bλ , aμ ] = 2δλμ , −[aλ , aμ ] + [bλ , bμ ] + [aλ , bμ ] − [bλ , aμ ] = 2δλμ . [x, y] = xy ∓ yx, where the upper/lower sign refers to Einstein/Fermi particles. [aλ , aμ ] + [bλ , bμ ] + [aλ , bμ ] + [bλ , aμ ] = 0, [aλ , aμ ] + [bλ , bμ ] + [aλ , bμ ] + [bλ , aμ ] = 0, [aλ , aμ ] + [bλ , bμ ] + [aλ , bμ ] + [bλ , aμ ] = 0, [aλ , aμ ] + [bλ , bμ ] − [aλ , bμ ] − [bλ , aμ ] = 0, [aλ , aμ ] + [bλ , bμ ] − [aλ , bμ ] − [bλ , aμ ] = 0, [aλ , aμ ] + [bλ , bμ ] − [aλ , bμ ] − [bλ , aμ ] = 0, [aλ , aμ ] − [bλ , bμ ] + [bλ , aμ ] − [aλ , bμ ] = 0, [aλ , aμ ] − [bλ , bμ ] + [bλ , aμ ] − [aλ , bμ ] = 0.

2.8.

CONTINUATION ψ =

P

=

&

mc mc (aλ + bλ ) uλ , 2 2 m c + h2 λ2 /4π 2 λ & m2 c2 + h2 λ2 /4π 2 hi (aλ − bλ ) uλ , 4π mc λ

14 @ In the original manuscript the simple formulas (a − ib)(a + ib) = a2 + b2 + i(ab − ba) and (a + ib)(a − ib) = a2 + b2 − i(ab − ba) are noted on the side.

95

QUANTUM ELECTRODYNAMICS

ψ =

&

λ

P

hi = − 4π λ

mc m2 c2

+ h2 λ2 /4π 2

(aλ + bλ ) uλ ,

& m2 c2 + h2 λ2 /4π 2 (aλ − bλ ) uλ . mc

From the commutation relations reported at the end of the previous Section, we deduce that: ( ' [aλ , aμ ] + bλ , bμ = 0, ( ' ( ' aλ , bμ + bλ , aμ = 0, ( ' [aλ , aμ ] + bλ , bμ = 0, ( ' [aλ , bμ ] + bλ , aμ = 0, [aλ , aμ ] + [bλ , bμ ] = 0, [aλ , bμ ] + [bλ , aμ ] = 0, ( ' [aλ , aμ ] + bλ , bμ = 0, [bλ , bμ ] + [aλ , bμ ] = 0; ( ' [aλ , aμ ] − bλ , bμ = 2δλμ , ( ' [aλ , aμ ] − bλ , bμ = −2δλμ . 0 = [a + ib, a + ib] = [a, a] − [b, b] + i[a, b] + i[b, a], 0 = [a − ib, a − ib] = [a, a] − [b, b] − i[a, b] − i[b, a], 0 = [a + ib, a − ib] = [a, a] + [b, b] − i[a, b] + i[b, a], 0 = [a − ib, a + ib] = [a, a] + [b, b] + i[a, b] − i[b, a]; [a, a] = [b, b] = [a, b] = [b, a] = 0.

2.9.

QUANTIZED RADIATION FIELD

The author again considered the quantization of the electromagnetic ﬁeld, but using now another expansion in a basis diﬀerent from that adopted in Sects. 2.4, 2.5. In the original manuscript, the present Section and the following four Sections are placed in the Quaderno 17 just after what has been here reported in Sect. 7.1. E=−

1 ∂C , c ∂t

1 ∂E 1 ∂2C 2 = ∇2 C = − . c2 ∂t2 c ∂t

96

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

Cx , −

Cy ,

Cz ;

Ey Ez Ex , − , − . 4πc 4πc 4πc

γ1 , γ2 , γ3 = 0, ±1, ±2, . . . ; c γ12 + γ22 + γ32 ; γ= k h h h px = γ1 , py = γ2 , pz = γ3 . k k k |ks | = 1,

ks = k−s .

s s s 1 f s = ks e2πi(γi x/k+γ2 y/k+γ3 z/k) √ . k3

[15 ] C= E=

) )

as f s , bs f s .

as = a ˜−s , bs = ˜b−s . as as − as as = 0, bs bs − bs bs = 0,

as˜bs − ˜bs as =

2hc δs,s . i

√ In the original manuscript, the normalization factor 1/ k3 is incorrectly treated as a denominator instead of a numerator. 15 @

97

QUANTUM ELECTRODYNAMICS

˙ = −c E = C

−c bs f s ;

˙ = −c ∇2 C = E

4π 2 ν 2 s

c

as f s .

a˙ s = −c bs , 4π 2 νs2 as . b˙ s = c

c c d as + bs = −c bs − 2πνs i as = −2πνs i as + bs , dt 2πνs i 2πνs i

c c d as − bs = −c bs + 2πνs i as = 2πνs i as − bs . dt 2πνs i 2πνs i

As = as +

c bs , 2πνs i

Bs = as −

c bs ; 2πνs i

A˙ s = −2πνs i As , B˙ s = 2πνs i Bs ; A˜s = B−s , ˜s = A−s . B As Bs − Bs As = 0, ˜s − B ˜s A˜s = 0, A˜s B ˜s − B ˜s As = 0, As B

2hc2 As A˜s − A˜s As = , πνs

˜s Bs = − ˜s − B Bs B

2hc2 . πνs

98

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

As At − At As = 0, Bs Bt − Bt Bs = 0, A˜s A˜t − A˜t A˜s = 0, ˜t B ˜t − B ˜s = 0, ˜s B B ˜t As = 0, ˜t − B As B A˜s Bt − Bt A˜s = 0,

2hc2 As A˜t − A˜t As = δst , πνs 2

˜t − B ˜t Bs = − 2hc δst . Bs B πνs 1 Zs = c

πνs As . 2h

Zs Zt − Zt Zs = 0, Z˜s Z˜t − Z˜t Z˜s = 0, Zs Z˜t − Z˜t Zs = δst . Z˜s Zs = ns . √ < ns |Zs |ns+1 >= ns + 1, √ < ns |Z˜s |ns−1 >= ns . [16 ] As = c

16 @

2h Zs , πνs

In the original manuscript, the unidentiﬁed Ref. 5.45 is here alluded to.

99

QUANTUM ELECTRODYNAMICS

As + A˜−s 2h Zs + Z˜−s as = =c , 2 πνs 2 bs =

Ws = = = =

2πνs i As − A˜s = i 2hπνs (Zs − Z˜−s ). c 2

1 ˜ 4π 2 νs2 a ˜s as bs bs + 8π c2 $ % 1 2hπνs (Z˜s − Z−s )(Zs − Z˜−s ) + (Z˜s + Z−s )(Zs + Z˜−s ) 8π 1 hνs {2Z˜s Zs + 2Z−s Z˜−s } 4

1 Z˜s Zs + Z−s Z˜−s hνs = hνs . ns + 2 2

fs = f −s =

1

s

k 3/2 1

s

e−2πi(γ1 x/k+γ2 y/k+γ3 z/k) ks = f s . s

k 3/2

s

e2πi(γ1 x/k+γ2 y/k+γ3 z/k) ks ,

fs =

s

s

1 2πiγ ·r /k s e ks , k 3/2

with r = (x, y, z).

c C= 2 s E=

2h (Zs f s + Z˜s f s ), πνs

i 2hπνs (Zs f s − Z˜s f s ). s

100

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

2hπ √ Zs Zt e2πi(γ s +γ t )·r /k ν ν k · k s t s t k 3 s,t +Z˜s Z˜t e−2πi(γ s +γ t )·r /k − Zs Z˜t e2πi(γ s −γ t )·r /k −Z˜ Z e2πi(−γ s +γ t )·r /k .

E 2 (r) = −

s t

[17 ] 2hπ √ Zs Zt e2πi(γ s +γ t )·r /k ν ν k · k s t s t k 3 s,t +Z˜s Z˜t e−2πi(γ s +γ t )·r /k − Zs Z˜t e2πi(γ s −γ t )·r /k −Z˜ Z e2πi(−γ s +γ t )·r /k .

H 2 (r) = −

s t

2.10.

WAVE EQUATION OF LIGHT QUANTA

Quantized ﬁelds of the electromagnetic interaction were again considered in these pages, with an emphasis (the name of this Section is the original one) on the deﬁnition of a wavefunction ψ for the photon. Matrix elements of the annihilation and creation operators Z, Z˜ were reported in the subsequent Section, along with quantum expressions for the photon energy and angular momentum. [18 ] C= as = c

as f s ,

2h Zs + Z −s , πνs 2

fs =

1 k 3/2

E=

bs f s ;

bs = i 2hπνs (Zs − Z −s ).

e2πiγ

s ·r/h

ks ,

f s = f −s . 17 C 18 @

∼ (e2πiγr/k , 0, 0), H ∼ (0, 2πi(γ/k) e2πγr/k , 0) . The original manuscript alludes here to the unidentiﬁed Ref. 11.20.

101

QUANTUM ELECTRODYNAMICS

γ s = (γ1s , γ2s , γ3s ), γ1 , γ2 , γ3 = 0, ±1, ±2, ±3, . . . ;

νs =

c s γ , k

ψ=

C=

s

E=

c

hνs =

hc s γ . k

Zs f s .

2h Zs + Z −s 2h Zs f s + Z s f s fs = , c πνs 2 πνs 2 s

i 2hπνs (Zs − Z −s )f s = i 2hπνs (Zs f s − Z s f s ). s

2.11.

s

CONTINUATION

∇ · C = 0. 1 ∂E = ∇ × ∇ × C = −∇2 C, c ∂t 1 ∂ 1 ∂H − ∇×E = ∇ × C. c ∂t c ∂t

C = ∂C ∂t

=

∇2 C = E = ∂E ∂t

=

h (Zs f s + Z˜s f s ), 2πνs h (Z˙ s f s + Z˜˙ s f s ), c 2πνs 2πνs 2hπνs (Zs f s + Z˜s f s ); c i 2hπνs (Zs f s − Z˜s f s ), i 2hπνs (Z˙ s f s − Z˜˙ s f s ). c

102

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

i 2hπνs (Z˙ s − Z˜˙ −s ) − 2πνs 2hπνs (Zs + Z˜−s ) = 0, h (Z˙ s + Z˜˙ −s ) + i 2hπνs (Zs − Z˜−s ) = 0. 2πνs Z˙ s − Z˜˙ −s = −2πiνs (Zs + Z˜−s ), Z˙ s + Z˜˙ −s = −2πiνs (Zs − Z˜−s ). Z˙ s = −2πiνs Zs ,

E2 dτ 8π

=

H2 dτ 8π

=

Z˜˙ s = 2πiνs Z˜s ,

Z˜˙ −s = 2πiνs Z˜−s .

hνs

(Zs − Z˜−s )(Z˜s − Z−s ) 4 hνs (Zs Z˜s + Z˜−s Z−s − Zs Z−s − Z˜−s Z˜s ) = 4 " # hνs Zs Z˜s + Z˜s Z˜s Zs Z−s + Z˜s Z˜−s − = . 2 2 2 hνs

(Zs + Z˜−s )(Z˜s + Z−s ) 4 hνs (Zs Z˜s + Z˜−s Z˜−s + Zs Z−s + Z˜−s Z˜s ) = 4 " # hνs Zs Z˜s + Z˜s Zs Zs Z−s + Z˜s Z˜−s = + . 2 2 2

Zs Z˜s + Z˜s Zs E2 + H 2 dτ = hνs . 8π 2 eiLx (0, 0, 1) = f s , iLeiLx (0, −1, 0) = ∇ × f s f −s × ∇ × f s = iL(1, 0, 0).

103

QUANTUM ELECTRODYNAMICS

Let us denote with r s a unitary vector along the propagation direction:

E×H dτ 4πc

hνs (Zs − Z˜−s )(Zs + Z˜−s )r s 2c hνs r s (Z˜s Zs − Z−s Z˜−s − Z−s Zs − Z˜s Z˜−s ) = 2c hνs Z˜s Zs + Zs Z˜−s rs . = c 2

=

−

Zs Z˜s − Z˜s Zs = 1. Z˜s Zs = X. Zs X − XZs = (Zs , X) = Zs , Z˜s X − X Z˜s = (Z˜s , X) = −Z˜s ,

Zik (Xk − Xi ) = 1, Z˜ik (Xk − Xi ) = −1.

< X|Z|X + 1 > = f (X), ˜ < X + 1|Z|X > = f˜(X). ˜ ˜ − 1 >< X − 1|Z|X >= |f (X − 1)|2 , < X|ZZ|X > = < X|Z|X ˜ ˜ < X|Z Z|X > = < X|Z|X + 1 >< X + 1|Z|X >= |f (X)|2 ; |f (X)|2 = X + 1, |f (X0 )|2 = 1, X0 = 0. |f (X)|2 = |f (X − 1)|2 + 1, |f (X0 )|2 = 1. ˜ < X0 |ZZ|X 0 > = 0, ˜ < X0 |Z Z|X0 > = |f (0)|2 .

Z˜s Zs = ns , < ns |Zs |ns + 1 > =

√

(ns = 0, 1, 2 . . .)

ns + 1,

√ ns . < ns + 1|Z˜s |ns > =

104

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

f s = f −s .

E2 + H 2 dτ 8π

2.12.

E×H dτ 4πc

1 = hνs ns + , 2

hνs 1 = r s ns + . c 2

FREE ELECTRON SCATTERING

The interaction between electrons and electromagnetic radiation was here studied in detail, and expressions for the matrix elements of the interaction energy (as well as for the transition probability) were explicitly obtained. Some care was also devoted to the kinematics of the process here considered. The material reported in this Section starts with that present in Quaderno 17 on the page following 151bis, but the complete study of the subject starts at page 133 of the same Quaderno.

e W + ρ1 σ · p + C + ρ3 mc ψ = 0. c c

Using Dirac coordinates: r r r 1 ψr = ur √ e2πi(Γ 1 x/k+Γ 2 y/k+Γ 3 z/k) . k3 ˜u ur = 1, Γ = Γ r1 + Γ r2 + Γ r3 . ur = (ur1 , ur2 , ur3 , urr ), u

Er = ±c m2 c2 +

h2 2 Γ . k2

H = H0 + I, H0 = −c ρ1 σ · p − ρ3 mc2 +

ns hνs ,

s

e I = −c ρ1 σ · C = −e ρ1 σ · C. c

105

QUANTUM ELECTRODYNAMICS

< . . . |H0 | . . . > = Er +

ns hνs ,

√ ec 2h < r; ns . . . |I|r ; ns + 1 . . . > = − ns + 1 2 πνs * × ψr ρ1 σ · f s ψr dτ,

√ ec 2h < r; ns . . . |I|r ; ns − 1 . . . > = − ns 2 πνs × ψ*r ρ1 σ · f −s ψr dτ.

ψr = ur

f −s

1

e2πiΓ

r

·r/k

,

r

·r/k , k 3/2 1 s = ks 3/2 e2πiγ ·r/k , k 1 s = f s 3/2 e−2πiγ ·r/k . k

ψr = ur fs

1 k 3/2

e2πiΓ

ks = k−s .

ψ*r ρ1 σ · f s ψr dτ = k −7/2 u ˜r ρ1 σ · ks ur r r × e2πi(Γ +γs −Γ )·r/k dτ =

u ˜r ρ1 σ · ks ur δ r r , Γ , Γ +γs k 7/2

u ˜r ρ1 σ · ks ur δ r r . ψ*r ρ1 σ · f −s ψr dτ = Γ , Γ −γs k 7/2

ec √ 2h < r; ns ns + 1 s + 1... > = − 3/2 πνs 2k ×u ˜r ρ1 σ · ks ur δ r r , Γ ,Γ +γs ec √ 2h < r; ns . . . |I|r ; ns − 1 . . . > = − 3/2 ns πνs 2k ×u ˜r ρ1 σ · ks ur δ r r . Γ ,Γ −γs . . . |I|r ; n

106

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

For t = 0: a1 = 1, a2 , . . . = 0. For t → 0: 2πi 2πi(Ei −E1 )t/h e Hi1 ; a˙ i = − h

ai = −

1 e2πi(Ei −E1 )t/h − 1 Hi1 . Ei − E1

H12 = 0.

a˙ 2

2πi −1 2πi(E2 −Ei )t/h 2πi(Ei −E1 )t/h = − e − 1 H2i Hi1 e h Ei − E1 i 2πi 1 = e2πi(E2 −E1 )t/h − e2πi(E2 −Ei )t/h H2i Hi1 ; h Ei − E1 i

a2 =

i

2

1 e2πi(E2 −E1 )t/h − 1 (Ei − E1 )(E2 − E1 ) 1 2πi(E2 −Ei )t H2i Hi1 . e − (E2 − Ei )(Ei − E1 )

electron

radiation

b

nt = 1

i, i r, r 1

a

n = 1, ns = 1 t

ns = 1

Γ a + γs = Γ b + γt = Γ r = Γ r + γs + γt

s, t label the incident and the scattered quanta, respectively.

107

QUANTUM ELECTRODYNAMICS

ec < b; 0, 1 . . . |I|r; 0, 0 > = − 3/2 2k ec < r ; 0, 0 . . . |I|a; 1, 0 > = − 3/2 2k

ec < b; 0, 1 . . . |I|r; 1, 1 > = − 3/2 2k ec < r ; 1, 1 . . . |I|a; 1, 0 > = − 3/2 2k

2h u ˜b ρ1 σ · kt ur , πνt 2h u ˜r ρ1 σ · ks ua , πνs 2h u ˜b ρ1 σ · ks ur , πνs 2h u ˜r ρ1 σ · kt ua . πνt

The probability for a transition at a time t to occur is (taking into account only the term with the resonance denominator equal to E1 − E2 in the expression for a2 ):

P12

2 sin2 [π(E2 − E1 )t/h] H2i Hi1 = · 4 . (E2 − E1 )2 Ei − E1 i

pa =

h a Γ , k

pr =

h a (Γ + γ s ), k

pb =

h b Γ , k

pr =

h b (Γ − γ t ). k

Γ = Γ a + γs = Γ b + γt, Γ b = Γ a + γs − γt.

h2 m2 c2 + 2 Γ a2 , k h2 2 Eb = c m2 c2 + 2 Γ b , k h2 Er = ±c m2 c2 + 2 (Γ a + γ s )2 , k 2 h2 Er = ±c m2 c2 + 2 Γ b − γ t . k Ea = c

108

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

m2 c2 +

E1 = c

E2 = c

m2 c2 +

h2 a2 Γ + hνs , k2 h2 a (Γ + γs − γt )2 + hνt , k2

Ei = ±c

m2 c2 +

Ei = ±c

m2 c2 +

h2 a (Γ + γs )2 , k2 h2 a (Γ − γ t )2 + hνs + hνt . k2

Let us denote by u the spin function for a plane wave with momentum px , py , pz and by u0 that for a wave of zero momentum. α·p 0 u , u = f1 ∓ f2 p p

where the upper/lower sign refers to positive/negative energy waves. & f1 =

1 + 1 + p2 /m2 c2 , 2 1 + p2 /m2 c2

& f2 =

−1 + 1 + p2 /m2 c2 ; 2 1 + p2 /m2 c2

|f12 | + |f22 | = 1.

α = ρ1 σ. α · pb 0 ub = f1b − f2b ub , pb a b α · pa u0a , u a = f1 − f2 pa We consider positive waves ua , ub .

α · pr 0 ur f1r ∓ f2r ur , pr r r α · pr u r f1 ∓ f2 u0r . pr

QUANTUM ELECTRODYNAMICS

109

[19 ] 1) Positive ur : ˜ r α · k s ua u ˜ b α · k t ur u

0 b b α · pb r r α · pr =u ˜b f1 − f2 α · kt f1 − f2 u0r pb pr

0 r r α · pr a a α · pa ×u ˜r f1 − f2 α · ks f1 − f2 u0a pr pa 0 b r α · pr b α · pb r =u ˜b f1 α · kt f2 + f2 α · kt f1 u0r pr pb 0 r a α · pa r α · pr a ×u ˜r f1 α · ks f2 + f2 α · ks f1 u0a . pa pr 2) Negative ur : ˜ r α · k s ua u ˜ b α · k t ur u α · pb α · pr 0 ur =u ˜0b f1b f1r α · kt − f2b f2r α · kt pb pr α · pa 0 0 r a r a α · pr ur . ×u ˜r f1 f1 α · ks − f2 f2 α · ks pr pa 3) Positive ur : ˜ r α · k t ua = . . . u ˜ b α · k s ur u [which is obtained from 1) with the replacements r → r , ks → kt , kt → ks ]. 4) Negative ur : ˜ r α · k t ua = . . . u ˜ b α · k s ur u [which is obtained from 1) with the replacements r → r , ks → kt , kt → ks ].

19 @

The original manuscript alludes here to the unidentiﬁed Ref. 10.40.

110

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

1)

u ˜ b α · k t ur u ˜ r α · k s ua

positive ur

α · pr α · pb =u ˜0b f1b f2r α · kt + f1r f2b α · kt pr pb α · pa r a r a α · pr + f2 f1 α · ks u0a × f1 f2 α · k s pa pr σ · pr σ · pb =u ˜0b f1b f2r σ · kt + f2b f1r σ · kt pr pb σ · pa r a r a σ · pr + f2 f1 σ · ks u0a . × f 1 f2 σ · k s pa pr

(σ · kt )(σ · pr ) = kt · pr + iσ · kt × pr , (σ · kt )(σ · pr )(σ · ks )(σ · pr ) = (kt · pr )(ks · pa ) + i(kt · pr )(σ · ks × pa ) + i(ks · pa )(σ · kp × pr ) − (σ · kt × pr )(σ · ks × pa ) =

(kt · pr )(ks · pa ) + i(kt · pr )(σ · ks × pa ) +i(ks · pa )(σ · kt × pr ) − (kt × pr )(ks × pa ) −i[σ, (kt × pr ) × (ks × pa )].

For ua = u0a , pa = 0: f1a = 1, f2a = 0. ——————– 1)

u ˜b positive ur =

α · k t ur u ˜ r α · k s ua u ˜0b

σ · pr σ · pr b r b r σ · pb + f2 f1 σ · kt f2r σ · ks u0a . f1 f2 σ · k t pr pb pr

For ks · pr = 0: (σ · kt )(σ · pr )(σ · pr )(σ · ks ) = p2r (σ · kt )(σ · ks ) = p2r (kt · ks ) + ip2r (σ · kt × ks ), (σ · pb )(σ · kt )(σ · pr )(σ · ks ) = (pb · kt + iσ · pb × kt ) iσ · pr × ks = −(pb × kt ) · (pr × ks ) + i(pb · kt )(σ · pr × ks ) − iσ · (pb × kt ) × (pr × ks ).

111

QUANTUM ELECTRODYNAMICS

2)

u ˜ b α · k t ur u ˜ r α · k s ua

negative ur

=

u ˜0b

σ · pr r b b b b σ · pb f1 f1 σ · k t − f2 f2 f1 σ · ks u0a . σ · kt pb pr

3)

u ˜b positive ur =

α · k s ur u ˜ r α · k t ua

u ˜0b

σ · pr σ · pr b r b r σ · pb − f2 f1 σ · ks f2r σ · kt u0a . f1 f2 σ · k s pr pb pr

4)

u ˜b negative ur =

α · k s ur u ˜ r α · k t ua u ˜0b

σ · pr r b r b r σ · pb f1 f1 σ · k s − f2 f2 f1 σ · kt u0a . σ · ks pb pr ——————–

Let us denote with η the average value with respect to u0b and u0a : ˜ 0 = 1u ˜ 0a = 1 [(AA) ˜ 22 ]. ˜ 11 + (AA) |˜ u0b Au0a |2 = u ˜0b Au0a Au ˜0b AAu b 2 4 A = A0 + iσ · B, AA˜ = [A0 + iσ · B][A0 − iσ · B] ˜ + iσ · B × B. = A0 A0 + iA0 σ · B − iA0 σ · B + B · B AA˜ = A0 A0 + B · B,

1 1 |˜ ub0 Aua0 |2 = A0 A0 + B · B. 2 2

γ s = (γs , 0, 0), ks = (0, 0, 1), γ t = (γt sin ϑ cos ϕ, γt sin ϑ sin ϕ, γt cos ϑ). Near the resonance we have: νs . νt = hνs 1+ (1 − sin ϑ cos ϕ) mc2 pr =

hνs (1, 0, 0), c

pr = −

hνt (sin ϑ cos ϕ, sin ϑϕ, cos ϑ), c

112

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

hνt pb = c

hνs (1 − sin ϑ cos ϕ), − sin ϑ sin ϕ, − cos ϑ . 1+ mc2 E1 = mc2 + hνs , E1 = ± m2 c4 + h2 νt2 + hνs + hνt , Ei = ± m2 c4 + h2 νs2 , Er ∼ E1 .

2.13.

BOUND ELECTRON SCATTERING

Let us consider f bound electrons; the unperturbed ) energy of the system interacting with an electromagnetic ﬁeld is En + s ns hνs . Denoting with ψa (q1 , . . . , qf ) the electron wavefunction corresponding to energy Ea , the interaction with the electromagnetic ﬁeld is described by: & h(ns + 1) 2πνs f × ψ˜a αi · f s (q1 ) ψf dτ,

< a; ns . . . |I|b; ns + 1 . . . > = −e c

i=1

hns < a; ns . . . |I|b; ns − 1 . . . > = −e c 2πνs f × ψ˜a αi , f s (q1 ) ψf dτ. i=1

αi = ρi1 σ i . In ﬁrst approximation, λ |qi |; fs (qi ) ∼ fs (0) =

ks . k 3/2

For coherent scattering , by labelling with S, t the incident and scattered quantum, respectively, with wave-vectors ks , kt , we have:

113

QUANTUM ELECTRODYNAMICS

ec < a; 0, 1, . . . |I|b; 0, 0 . . . > = − 3/2 k ec < b; 0, 1, . . . |I|a; 1, 0 . . . > = − 3/2 k

h 2πνt h 2πνs

ψ˜a

f

αi · kt ψb dτ,

i=1

ψ˜b

f

αi · ks ψa dτ,

i=1

for resonant scattering, or otherwise

ec < a; 0, 1, . . . |I|b; 1, 1 . . . > = − 3/2 k ec < b; 1, 1, . . . |I|a; 1, 0 . . . > = − 3/2 k

h 2πνs h 2πνt

ψ˜a

f

αi · ks ψb dτ,

i=1

ψ˜b

f

αi · kt ψa dτ.

i=1

For t = 0: a1 = 1, a2 = 0, ni = 0; H12 = 0, H1i , H2i = 0. For t ∼ 0: a˙ i = −

ai = −

2πi 1 Hi1 e2πi(Ei −E1 )t/h − ai . h 2T

e−t/2T e2πi(Ei −E1 )t/h+t/2T − 1 Hi1 . Ei − E1 + (h/4πiT )

tT :

a˙ 2 =

ai =

−Hi1 e2πi(Ei −E1 )t/h . Ei − E1 + (h/4πiT )

2πi H2i Hi1 e2πi(Ei −E1 )t/h . h Ei − E1 + (h/4πiT ) i

" a2 =

i

H2i Hi1 Ei − E1 + (h/4πiT )

#

——————–

e2πi(E2 −E1 )t/h − 1 . E2 − E1

114

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

When a variable magnetic ﬁeld H = H(t) is included in the interaction, we have to consider also the diagonal magnetic moments μi . For Hx = Hy = 0, Hz = H(t): 2πi H(t) μ1 a1 , h (2πi)/h μ1 Hdt a1 = e . a˙ 1 =

a˙ i = −

(2πi)/h μ1 Hdt 2πi 2πi 1 Hi1 e2πi(Ei −E1 )t/h e ai + H μi ai . − h 2T h

R 2πi ai = e−t/2T e(2πi)/h μi Hdt − Hi1 h ⎤ ⎡ 2πi(E − E )t/h + t/2T + (2πi)/h (μ − μ ) Hdt i 1 1 i ⎥ ⎢ dt + C ⎦ . ×⎣ e

a˙ 2 = −

2πi 2πi Hμ2 a2 . H2i e2πi(E2 −E1 )t/h ai + h h i

(2πi/h) μ2 Hdt a2 = ⎡ − 2πi h e ⎤ t 2πi(E2 − Ei )t/h − (2πi/h) μ2 Hdt ⎥ ⎢ e ai dt⎦ . ×⎣ H2i 0

H = H0 cos 2πνt, Hdt =

H0 sin 2πνt, 2πν

H0 (μ1 − μ − i) 2π (μ1 − μi ) Hdt = sin 2πνt, h hν

115

QUANTUM ELECTRODYNAMICS

(2πi/h)(μ1 − μi ) Hdt e = ei[H0 (μ1 − μi )/hν] sin 2πνt = eiAi sin 2πνt ,

H0 (μ1 − μi ) . hν

Ai =

[20 ] eiAi sin 2πνt = ci0 + cii e2πνit + ci−1 e−2πνit + ci2 e4πνit + ci−2 e−4πνit + . . . . ω = 2πνt: eiAi sin ω = ci0 + ci1 eiω + ci−1 e−iω + ci2 e2iω + ci−2 e−2iω + . . . .

ci0 ζ = eiω ,

1 = 2π

sin ω =

2π

eiAi sin ω dω. 0

ζ − ζ −1 , 2i

e

Ai (ζ−ζ −1 )/2

1 2πi

1

ζ −ζ

−1 n

=

n r=0

20 @

dζ ; ζ

1 Ai (ζ−ζ −1 )/2 e dζ. ζ

ζ − ζ −1 A2i = 1 + Ai + 2 2!

dω = −i

1 Ai (ζ−ζ −1 )/2 e dζ. iζ

eiAi sin ω dω = ci0 =

dζ = iζdω,

ζ − ζ −1 2

ζ

n−2r

2

n r

A3 + i 3!

ζ − ζ −1 2

(−1)r ,

The original manuscript alludes here to the unidentiﬁed Ref. 11.05.

3 +....

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E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

ζ −ζ

−1 2n

2n 2n r 2n−2r = (−1) ζ r r=0

n 2n 2n ζ −2s (−1)s (−1)n (−1)n = n+s n s=−n

=

(2n)! (−1)n . n!2

ci0 = 1 −

2.14.

A2i A4i + − . . . = I0 (Ai ). 1 · 22 2!2 · 24

RETARDED FIELDS

The possibility is considered, in the following pages, of introducing an intrinsic constant time delay τ (or an intrinsic space constant ε = cτ ) in the expressions for the electromagnetic retarded ﬁelds, generically denoted with f (x, y, z, t). f = f (x, y, z, t). r = f (x, y, z, t). ϕ(x, y, z, t) = f x, y, z, t − c x r r − ft x, y, z, t − ϕx (x, y, z, t) = fx x, y, z, t − c rc c x = fx (x, y, z, t) − ft (x, y, z, t), rc r 2x r − fxt x, y, z, t − ϕx (x, y, z, t) = fx2 x, y, z, t − c rc c 2 2 2 x r −x r r + 2 2 ftt x, y, z, − − f x, y, z, t − r c c r3 c t c 2 2x x (x, y, z, t) − f (x, y, z, t) + 2 2 ftt (x, y, z, t) = fxx rc xt r c r2 − x2 − 3 ft (x, y, z, t). r c r = ft (x, y, z, t), ϕt (x, y, z, t) = ft x, y, z, t − c r = ftt (x, y, z, t). ϕtt (x, y, z, t) = ft x, y, z, t − c

117

QUANTUM ELECTRODYNAMICS

= ∇2 −

1 ∂2 : c2 ∂t2

r 2 ∂2 ϕ(x, y, z, t) = ∇2 f x, y, z, t − − (x, y, z, t) c c ∂r∂t 2 − ft (x, y, z, t), rc x r 1 r ∂ ϕ(x, y, z, t) = fx x, y, z, t − − ft x, y, z, t − ∂r r c c c x = ∂2 ϕ(x, y, z, t) = ∂r∂t

ϕ +

∂ 1 f (x, y, z, t) − ft (x, y, z, t), ∂z c ∂2 1 f (x, y, z, t) − ft (x, y, z, t). ∂r∂t c

2 2 2 ∂2 ϕ = ∇2 f − 2 ft − ft c ∂z∂t c rc 1 2 = f − 2 ft − ft . c rc

f = ∇2 ϕ +

2 2 ∂2 ϕ + ϕ. rc c ∂z∂t

——————– " ϕ(x, y, z, t) = f

√

x, y, z, t −

r 2 + ε2 c

# = f*(x, y, z, t).

# r 2 + ε2 , f (x, y, z, t) = ϕ x, y, z, t − c # " √ 2 + ε2 r fx (x, y, z, t) = ϕx x, y, z, t − c " # √ 2 + ε2 x r + √ ϕt x, y, z, t + , c2 c r 2 + ε2 "

√

118

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

"

# r 2 + ε2 fxx (x, y, z, t) = ϕx x, y, z, t + c " # √ r 2 + ε2 2x ϕxt x, y, z, t + + √ c2 c r 2 + ε2 " # √ r 2 + ε2 r2 + ε2 − x2 ϕ x, y, z, t + + 2 c c(r + ε2 )3/2 t # " √ r 2 + ε2 x2 ϕ , + 2 2 x, y, z, t + c (r + ε2 ) tt c

√

"

ftt (x, y, z, t) = ϕtt

√ x, y, z, t +

"

r 2 + ε2 c

# .

# r 2 + ε2 ftt (x, y, z, t) = ∇ ϕ x, y, z, t + c # " √ r 2 + ε2 ε2 ϕ − 2 2 x, y, t + c (r + ε2 ) tt c # " √ 2 + ε2 2r2 + 3ε2 r + √ ϕt x, y, z, t + c c( r2 + ε2 )3 " # √ r 2 + ε2 2r ∂2 + √ ϕ x, y, z, t + . c c r2 + ε2 ∂r∂t

2 = ∇2 ϕ − f

2.14.1

√

2

2r2 + 3ε2 ε2 2z ∂2 √ ϕ ¨ + ϕ. ϕ ˙ + c2 (r2 + ε2 ) c(r2 + ε2 )3/2 c r2 + ε2 ∂r∂t

Time Delay

With the introduction of a time delay τ , which is a universal constant (classically τ = 0), by setting ε = τc ,

QUANTUM ELECTRODYNAMICS

we get:

Φ=

119

√

z2 + ε 2 1 √ S t− , x, y, z dx dy dz, c r 2 + ε2

and, for ε → 0: r 1 S t − , x, y, z dx dy dz Φ = r c r 1 , x, y, z dx dy dz S t − −ε2 2r3 c r 1 ˙ S t − , x, y, z dx dy dz + . . . . + 2r2 c c

2.15.

MAGNETIC CHARGES

A modiﬁcation of the classical Maxwell equations was considered in the following pages, in order to include also the eﬀect of magnetic charges. ∇ · g(q ) 1 dq . A(q) = − 4π r ∇ · g(q ) 1 0 ∇ dq , g = − 4π r g1 = g − g0. g = (δ(q − q0 ); 0; 0), ∇ · g = δ (x − x0 ) δ(y − y0 ) δ(z − z0 ). r = |q − q|:

δ (x − x0 ) δ(y − y0 ) δ(z − z0 ) dq r δ (x − x0 ) = dx 2 2 2 (y0 − y) + (z0 − z) + (x − x) x − x = δ(x − x0 ) dx 3/2 2 2 2 [(y0 − y) + (z0 − z) + (x − x) ] x − x0 x − x0 =− =− . 3/2 R3 [(x − x0 )2 + (y − y0 )2 + (z − z0 )2 ]

120

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

g10 =

3(x − x0 )2 − R2 , R5 g30 =

g11 = δ(q − q0 ) −

g20 =

3(x − x0 )(y − y0 ) , R5

3(x − x0 )(z − z0 ) ; R5

3(x − x0 )2 − R2 , R5 g31 = −

g21 = −

3(x − x0 )(y − y0 ) , R5

3(x − x0 )(z − z0 ) . R5

——————–

E=

4πI

+

−4πI −

E + E , 2

1 ∂E = ∇ × H , c ∂t

1 ∂H = ∇ × E, c ∂t

H=

H + H . 2

4πI +

1 ∂E = ∇ × H , c ∂t

4πI −

1 ∂H = ∇ × E , c ∂t

∇ · E = 4πρ,

∇ · E = 4πρ,

∇ · H = 4πρ,

∇ · H = −4πρ.

⎧ 1 ∂(E − iH ) ⎪ ⎨ 4πI (1 − i) + = i ∇ × (E − iH ), c ∂t ⎪ ⎩ ∇ · (E − iH ) = 4πρ (1 − i), ⎧ 1 ∂(E − iH ) ⎪ ⎨ 4πI (1 + i) + = i ∇ × (E − iH ), c ∂t ⎪ ⎩ ∇ · (E − iH ) = 4πρ (1 + i),

121

QUANTUM ELECTRODYNAMICS

⎧ 1 ∂(E + iH ) ⎪ ⎨ 4πI (1 + i) + = −i ∇ × (E + iH ), c ∂t ⎪ ⎩ ∇ · (E + iH ) = 4πρ (1 + i), ⎧ 1 ∂(E + iH ) ⎪ ⎨ 4πI (1 − i) + = −i ∇ × (E + iH ), c ∂t ⎪ ⎩ ∇ · (E + iH ) = 4πρ (1 − i), For E = −H , H = E we re-obtain the Maxwell equations: E=

E + H , 2

H=

H − E . 2

[21 ]

Appendix: Potential experienced by an electric charge: a particular case For a charge-1 particle: 1 dV =− 2 2 dt 2(a + t)(a + t)(c2 + t)

=−

1 √ , 2(a2 + t) c2 + t

21 @

The page ended with an attempt to generalize the above results to arbitrary linear combinations of the E and H ﬁelds (with space-time dependent coeﬃcients), in the case of Maxwell equations without sources: E = αE + βH,

H = −βE + αH;

1 ∂E = ∇ × H, c ∂t

−

α = α(q, t), β = β(q, t);

∇ · E = 0,

1 ∂H = ∇ × E, c ∂t

∇ · H = 0;

∇ · E

=

∇ α · E + ∇ β · H,

=

−∇ β · E + ∇ α · H.

∇· H

122

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

∞ dt 1 − =V = c 2(a2 + t) (c2 + t) 0

∞ 1 c dz π = =√ − arctan √ z 2 + (a2 − c2 ) a2 − c2 2 a2 − c2 c √ 2 2 a −c 1 . arctan =√ c a2 − c2 z z2 dt t 2 a +t

= = = = =

c = a 1 − β2,

c2 + t, 2 c + t, 2z dz, z 2 − c2 , z 2 + (a2 − c2 ). a2 − c2 = a2 β 2 .

1 1 β 1 = =V = arctan √ arcsin β. c aβ aβ 1 − r2

c=a

1 1 arcsin β β ; V = = . arcsin β c a β

2

∂Cy ∂Cx − ∂x ∂y

2

∂Cy ∂Cz − + + ∂y ∂z ∂Cx ∂Cy . = |∇ Cx |2 + |∇ Cy |2 + |∇ Cz |2 − ∂y ∂x xy ∂Cz ∂Cx − ∂z ∂Cx

2

PART II

3 ATOMIC PHYSICS

3.1.

GROUND STATE ENERGY OF A TWO-ELECTRON ATOM

Let us consider a nucleus of charge Z with two electrons. In electronic units we have: ∇2 ψ + 2(E − V )ψ = 0, Z Z 1 V =− − + . r1 r2 r3 In the same units, but denoting with W the energy in rydberg, we have W = 2E and thus: W ψ = V ψ − ∇2 ψ, that is: W ψ = −2Z

3.1.1

1 1 + r1 r2

ψ+

2 ψ − ∇2 ψ = Hψ. r3

Perturbation Method

In ﬁrst approximation, neglecting the interaction and up to a normalization constant, we have:

125

126

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

ψ = e−Zr1 e−Zr2 , and: H0 ψ = W0 ψ = −2Z 2 ψ, where H0 is the unperturbed Hamiltonian: 1 1 H0 = −2Z + − ∇2 . r1 r2 In fact: ∂2ψ ∂2ψ 2 ∂ψ 2 ∂ψ ∇ ψ= + + + = 2Z 2 ψ − 2Z 2 2 r1 ∂r1 r2 ∂r2 ∂r1 ∂r2 2

1 1 + r1 r2

ψ.

In ﬁrst approximation, assuming a normalized ψ, we have: 2 2 ψ dτ, dW = r3 and since, evidently, W0 = ψH0 ψdτ, more expressively we can write: 2 W = W0 + ΔW = ψ H0 + ψdτ = ψHψdτ. r The correct value W appears, then, to be the mean value of the energy relative to the function ψ that, in ﬁrst approximation, coincides with the energy eigenfunction. This will be useful in comparing the results obtained with the perturbation method with those of the variational method.1 We thus have: 2 −2Z(r1 +r2 ) e dτ r3 . dW = e−2Z(r1 +r2 ) dτ The integration with respect to the angular coordinates gives: 1 @ In the original manuscript, the variational method is appropriately called the “minimum method”.

127

ATOMIC PHYSICS

2

dW =

r12 r22

1 −2Z(r1 +r2 ) e dr1 dr2 ρ

r12 r22 e−2Z(r1 +r2 ) dr1 dr2

,

where ρ is the greater value between r1 and r2 . By restricting the double integration ﬁeld to the region r1 ≤ r2 , the numerator and the denominator will be divided by a factor two, so that: ∞ r2 −2Zr2 2r2 e dr2 r12 e−2Zr1 dr1 0r2 . dW = 0 ∞ 2 −2Zr2 r2 e dr2 r12 e−2Zr1 dr1 0

0

Now we have:

r12 −2Zr1 1 e r1 e−2Zr1 dr1 + 2Z Z r12 −2Zr1 r1 −2Zr1 1 e =− − e + e−2Zr1 dr1 2Z 2Z 2 2Z 2 r12 r1 1 e−2Zr1 , − = − − 2Z 2Z 2 4Z 3

so that:

r2

r12

0

r12 e−2Zr1 dr1 = −

−2Zr1

e

1 dr1 = − 4Z 3

We thus have: dW =

= D =

=

∞

e−2Zr2 .

N , D

∞ ∞ r2 −2Zr2 r2 −4Zr2 e dr2 − e dr2 − 2Z 3 2Z 3 0 0 0 ∞ 2 r2 −4zr2 − dr2 e Z 0 1 1 1 3 5 − − − = , 5 5 5 5 8Z 32Z 32Z 128Z 128Z 5 ∞ 2 ∞ ∞ 2 r2 −2Zr2 r2 −4Zr2 e dr2 − e dr2 − 4Z 3 4Z 3 0 0 0 ∞ 4 r2 −4Zr2 e − dr2 2Z 0 1 1 3 3 1 − − − = , 16Z 6 128Z 6 256Z 6 256Z 6 32Z 6

N =

1 r2 r22 + + 4Z 3 2Z 2 2Z

r22 −4Zr2 e Z2

r23 −4Zr2 e dr2 2Z 2

128

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

5 dW = Z, 4 and therefore: 5 W = W0 + ΔW = −2Z 2 + Z. 4 The ionization energy consequently is:2 5 Wj = −Z − W = Z − Z 4 2

2

5 2 25 . − = Z− 8 64

For the helium atom we thus have:3 5 3 · 2 = = 20.31 V. 4 2 For the lithium atom, the second ionization potential is: Wj = 4 −

Wj = 9 −

3.1.2

5 21 ·3= = 71.08 V. 4 4

Variational Method

The ground state energy is the minimum value of the expression ϕHϕ dτ , 2 ϕ dτ i.e., the minimum value assumed by the mean value of the energy with respect to any wavefunction ϕ. If we consider only a given set of functions ϕ, the minimum will correspond to an approximate value. The given approximation improves when the set is enlarged. When this set reduces to the only unperturbed wavefunction considered in the perturbation method, we obtain the same result given by that method. If the set is composed also of further wavefunctions besides the unperturbed wavefunction, in general we will have a better approximation. 2@

Note that, in the following, the author uses to write volt instead of eV for the energy unit. 3 @ Here and in the following pages, Majorana usually employed the electron-volt as energy unit. The symbol used by him was V (the same as for volt) rather than eV.

129

ATOMIC PHYSICS

3.1.2.1

First case.

To this end, we consider the functions ϕ = e−k(r1 +r2 )

with arbitrary k. We have: Hϕ = −2k ϕ + 2(k − Z) 2

1 1 + r1 r2

∞

0

∞

ϕHϕ dτ = −2k + 4(k − r) 2

ϕ2 dτ

ϕ+

2 ϕ, r3

r1 e−2kr1 dr1 r12 e−2kr1 dr1

5 + k 4

0

5 = −2k 2 + 4(k − Z)k + k, 4 that is: 5 Wmean = 2k 2 − 4kZ + k. 4 The minimum will be reached when: 4k − 4Z +

5 = 0, 4

that is: k=Z−

5 . 16

In this case we have: 5 5 2 5 5 − 4Z Z − W =2 Z− + Z− , 16 16 4 16 that is: 5 25 5 2 = −2k 2 . = −2 Z − W = −2Z + Z − 4 128 16 The ionization energy will be 2

25 5 . Wj = −Z 2 − W = Z 2 − Z + 4 128 For the helium atom: Wj =

217 = 22.95 V. 128

130

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

3.1.2.2 Second case. Let ϕ be an arbitrary function; the wavefunction of the ground state can be approximated by an expression of the form: y = aϕ + bHϕ, so that we have: (aϕ + bHϕ)(aHϕ + bH 2 ϕ) dτ yHy dτ = Wmean = 2 y dτ (aϕ + bHϕ)2 dτ 2 2 2 a ϕHϕ dτ + b Hϕ · H ϕ dτ + ab Hϕ · Hϕ dτ + ab Hϕ · Hϕ dτ = 2 2 2 a ϕ dτ + b Hϕ · Hϕ dτ + 2ab ϕ · dτ By noting that 2 Hϕ · Hϕ − ϕ H ϕ dτ = [(Hϕ)Hϕ − ϕH(Hϕ)] dτ = 0 or:

Hϕ · Hϕ dτ =

ϕ · H 2 ϕ dτ,

and, in general,

H ϕ · H ϕ dτ = m

n

ϕH m+n ϕ dτ,

we get: Wmean =

a2 A + 2abB + b2 C , a2 + 2abA + b2 B

where

A=

ϕ · Hϕ dτ , 2 ϕ dτ

ϕ · H ϕ dτ , 2 ϕ dτ 2

B=

C=

ϕ · H 3 ϕ dτ . 2 ϕ dτ

If we consider the generalized trial function y = a0 ϕ + a1 Hϕ + a2 H 2 ϕ + . . . + an H n ϕ,

131

ATOMIC PHYSICS

we analogously get: n

Wmedia =

ai ak Ai+k+1

i,k=0 n

, ai ak Ai+k

i,k=0

where:

Ar =

ϕH r ϕ dτ , ϕ2 dτ

and W will be the smallest root of

A1 − W A 2 − A1 W

A 2 − A1 W A 3 − A2 W

A 3 − A2 W A 4 − A3 W

...

An − An−1 W An+1 − An W For n = 1, we simply have:

A1 − W

A2 − A 1 W

the following equation: ... ... ... ...

= 0.

A2n−1 − A2n−2 W

An − An−1 W An+1 − An W An+2 − An+1 W

A2 − A1 W

= 0. A 3 − A2 W

Often, this procedure does not converge, because, starting from a given value of n, quantity H n ϕ exhibits too many singularities, which forces us to consider only combinations of the form y = a0 ϕ + a1 Hϕ + . . . + an−1 H n−1 ϕ. The inclusion of additional terms is not useful, since the corresponding a coeﬃcients would necessarily vanish.

3.1.2.3 Third case. In our eﬀorts for the search of the minimum value, let us consider the set of functions of the form: ϕ = e−kr1 e−kr2 er3 , with arbitrary k and . A particular case of this set ( = 0) has been considered in Sect. 3.1.2.1; then, we will certainly obtain a better approximation. We get:

132

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

Wmean =

e−k(r1 +r2 )+r3 He−k(r1 +r2 )+r3 dτ . e−2k(r1 +r2 )+2r3 dτ

Now we have: ∇2 ϕ = ∇2 e−k(r1 +r2 )+r3 = 2k 2 ϕ + 22 ϕ − 2kϕ cos r1 · r3 − 2kϕ cos r2 · r3 2k 2k 4 − ϕ − ϕ + ϕ, r1 r2 r3 or, by setting: α13 = cos r 1 r3 ,

a23 = cos r 2 r3 ,

and, remembering the expression for H, we obtain: Hϕ = −2k 2 ϕ−22 ϕ+2kϕα13 +2kϕα23 −2

Z −k Z −k 2 − 4 ϕ−2 ϕ+ ϕ. r1 r2 r3

It follows that

2

ϕ2 α23 dτ ϕ α13 dτ + 2k Wmean = −2k − 2 + 2k 2 ϕ dτ ϕ2 dτ 1 1 2 1 2 ϕ dτ dτ ϕ dτ r r r − 2(Z − k) 2 + (2 − 4) 3 . −2(Z − k) 1 2 2 2 ϕ dτ ϕ dτ ϕ dτ 2

2

Due to the symmetry of function ϕ for the two electrons, the third and fourth term above in the r.h.s are equal, as well as the ﬁfth and the sixth terms. Moreover, by observing that

α13 =

r12 + r32 − r22 , 2r1 r3

α23 =

r22 + r32 − r12 , 2r2 r3

133

ATOMIC PHYSICS

we have:

Wmean = −2k 2 − 22 + k

r1 2 ϕ dτ r3

+ k

r2 2 ϕ dτ r3

2

ϕ2 dτ

ϕ dτ +k

r3 2 ϕ dτ r1

+ k

2

2

ϕ dτ −2(Z − k)

r3 2 ϕ dτ r2

− k

2

ϕ dτ

1 2 ϕ dτ r1 2

ϕ dτ

r22 2 ϕ dτ r1 r3

− k

ϕ2 dτ

ϕ dτ

− 2(Z − k)

r12 2 ϕ dτ r2 r3

1 2 ϕ dτ r2 2

+ (2 − 4)

ϕ dτ

1 2 ϕ dτ r3

.

2

ϕ dτ

[4 ]

3.2.

WAVEFUNCTIONS OF A TWO-ELECTRON ATOM

The author again considered two-electron atoms, but now he focused on possible expressions for their wavefunctions. The notation is similar to that of the previous Section. 1 y = 1 − 2r1 − 2r2 + r3 + a(r12 + r22 ) + br32 + cr1 r2 + d(r1 + r2 )r3 + . . . , 2 ∂y = −2 + 2ar1 + cr2 + dr3 + . . . . ∂r1 3 yr1 =0,r2 =r3 =R = 1 − R + . . . , 2 ∂y = −2 + (c + d)R + . . . . ∂r1 r1 =0, r2 =r3 =R c + d = 3. [5 ] 4@

This Section is left incomplete in the original manuscript, which continues as follows: “By performing a ﬁrst integration on the 4-dimensional surface r1 = const., r2 = const., apart from a common factor in the numerator and in the denominator of the fractional terms above, and observing that on the considered surface we have the mean values of the following expressions, we ﬁnd that . . . ”. 5 @ The numerical values for the coeﬃcients c, d are deduced by requiring that y and its ﬁrst derivative have a node at the same position when the two-electron system collapses into a one-electron one [r1 = 0 (or r2 = 0) and r3 = 0].

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E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

1 ∂y = − + 2br3 + d(r1 + r2 ) + . . . ; ∂r3 2 yr3 =0, r1 =r2 =R = 1 − 4R + . . . , 1 ∂y = + 2dR + . . . . ∂r2 r3 =0, r1 =r2 =R 2 d = −1. 1 y = 1 − 2r1 − 2r2 + r3 + a(r12 + r22 ) + br32 + 4r1 r2 − (r1 + r2 )r3 + . . . . 2 [6 ] r12 + r32 − r22 r2 + r32 − r12 , 2 cos α2 = 2 ; r1 r3 r2 r3 r1 + r2 r3 r3 r2 r2 2 cos α1 + 2 cos α2 = + + − 2 − 1 . r3 r1 r2 r1 r3 r2 r3 2 cos α1 =

——————–

λψ = Lψ, L =

4 4 2 ∂2 ∂2 2∂ 2 2 ∂ 2 ∂ 4 ∂ + − + 2+ 2+ 2+ + + r1 r2 r3 ∂r1 r1 ∂r1 r2 ∂r2 r3 ∂r3 ∂r2 ∂r3 2 2 ∂ ∂ +2 cos α1 + 2 cos α2 . ∂r1 ∂r3 ∂r2 ∂r3

−2r1 −(2−2α)r2 e−(2−2α)r1 −2r2 e 1 , + ψ = 1 + r3 2 1 + 2αr2 1 + 2αr1 ∂ψ 1 −2 = 1 + r3 e−2r1 −(2−2α)r2 ∂r1 2 1 + 2αr2

2α −(2 − 2α) −(2−2α)r1 −2r2 e , + − 1 + 2αr1 (1 + 2αr1 )2 1 2α −(2 − 2α) ∂ψ = 1 + r3 − e−2r1 −(2−2α)r2 ∂r2 2 1 + 2αr2 (1 + 2αr3 )2 −2 + e−(2−2α)r1 −2r2 , 1 + 2αr1

6@

With reference to the ﬁgure on page 125, α1 [α2 ] is the angle between r1 [r2 ] and r3 .

ATOMIC PHYSICS

135

1 e−2r1 −(2−2α)r2 e−(2−2α) r1 − 2r2 ∂ψ = + , ∂r3 2 1 + 2αr2 1 + 2αr1 1 ∂2ψ 4 = 1 + r3 e−2r1 −(2−2α)r2 2 2 1 + 2αr2 ∂r1

(2 − 2α)2 4α(2 − 2α) 8α2 −(2−2α)r1 −2r2 + + + e , 1 + 2αr1 (1 + 2αr1 )2 (1 + 2αr1 )3 1 4α(2 − 2α) (2 − 2α)2 ∂2ψ r = 1 + + 3 2 2 1 + 2αr2 (1 + 2αr2 )2 ∂r2 8α2 4 −2r1 −(2−2α)r2 −(2−2α)r1 −2r2 e , + + e (1 + 2αr2 )3 1 + 2αr1 ∂2ψ = 0, ∂r32 −1 ∂2ψ = e−2r1 −(2−2α)r1 ∂r1 ∂r3 1 + 2αr2

−(1 − α) α e−(2−2α)r1 −2r2 , + − 1 + 2αr1 (1 + 2αr1 )2

−(1 − α) α ∂2ψ = − e−2r1 −(2−2α)r2 ∂r2 ∂r3 1 + 2αr2 (1 + 2αr2 )2 −1 + e−(2−2α)r1 −2r2 . 1 + 2αr1 Lψ = P (r1 , r2 , r3 )e−2r1 −(2−2α)r2 + P (r2 , r1 , r3 )e−2(2−2α)r1 −2r2 ,

4 2 4α(2 − 2α) 1 + r3 /2 4 + + + 4 + (2 − 2α)2 + P= 1 + 2αr2 r1 r2 r3 1 + 2αr2 2 2 8α 4 4 − 4α 4α + + − − − 2 (1 + 2αr2 ) r1 r2 r2 (1 + 2αr2 ) r3 (1 + r3 /2) cos α2 2 cos α1 2α − − 2 − 2α + 1 + r3 /2 1 + 12 r3 1 + 2αr1

2 1 + r3 /2 8α 1 cos α1 = − − 4 + (2 − 2α)2 + 1 + 2αr2 1 + 2αr2 1 + r3 /2 1 + r3 /2 2α cos α2 2 − 2α + . − 1 + r3 /2 1 + 2αr1

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E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

3.3.

CONTINUATION: WAVEFUNCTIONS FOR THE HELIUM ATOM ψ = e−p ,

1 2r1 + 2r2 − r3 + a(r12 + r22 ) + br1 r2 + cr32 + d(r1 + r2 )r3 2 . p= 1 + e(r1 + r2 ) + f r3 4 2 4 + − + ∇2 r1 r2 r3 4 4 2 ∂2 ∂2 ∂2 2 ∂ 2 ∂ 4 ∂ = + − + 2 + 2 +2 2 + + + r1 r2 r3 ∂r1 r ∂r r ∂r r ∂r2 ∂r3 1 1 2 2 3 ∂r3 ∂2 ∂2 +2 cos α1 · + 2 cos α2 · . ∂r1 ∂r3 ∂r2 ∂r3

λ =

α1 = OP1 − P2 P1 ;

α2 = OP2 − P1 P2 .

ψ0 = e−2r1 −2r2 + 2 r3 ; 1

λ0 = =

∂ = −2, ∂r1

∂ = −2, ∂r2

∂ 1 = , ∂r3 2

∂2 = 4, ∂r12

∂2 = 4, ∂r22

∂2 1 = , 2 4 ∂r3

∂2 = 4, ∂r1 ∂r2

∂2 = −1, ∂r1 ∂r3

∂2 = −1. ∂r2 ∂r3

4 4 2 1 4 4 2 + − +4+4+ − − + − 2 cos α1 − 2 cos α2 r1 r2 r3 2 r1 r2 r3 17 − 2 cos α1 − 2 cos α2 , 2 = 8.5, λmax 0

λmin = 4.5. 0

137

ATOMIC PHYSICS

2 =

∂p ∂r1

r1 =0,r2 =r3 =R

(2 + bR + dR)(1 + eR + f R) − e 2R − 12 R + aR2 + cR2 + dR2 = (1 + eR + f R)2 2 + R b + d + 2e + 2f − 32 e + R2 (be + bf + de + df − ae − ce − de) = 1 + R(2e + 2f ) + R2 (e2 + f 2 + 2ef ) 2 + R b + d + 12 e + 2f + R2 (−ae + be + bf − ce + df ) . = 1 + R(2e + 2f ) + R2 (e2 + f 2 + 2ef ) 1 b + d + e + 2f = 4e + 4f, 2 −ae + be + bf − ce + df = 2e2 + 2f 2 + 4ef ; 7 b + d − e − 2f = 0, 2 ac − be − bf + ce − df + 2e2 + 4ef + 2f 2 = 0. ∂p 1 − = 2 ∂r3 r3 =0,r1 =r2 =R 1 − 2 + 2dR (1 + 2eR) − f (2R + 2R + aR2 + aR2 + bR2 ) = (1 + 2eR)2 − 12 + R(2d − e − 4f ) + R2 (4de − 2af − bf ) = . 1 + 4eR + 4e2 R2 2d − e − 4f = −2e, 4de − 2af − bf = −2e2 . 2d + e − 4f = 0, 2b + 2d − 7e − 4f = 0, 2af + bf − de − 2e2 = 0, ae − be − bf + ce − df + 2e2 + 4ef + 2f 2 = 0.

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E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

e = A, f = B, A d = − + 2B, 2 b = 4A, a = 0, 1 c = 2A − B. 2 2r1 + 2r2 − 12 r3 + 4Ar1 r2 + 2A − 12 B r32 + 2B − 12 A (r1 + r2 )r3 p0 = . 1 + A(r1 + r2 ) + Br3 ψ0 = e−p ,

λ=

∇2 ψ0 = (−∇2 p + (∇ p)2 )ψ0 .

4 4 2 ∂2p ∂2p ∂2p 2 ∂p 2 ∂p 4 ∂p + − − 2 − 2 −2 2 − − − r1 r2 r3 ∂r1 r1 ∂r1 r2 ∂r2 r3 ∂r3 ∂r2 ∂r3

∂2p ∂2p −2 cos α1 − 2 cos α2 + ∂r1 ∂r3 ∂r2 ∂r3 +2 cos α1

∂p ∂r1

2

+

∂p ∂r2

2

+2

∂p ∂r3

2

∂p ∂p ∂p ∂p + 2 cos α2 . ∂r1 ∂r3 ∂r2 ∂r3

p= 1 ∂p = 2 ∂r1 S

∂2p ∂r12

=

=

R . S

dS ∂p 1 ∂S ∂R ∂R S− R , = 2 S− R , ∂r1 dr1 ∂r2 S ∂r2 ∂r2 ∂R 1 ∂S ∂p = 2 S− R . ∂r3 S ∂r3 ∂r3

∂R ∂S ∂S ∂R ∂2S ∂2p S+ − − 2 S ∂r1 ∂r1 ∂r1 ∂r1 ∂r12 ∂r1 ∂S ∂S ∂R S− R −2 ∂r1 ∂r1 ∂r1 2 2 ∂ S ∂S ∂ R ∂S ∂R 1 −R 2 −2 −R S S ; S3 ∂r1 ∂r1 ∂r1 ∂r12 ∂r1 1 S3

139

ATOMIC PHYSICS

∂2p ∂r1 ∂r2

=

=

2 ∂R ∂S ∂S ∂R ∂2S ∂ R + − −R S ∂r1 ∂r2 ∂r1 ∂r2 ∂r1 ∂r2 ∂r1 ∂r2 ∂S ∂R ∂S −2 −R S ∂r2 ∂r1 ∂r1 2 ∂2S ∂R ∂S ∂R ∂S ∂ R 1 S −R − − S3 ∂r1 ∂r2 ∂r1 ∂r2 ∂r1 ∂r2 ∂r2 ∂r1 ∂S ∂S +2R . ∂r1 ∂r2 1 S3

1 1 R = 2r1 + 2r2 − r3 + 4Ar1 r2 + 2A − B r32 2 2 1 + 2B − A (r1 + r2 )r3 , 2 S = 1 + A(r1 + r2 ) + Br3 . 1 ∂R 1 ∂R = 2 + 4Ar2 + 2B − A r3 , = 2 + 4Ar1 + 2B − A r3 , ∂r1 2 ∂r2 2 1 1 ∂R = − + 2B − A (r1 + r2 ) + (4A − B)r3 ; ∂r3 2 2 ∂2R = 0, ∂r12 ∂2R = 4A, ∂r1 ∂r2

∂2R , ∂r22

∂2R = 4A − B; ∂r32

∂2R 1 = 2B − A, ∂r1 ∂r3 2

1 ∂2R = 2B − A. ∂r2 ∂r3 2

∂S = A, ∂r1

∂S = A, ∂r2

∂S = B; ∂r3

∂2S = 0, ∂r12

∂2S = 0, ∂r22

∂2S = 0; ∂r32

∂2S = 0, ∂r1 ∂r2

∂2S = 0, ∂r1 ∂r3

∂2S = 0. ∂r2 ∂r3

p = p0 . [7 ] 7 @ The original manuscript then continues with some calculations aimed at evaluating the derivatives of p. In the following we report only the ﬁnal results.

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E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

−4Ar1 − 2A2 r12 + 2A2 r22 − 2A2 r32 − 4A2 r1 r2 − 4ABr1 r3 ∂p =2+ , ∂r1 [1 + A(r1 + r2 ) + Br3 ]2 −4Ar2 − 2A2 r22 + 2A2 r12 − 2A2 r32 − 4A2 r1 r2 − 4ABr2 r3 ∂p =2+ , ∂r2 [1 + A(r1 + r2 ) + Br3 ]2 ∂p ∂r3

=

1 1 [1 + a(R1 + R2 ) + bR3 ] − 2 [1 + A(r1 + r2 ) + Br3 ] 2 1 + 2B − A (r1 + r2 ) + (4A − B)r3 2 1 1 −B 2r1 + 2r2 − r3 + 4Ar1 r2 + 2A − B r32 2 2 1 + 2B − A (r1 + r2 )r3 . 2 ——————– 1 ψ0 = 1 + r3 e−2r1 −2r2 . 2 ∂ψ0 = −2ψ0 , ∂r2

∂ψ0 = −2ψ0 , ∂r1 ∂ 2 ψ0 = 4ψ0 , ∂r12 ∂ 2 ψ0 = 4ψ0 , ∂r1 ∂r3

∂ψ0 1 = e−2r1 −2r2 ; ∂r3 2

∂ 2 ψ0 = 4ψ0 , ∂r22

∂ 2 ψ0 = e−2r1 −2r2 , ∂r1 ∂r3

∂ 2 ψ0 = 0; ∂r32 ∂ 2 ψ0 = e−2r1 −2r2 . ∂r32

4 4 2 4 4 1 2 + − +4+4− − + 1 r1 r2 r3 r1 r2 1 + 2 r3 r3 2 2 − cos α1 − cos α2 1 + 12 r3 1 + 12 r3 1 2 = 8− − (cos α1 + cos α2 ) , 1 1 + 2 r3 1 + 12 r3

λ0 =

= 8, λmax 0

λmin = 3. 0

——————–

141

ATOMIC PHYSICS

λψ = Lψ,

L= χ=

√

r1 r2 r3 ψ.

λχ = L χ = L =

√

4 4 2 + − + ∇2 . r1 r2 r3

√

r1 r2 r3 Lψ,

r1 r2 r3 L √

1 . r1 r2 r3

2 2 ∂ ∂ 4 4 2 1 ∂ 3 1 ∂ 3 + − + − + − + L = + r1 r2 r3 ∂r12 r1 ∂r1 4r12 ∂r22 r2 ∂r2 4r22 2 2 ∂ 3 1 2 ∂ ∂ + − + + 2 2− r3 ∂r3 2r32 r1 ∂r1 r12 ∂r3 1 2 4 ∂ 2 ∂ − − + + r2 ∂r2 r32 r3 ∂r3 r33 1 ∂ 1 ∂ 1 ∂2 − − + +2 cos α1 ∂r1 ∂r3 2r1 ∂r3 2r3 ∂r1 4r1 r3 1 ∂ 1 ∂ 1 ∂2 − − + . +2 cos α2 ∂r2 ∂r3 2r2 ∂r3 2r3 ∂r2 4r2 r3

1 1 ∂ 1 √ =− √ , ∂r1 r1 2r r1

∂2 1 3 1 √ = 2 √ , 2 4r r1 ∂r1 r1

∂2 1 1 = . √ ∂r1 ∂r3 r1 r3 4r1 r3

∂2 1 ∂ 3 ∂2 −→ − + , ∂r12 ∂r12 r1 ∂r1 4r12 ∂ ∂ 1 −→ − , ∂r1 ∂r1 2r1 ∂2 ∂2 1 ∂ 1 ∂ 1 −→ − − + . ∂r1 ∂r3 ∂r1 ∂r3 2r1 ∂r3 2r3 ∂r1 4r1 r3

3.4.

SELF-CONSISTENT FIELD IN TWO-ELECTRON ATOMS

A self-consistent ﬁeld method is here applied to the problem of twoelectron atoms with nuclear charge Z. The quantities r1 and r2 are

142

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

the distance of the two electrons from the nucleus, while r12 denotes the inter-electron distance. Z 2 Z Eϕ = Hϕ = −2 − 2 + ψ − ∇2 ϕ. r1 r2 r12 ϕHψdτ . W = ϕ2 dτ

ϕ(H − W )ϕdτ = 0.

δ

ϕ(H − W )ϕdτ = 0, δϕ = αϕ: δ

ϕ(H − W )ϕdτ = 2α

ϕ(H − W )ϕdτ = 0;

W = W. ——————– δ

ϕ(H − W )ϕdτ = 0.

(1)

ϕ(r1 , r2 , r12 ) = y(r1 )y(r2 ), δϕ = y(r1 )δy(r2 ) + y(r2 )δy(r1 ). ϕ(H − W )ϕdτ = 2 [y(r1 )δy(r2 ) + y(r2 )δy(r1 )](H − W )y(r1 )y(r2 )dτ = 4 y(r2 )δy(r1 )(H − W )[y(r1 )y(r2 )]dq2 = 0 δ

(2)

2Z z 2Z y(r1 )y(r2 ) − y(r1 ) y(r2 ) + y(r1 ) y(r2 ) − y(r1 )∇2 y(r2 ) r1 r2 r12 − ∇2 y(r1 ) · y(r2 ) − W y(r1 )y(r2 ),

=−

143

ATOMIC PHYSICS

δ

ϕ(H − W )ϕdτ

2Z 2Zy 2 (r1 ) = 4 δy(r1 ) − − dq2 r1 r2 2y 2 (r2 ) 2 + dq2 − y(r2 )∇ y(r2 )dτ − W r12 −∇2 y(r1 ) dq1 .

2Zy 2 (r2 ) 2y 2 (r2 ) 2Z − dq2 + dq2 − r1 r2 r12 − y(r2 )∇2 y(r2 )dq2 − W y(r1 ) − ∇2 y(r1 ) = 0, −

2Zy 2 (r1 ) 2y 2 (r1 ) 2Z − dq1 + dq1 r2 r1 r12 2 − y(r1 )∇ y(r1 )dq1 − W y(r2 ) − ∇2 y(r2 ) = 0. −2Zy 2 (r2 ) 2 + y(r2 )∇ y(r2 ) dq2 = A r2 −2Zy 2 (r2 ) − y(r1 )∇2 y(r2 ) dq1 . = r1

2y 2 (r2 ) 2Z + dq2 − W + A y(r1 ) − ∇2 y(r1 ) = 0, − r1 r12 (W − A)y(r1 ) =

2z − + r1

2y 2 (r2 ) dq2 y(r1 ) − ∇2 y(r1 ). r12

W − A = B, r1 = r: d2 y 2 dy 2Z 2y 2 (r2 ) ; + dq2 y(r) − 2 − By(r) = − r1 r12 dr r dr 2y 2 (r2 ) 1 2 2Z + dq2 − y − y . B=− r1 r12 y ry P = ry: 2Z + B=− r

2y 2 (r2 ) P . dq2 − r12 P

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E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

0 = −8π

P 2 1 d2 rP , − r2 r dr2 P

8πP 2 + r

d2 rP = 0, dr2 P

−8πP

2

3.5.

d rP + P rP P = r − dr P P2 rP + 2P 2P P − 2rP P − rP 2 2rP 2 P − + . = r P P2 P2

2s TERMS FOR TWO-ELECTRON ATOMS

An approximate expression for the energy (in rydbergs) W (which is equal to half the mean value of the potential energy) of the 2s terms of two-electron atoms with charge Z is given. For further details, see Sect. 15 of Volumetto III. −W

=

=

3.6.

32 1 306 ∓ 32 5 2 34 Z − ± Z = Z2 + Z2 + Z 4 81 729 4 729 ⎧ 1 1 ⎪ Z 2 + Z 2 − 0.3759Z = Z 2 + (Z 2 − 1.5034Z), ⎪ ⎪ ⎪ 4 4 ⎪ ⎪ ⎨ for ortho-states, ⎪ ⎪ 1 1 ⎪ ⎪ Z 2 + Z 2 − 0.4636Z = Z 2 + (Z 2 − 1.8546Z), ⎪ ⎪ 4 4 ⎩ for para-states.

ENERGY LEVELS FOR TWO-ELECTRON ATOMS

In the following pages, the author evaluates the energies for a number of terms in two-electron atoms, by using certain expressions for the corresponding wavefunctions. The numerical values are grouped in few tables.

145

ATOMIC PHYSICS

(m = 1, 0, −1),

rψ1 = y1 ϕm 1 m

(m = 1, 0, −1).

rψ2 = y2 ϕ1 dτ =

dxdydz . 4π 1 0 ϕ11 ϕ11 = ϕ11 ϕ−1 1 = 1 − √ ϕ2 , 5 2 ϕ01 ϕ01 = (ϕ01 )2 = 1 + √ ϕ02 . 5

For y1 ϕ11 : 1 V (r2 ) = r2

r2 0

−

For y1 ϕ01 : 1 V (r2 ) = r2

y12 dr1

+

1 √ 3 5 5r1

r1 0

r2

2 √ 3 5 5r2

r12 y12 dr1

∞

+ 0

A= B=

1 2 y dr1 r1 1

0

with ri ≥ rk .

r2

y12 dr1

∞

+

r2 r2

r2 + √2 5 5

1 2 y dr1 r1 1

r12 y12 dr1

2r2 + √2 5 5

y12 y22 dr1 dr2 , ri rk2 y12 y22 dr1 dr2 , ri3

∞ r2

∞ r2

1 2 y dr1 r13 1

1 2 y dr1 r13 1

ϕ02 .

ϕ02 .

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E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

Electrostatic energy y1 ϕ11

y1 ϕ01

y1 ϕ−1 1

y2 ϕ11

A+

1 2 1 B A− B A+ B 25 25 25

y2 ϕ02

A−

2 4 2 B A+ B A− B 25 25 25

y2 ϕ−1 A+ 2

1 2 1 B A− B A+ B 25 25 25

Electrostatic energy A+

1

1 A− B 5

0

2 A+ B 5

E2 = S, E1 + E2 = 2T, E0 + E1 + E2 = 2S + R,

2p2p

Z→∞ one electron

1 B 25

2

1 B 25

1D

: A+

3P

1 : A− B 5

1S

:

3 A+ B 5

E1 = 2T − S, E0 = 2S + R − 2T.

1 4 93 93 A= = Z 512 Z 128 = 0.181640625 = 0.7265625

45 1 B= Z 512 = 0.08789005

45 4 B= Z 128 = 0.3515625

147

ATOMIC PHYSICS

Z=1

1 237 B= = 0.18515625 25 1280

1D

A+

3P

1 21 A− B = = 0.1640625 5 128

1S

111 2 = 0.216796875 A+ B = 5 512

For y = x2 e− 2 x , y 2 = x4 e−x , N = 24 we have in fact: 2 2 ∞ ∞ 2 y1 y2 1 2 y2 A= 2 dx1 dx2 = 2 y12 dx1 dx2 , N xi N 0 x1 x2 2 y2 dx2 = x32 e−x2 dx2 = −(x32 + 3x22 + 6x2 + 6)e−x2 , x2 ∞ 2 y2 dx2 = (x31 + 3x21 + 6x1 + 6)e−x1 , x 2 x1 1

2

∞

(x71 + 3x61 + 6x51 + 6x41 )e−2x1 dx1 0 7! 6! 5! 4! 315 135 45 837 = 2 +3 7 +6 6 +6 5 = + + +9= , 8 2 2 2 2 8 4 2 8

N A = 2

A=

1 B= 2 N

93 93 837 = = . 8 · 576 8 · 64 512

2 x21 y12 y22 dx1 dx2 = 2 3 N xi y22 dx2 = x32 ∞ x1

∞

∞

x21 y12 dx1

0

∞ x1

y22 dx2 , x32

x2 e−x2 dx2 = −(x2 + 1)e−x2 ,

y22 dx2 = (x1 + 1)e−x1 , x32

(x71 + x61 )e−2x1 dx1 0 7! 6! 405 315 45 = 2 + 7 = + = = 50.625, 8 2 2 8 4 8

N 2B = 2

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E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

B=

2s2s

1Σ

45 45 405 = = . 8 · 576 8 · 64 512

: As

77 77 1 4 As = As = Z 512 Z 128 = 0.150390625 = 0.6015625

For y = (x2 − 2x)e− 2 x , y 2 = (x4 − 4x3 + 4x2 )e−x , N = 24 − 24 + 8 we have in fact: 1

A=

y22 dx2 = x2

1 N2

y12 y22 2 dx1 dx2 = 2 xi N

∞

y12 dx1

0

∞ x1

y22 dx2 , x2

(x32 − 4x22 + 4x2 )e−x2 dx2 = −(x32 − x22 + 2x2 + 2)e−x2 ,

∞ x1

N 2A = 2

y22 dx2 = (x31 − x21 + 2x1 + 2)e−x1 , x2

∞

0 ∞

(x41 − 4x31 + 4x21 )(x31 − x21 + 2x1 + 2)e−2x1 dx1

(x71 − 5x61 + 10x51 − 10x41 + 8x21 )e−2x1 dx1 0 6! 5! 4! 2! 7! −5 + 10 − 10 + 8 = 2 256 128 64 32 8 77 315 225 75 − + − 15 + 4 = = 9.625. = 8 4 2 8 = 2

3.6.1

Preliminaries For The X And Y Terms

(2s)2 1 S + 2p2p 1 S = X + Y .

dτ1 =

dx1 dy1 dz1 , 4π

dτ = dτ1 dτ2 .

149

ATOMIC PHYSICS

2p2p 1 S :

u = (x1 x2 + y1 y2 + z1 z2 )e− 2 (z1 +z2 ) ;

(2s)2 1 S :

v = (r1 − 2)(r2 − 2)e− 2 (r1 +r2 ) ;

1

1

uv = (x1 x2 + y1 y2 + z1 z2 )(r1 r2 − 2r1 − 2r2 + 4)e−r1 −r2 .

2

u dτ

(x1 x2 + y1 y2 + z1 z2 )2 e−r1 −r2 dτ = 3 x21 x22 e−r1 −r2 dτ 2 2 1 = 3 r12 e−r1 dτ1 x21 er1 dτ1 = 2 3 1 1 r14 e−r1 dr1 = 242 = 192, = 3 3

=

2

v dτ

= =

r2 >r1

uv dτ2 = r12 =

uv dτ r12

−r1

(r12

− 4r1 + 4)e

(r14

−

uv =2 r12

4r13

+

2 dτ1

4r12 )e−r1 dr1

dτ1 r2 >r1

2 = 64.

uv dτ2 , r12

1 2 −r1 ∞ r e r2 (r1 r2 − 2r1 − 2r2 + 4)e−r2 dr2 3 1 r1 1 2 −2r1 r e r1 (r12 + 2r1 + 2) − 2r1 (r1 + 1) 3 1 −2(r12 + 2r1 + 2) + 4(r1 + 1) ,

1 4 2 6 −2r1 r − r e = 2 dr1 3 1 3 1 0 1 7! 2 6! 105 15 45 = 2 − = − = : 3 28 3 27 8 2 8

∞

150

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

1 √ Nu Nv

√ 1 45 15 3 uv dτ = √ = = 0.050743676003. r12 512 64 3 8

77 512 √ 15 3 512

√ 15 3 512

.

111 512

a−E

c ,

c

b−E

E 2 − (a + b)E + ab − c2 = 0, a+b ± E= 2

a−b 2

2 + c2 .

a+b 47 a−b 17 = , − = , 2 256 2 512 172 289 675 a−b 2 = = , c2 = , 2 2 2 512 512 5122 √ + a−b 2 964 964 a−b 2 c = , + c2 = . 2 (512)2 2 (512)2

E1 = E2 =

√ √ 94 − 964 47 − 241 = , 512 256 √ √ 94 + 964 47 + 241 = . 512 256

√ 47 − 241 X E1 = 0.122952443 256 √ 47 + 241 Y E2 = 0.244235057 256 47 E1 + E2 = 0.3671975 = . 128

151

ATOMIC PHYSICS

a − E1

c

c

b − E1

=

√ −17 + 964 512 √ 15 3 512

√ 15 3 512 √ 17 + 964 512

0.027438182 0.050743676 =

, 0.050743676 0.093844432

a − E2

c

c

b − E2

=

√ −17 − 964 512 √ 15 3 512

√ 15 3 512 √ 17 − 964 512

0.093844432 0.050743676 =

. 0.050743676 0.027438182

p1 + p2 = 1. p1 = p2 =

3.6.2

X=

√

Y =

√

p1 (2s)2 1 S −

p2 (2s)2 1 S +

√ √

p2 2p2p 1 S,

p1 2p2p 1 S,

√ 964 + 17 964 675 √ = = 0.774, 1928 1928 − 34 964 √ 964 − 17 964 675 √ = = 0.226. 1928 1928 + 34 964

Simple Terms 2s2s 2p1 2s 2p0 2s 2p−1 2s

2s2p1 2p1 2p1 2p0 2p1 2p−1 2p1

2s2 p0 2p1 2p0 2p0 2p0 2p−1 2p0

——————–

2s2p−1 2p1 2p−1 2p0 2p−1 2p−1 2p−1

152

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

2p1 2p1 m=2 singlets

237

1280

2p1 2p1

2p2p 1 D =

237 1280

——————–

2s2p1 2p1 2p0 m=1 singlets

0

2s2p1 2p1 2p0

237 1280

0

2p2p 1 D =

237 1280

——————–

2s2p1 2p1 2p0 m=1 triplets

2s2p1

17 128

0

2s2p 3 P =

17 128

2p1 2p0

0

21 128

2p2p 3 P =

21 128

——————–

2s2p0 2s2s 2p1 2p−1 2p0 2p0 2s2p0

m=0 singlets

49 256

2s2s

0

2p1 2p−1

0

2p0 2p0

0

With a suitable change of states:

0 77 512 √ 15 2 512 15 512

0 √ 15 2 512 33 160 √ 27 2 2560

0 15 512 √ 27 2 2560 501 2560

153

ATOMIC PHYSICS

1 2 2p1 2p−1 − 2p0 2p0 , 2p2p 1 D = 3 3 2 1 2p2p 1 S = 2p1 2p−1 + 2p0 2p0 , 3 3 we have:8 2s2p0 2s2s 2p2p 1 D 2p2p 1 S 49 256

0

2s2s

0

77 512

0

2p2p 1 D

0

0

237 1280

0

0

111 512

2s2p0

2p2p 1 S

X = Y

=

√ 15 3 0 512 237 2p2p1 D = 1280

√ 47 − 241 256 √ 47 + 241 256

0

0 √ 15 3 512

(2s)2 1 S , 2p2p 1 S ,

[9 ] ——————– 2s2p0 2p1 2p−1 m=0 triplets

0

2s2p0 2p1 2p−1

0

21 128

2p2p 3 P =

21 128

——————– 8@

In the table below we have preferred to denote with the shorthand notations 2p2p 1 D and 2p2p 1 S (used even elsewhere in the original manuscript) what the author reported in the full expressions given above. 9 @ With X and Y the author denotes the eigenvalues of the subsystem formed by 2s2s and q q 2p2p 1 S =

2 2p1 2p−1 3

+

1 2p0 2p0 . 3

154

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

2p2p 1 D :

237 1280

= 0.185.156.250;

2p2p 3 P :

21 128

= 0.164.062.500;

2s2p 3 P :

17 128

= 0.132.812.500;

2s2p 1 P :

49 256 √ 47 + 241 256 √ 47 − 241 256

√ √ − p2 2s2s 1 S + p1 2p2p 1 S = Y : √

p1 2s2s 1 S +

√ p2 2p2p 1 S = X :

= 0.191.406.250;

= 0.244.235.057;

= 0.122.952.443.

155

ATOMIC PHYSICS

3.6.3

Electrostatic Energy Of The 2s2p Term ys2 = (r12 − 2r1 )2 e−r1 ,

Ns = 8;

yp2 = r24 e−r2 ,

Np = 24.

ri ≥ r1 , r2 :10

2 yp2 ys dr2 = yp2 dr2 ri ri ∞ ∞ ∞ 2 ∞ 2 yp ys 2 2 = ys dr1 dr2 + yp dr2 dr1 , 0 r1 r2 0 r2 r1

ys2 yp2 dr1 dr2 = ri

ys2 dr1

yp2 1 dr2 = ri r1

r1 0

yp2 dr2

∞

+ r1

yp2 dr2 , r2

r24 e−r2 dr2 = −(r24 + 4r24 + 12r22 + 24rs + 24)e−r2 , 1 2 y dr2 = r23 er2 dr2 = −(r24 + 3r22 + 6r2 + 6)e−r2 , r2 p yp2 dr2

0 ∞

r1

1 r1

10 @

0

r1

r1

=

yp2 dr2 = 24 − (r14 + 4r13 − 12r12 + 24r1 + 24)e−r1 ,

1 2 y dr2 = (r13 + 3r12 + 6r1 + 6)e−r1 , r2 p

yp2 dr2

∞

+ r1

24 24 2 − + 18 + 6r1 + r1 e−r1 r1 r1 2 yp = dr2 = Vp . ri

1 2 y dr2 = r2 p

In the original manuscript it is noted that: Z Z Z Z 2 Z 2 Z yp ys ys2 dr1 dr2 = yp2 dr2 = Vs yp2 dr2 , Vp ys2 dr12 = ri ri

where V denotes the electrostatic potential energy of the p or s state.

156

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

ys2 Vp dr1

=

∞

24r13 − 96r12 + 96r1 )er1

0 −(r16 + 2r15 − 2r14 − 24r13 − 24r12 + 96r1 e−2r1 dr1 720 240 48 144 48 96 + + + + − . = 144 − 192 + 96 − 128 64 32 16 8 4

1 Ns Np

ys2 yp2 1 15 5 1 3 1 1 3 − + + + − dr1 dr2 = − 1 + − ri 4 2 512 256 128 64 32 8 83 = M. = 512

yp2 ys2 3 2 1 2 y − y = 12 − = (r14 − 6r13 + 6r12 )e−r1 = t1 . 2 s 2 p Ns2 Np2 ∞ ∞ t1 t2 t2 dr1 dr2 = 2 t1 dr1 dr2 , ri 0 r1 r2 t2 dr2 = (r23 − 6r22 + 6r2 )e−r2 dr2 = −(r23 − 3r22 )e−r2 . r2 Es + Ap − 2M

1 t1 t2 dr1 dr2 144 ri ∞ 1 = (r14 − 6r13 + 6r12 )(r13 − 3r12 )e−2ri dr1 72 0 ∞ = (r17 − 9r16 + 24r15 − 18r14 )e−2r1 dr1

=

0

5040 9 · 720 24 · 120 18 · 24 − + − 256 128 64 32

=

1 72

=

45 5 3 1 35 − + − = . 128 84 8 16 128

Es =

77 ; 512

M=

Es + Ap 1 83 − = = 0.162109375. 2 256 512

Ap =

93 ; 512

157

ATOMIC PHYSICS

3.6.4

Perturbation Theory For s Terms ψ = er1 −r2 .

H0 = −

1 1 1 1 − − ∇21 − ∇22 . r1 r2 2 2

1 1 1 1 1 1 − − H0 ψ0 = − − − − ψ0 = −ψ0 . r1 r2 2 r1 2 r2

Then: E0 = −1. For λ → 0: H = H0 + λH1 ,

H1 =

1 . r12

ψ = ψ0 + λψ1 + λ2 ψ2 + . . . , E = E0 + λE1 + λ2 E2 + . . . . 0 = (H − E)ψ = (H0 + λH1 − E0 − λE1 − λ2 E2 . . .)(ψ0 + λψ1 + λ2 ψ + . . .) ∞ ∞ i λ Ei λ k ψk ; = H0 + λH1 − i=0

k=0

(H0 − E0 )ψn = (E1 − H1 )ψn−1 + E2 ψn−2 + E3 ψn−3 + . . . + En ψ0 . (H0 − E0 )ψ0 = 0, (H0 − E0 )ψ1 = (E1 − H1 )ψ0 , 5 E1 = . 8 By setting: ψ1 = y e−r1 −r2 ,

158

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

we have: 2 2 ∂y ∂y 2 −r1 −r2 − −2 + ∇ ψ1 = 2 − ye e−r1 −r2 +∇2 y e−r1 −r2 , r1 r2 ∂r1 ∂r2

(H0 − E0 )ψ1

1 1 1 2 = 1− − − ∇ ψ1 r1 r2 2 1 ∂y ∂y 1 1 5 −r1 −r2 − = + + − y e = er1 −r2 , ∂r1 ∂r2 2 2 8 r12

∂y 1 5 ∂y 1 + − ∇2 y = − . ∂r1 ∂r2 2 8 r12

y=

∞

P (cos θ) fl (r1 , r2 ).

=0

3.6.5

2s2p 3 P Term

Let us consider the functions: (r1 − 2)e− 2 r1 , 1

r2 e− 2 r2 . 1

1 ψ = e− 2 (r1 +r2 ) (r1 − 2)r2 ϕ01 (q2 ) − (r2 − 2)r1 ϕ02 (q1 ) , ψ 2 = e−(r1 +r2 ) (r1 − 2)2 r22 + (r2 − 2)2 r12 −2r1 r2 (r1 − 2)(r2 − 2)ϕ01 (q1 )ϕ01 (q2 ) 2 2 2 2 0 2 2 0 + √ (r1 − 2) r2 ϕ2 (q2 ) + √ (r2 − 2) r1 ϕ2 (q1 ) , 5 5 2 2 where we have used: ϕ01 = 1 + √ ϕ02 . 5 N = 384 = 2 · 8 · 24.

159

ATOMIC PHYSICS

4 3

∞ 0

4 = 3

(r15

∞

− 2r )e dr1 4

∞

r2 (r2 − 2)e−r2 dr2

r1

(r17

−

2r16 )e−2r1 dr1

=

0

Isp =

3.6.6

r1

45 . 4

45 15 = . 4 · 384 512

X Term

Z = 2.

dτ = [11 ]

dxdydz . 4π y1 y2 y3 y4

∞

= r1 r2 e−r1 −r2 , = (r1 + r2 )e−r1 −r2 , = e−r1 −r2 , = (x1 x2 + y1 y2 + z1 z2 )e−r1 −r2 . 2

2

9 , 16 0 ∞ ∞ 2 4 −2r1 2 −2r r1 e dr1 r2 e dr2 + 2 y2 dτ = 2 y12 dτ

=

0

r14 e−2r1 dr1

=

0

24 32

=

∞

2 r13 e2r1 dr1

0

2 21 3 1 3 = , =2· · +2· 4 4 8 32 2 2 ∞ 1 1 r12 e−2r1 dr1 = = , y32 dτ = 4 16 0 2 2 1 1 3 3 r14 e−2r1 dr1 = = , y42 dτ = 3 3 4 16 ∞ ∞ 9 3 3 y1 y2 dτ = 2 r14 e2r1 dr1 r23 e−2r2 dr2 = 2 · · = , 4 8 16 0 0 11 @

Remember that the X term is a superposition of the 2s2s 1 S and 2p2p 1 S ones.

,

160

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

2 3 9 = = , y1 y3 dτ = 8 64 0 3 3 1 y2 y3 dτ = 2 r13 e−2r1 dr1 r22 e−2r2 dr2 = 2 · · = . 8 4 16 1 Kinetic energy: T = − ∇2 . 2 2 1 2 + . Potential energy: U = − − r1 r2 r3 2 4 2 2 −r1 r1 = r1 − 4 + + e = 1− r1 e−r1 , ∇1 r1 e r1 r1 r12 1 2 1 1 r1 1 2 −r1 −r1 = − + − − +2− e r1 e−r1 . − ∇1 r1 e 2 2 r1 2 r1 r12

y1 T y1 dτ

∞

r13 e−2r1 dr1

2

∞ r14 3 2 = 2 r24 e−2r2 dr2 − + 2r1 − r1 e−2r1 dr1 2 0 0 3 3 3 3 1 · = , = 2· − + − 8 4 4 4 16

∞

∞ r14 3 2 r23 e−2r2 dr2 y2 T y1 dτ = 2 − + 2r1 − r1 e−2r1 dr1 2 0 0 ∞ 3 ∞ r1 2 −2r1 +2 dr1 r34 e−2r2 dr1 − + 2r1 − r1 e 2 0 0 1 3 3 1 1 3 3 3 3 · = + = . = 2· · +2 − + − 8 8 16 2 4 4 32 32 16 r1 1 1 2 −r1 = − +2− e−r1 , − ∇ r1 e 2 2 r1 1 1 2 1 −r1 = − + − ∇ r1 e e−r1 , 2 2 r1 r1 1 1 1 −r1 −r2 T y2 = − + 2 − + − + e r2 e−r1 −r2 . 2 r1 2 r1

∞

∞ r14 3 3 −2r1 = 2 dr1 r23 e−2r1 dr2 − + 2r1 − r1 e 2 0 0 ∞ ∞ 1 3 2 +2 r24 e−2r1 dr2 − r1 + r1 e−2r1 dr1 2 0 0 1 3 1 3 3 = 2· · +2· · = , 8 8 16 4 16

y1 T y2 dτ

∞

161

ATOMIC PHYSICS ∞

y3 T y1 dτ

−

= 2 0

= 2·

∞ r13 + 2r12 − r1 e−2r1 dr1 r23 e−2r2 dr2 2 0

1 3 · , 16 8 y4 T y dτ = 0,

∞ r14 2 2 −2r1 = 2 dr1 r22 e−2r2 dr2 − + 2r1 − r1 e 2 0 0 ∞ 3 ∞ r1 +2 r23 e−2r2 dr2 − + r12 e−2r1 dr1 2 0 0 ∞ 3 ∞ r +2 r23 e−2r2 dr2 − 1 + 2r12 − r1 e−2r1 dr1 2 0 0 ∞ 2 ∞ r1 +2 r24 e−2r2 dr2 − + r1 e−2r1 dr1 2 0 0 1 1 1 3 1 3 1 3 = 2· · +2· · +2· · +2· · 8 4 16 8 16 8 8 4 3 3 3 11 1 + + + = , = 16 64 64 16 32

y2 T y2 dτ

∞

∞ r13 2 −2r1 − + r1 e = 2 dr1 r22 e−r2 dr2 2 0 0 ∞ 2 ∞ r1 −2r1 +2 dr1 r23 e−2r2 dr2 − + r1 e 2 0 0 1 3 1 3 1 1 1 · +2· · = + = , = 2· 16 4 8 8 32 32 8

y2 T y3 dτ

∞

∞ r12 −2r1 = 2 dr1 r22 e−r2 dr2 − + r1 e 2 0 0 1 1 1 = 2· · = . 8 4 16

y3 T y3 dτ

∞

T1 (x1 x2 + y1 y2 + z1 z2 )e−r1 −r2 1 1 = (x1 x2 + y1 y2 + z1 z2 ) − + e−r1 −r2 2 r1 1 + (x1 x2 + y1 y2 + z1 z2 ) e−r1 −r2 . r1

162

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

1 2 = 2 (x1 x2 + y1 y2 + r1 r2 ) − + e−2r1 −2r2 dτ 2 r1 4 ∞ 2 ∞ r1 3 −2r1 dr1 r24 e−r2 dr = − + 2r1 e 3 0 2 0 2 3 3 3 = · · = . 3 8 4 16

y4 T y4 dτ

2

y1 U y1 dτ

= −4

∞ 0

∞

+2

r13 e−2r1 dr1

∞

∞

0

r14 e−2r1 dr1

0

r24 e−2r1 dr2 r23 e−2r2 dr2

r1

3 3 837 9 837 3771 =− + =− , = −4 · · + 2 · 8 4 8192 8 4096 4096 given that:

1 3 3 2 3 3 −2r2 =− , r + r + r2 + e 2 2 4 2 4 8 ∞ 1 3 3 2 3 3 −2r1 r23 e−2r2 dr2 = , r1 + r1 + r1 + e 2 4 4 8 r1 ∞ 1 7 3 6 3 5 3 4 4r1 r + r + r + r e dr1 2 1 4 1 4 1 8 1 0 3 720 3 120 3 24 1 5040 + + + = 2 4096 · 16 4 1024 · 16 4 4096 8 1024 135 45 9 837 315 + + + = . = 8192 4096 2048 1024 8192

r23 e−2r2 dr2

U y1 y2 dτ

= −4 −4

∞

0

0

∞ ∞

+2 0 ∞ +2 0

r13 e−2r1 dr1 r14 e−2r1 dr1 r14 e−2r1 dr

r23 e−2r2 dr2

∞ 0

∞

0 ∞

r23 e−2r2 dr2 r22 e−2r2 dr2

r22 e−2r2 dr2

r1 ∞

r13 e−2r1 dr1

r2

3 1 87 9 3 3 · −4· · +2· +2· 8 8 4 4 2048 128 9 3 87 9 1113 = − − + + =− , 16 4 1024 64 1024 = −4 ·

163

ATOMIC PHYSICS

because:

1 2 1 1 −2r1 = , r + r1 + e 2 1 2 4 r1 ∞ 1 3 3 2 3 3 −2r2 3 −2r1 r + r + r2 + e r1 e dr1 = , 2 2 4 2 4 8 r2 ∞ 1 6 1 5 1 4 −4r2 r + r + r e dr1 2 1 2 1 4 1 0 1 720 1 120 1 24 87 + + = , = 2 16384 2 4096 4 1024 2048 ∞ 1 6 3 5 3 4 3 3 −4r2 r + r + r + r e dr2 2 2 4 2 4 2 8 2 r2 1 720 3 120 3 24 3 9 9 + + + = . = 2 16384 4 4096 4 1024 8 256 128

∞

r22 e−2r2 dr2

y3 y1 U dτ

= −4

∞ 0

+2

∞

r12 e−2r1 dr1

since:

∞

∞

0

r13 e−2r1 dr1

0

= −4 ·

r22 e−2r2 dr2 r22 e−2r2 dr2

r1

1 3 33 3 33 159 · +2· =− + =− , 2 8 1024 8 512 512

1 2 1 1 −2r1 r + r1 + e = , 2 1 2 4 r1 ∞ 1 5 1 4 1 3 −4r2 r + r + r e dr1 2 1 2 1 4 1 0 1 120 1 24 1 6 15 3 3 33 = + + = + + = . 2 4096 2 1024 4 256 1024 256 512 1024

∞

U y2 y2 dτ

r22 e−2r2 dr2

∞

∞

= −12 r22 e−2r2 dr2 0 0 ∞ ∞ −2r1 −4 r1 e dr1 r23 e−2r2 dr2 0 ∞ 0 ∞ 4 −2r1 +2 r1 e dr1 r2 e−2r2 dr2 0 ∞ r1∞ +2 r22 e−2r2 dr2 r13 e−2r1 dr2 0 ∞ r2∞ 3 −2r1 +4 r1 e dr1 r22 e−2r2 dr2 0

r13 e−2r1 dr1

r1

164

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

3 1 1 3 21 63 33 · −4· · +2· +2· +4· 8 4 4 4 1024 1024 1024 21 63 33 15 95 405 9 3 + + =− + =− , = − − + 8 4 512 512 256 8 256 256 = −12 ·

because: 1 1 −2r1 r1 + e r2 e−2r2 dr2 = , 2 4 r1 ∞ 1 5 1 4 −4r2 r + r e dr1 2 1 4 1 0 1 24 15 3 21 1 120 + = + = , = 2 4096 4 1024 1024 512 1024 ∞ 1 3 3 2 3 3 −2r2 r2 + r2 + r2 + e r13 e−2r1 dr1 = , 2 4 4 8 r2 ∞ 1 3 3 2 3 3 −4r2 dr2 r + r + r2 + e 2 2 4 2 4 8 0 1 120 3 24 3 6 3 2 + + + = 2 4096 4 1024 4 256 8 64 9 9 3 63 15 + + + = , = 1024 512 512 256 1024 ∞ 1 2 1 1 −2r1 r22 e−2r2 dr2 = , r1 + r1 + e 2 2 4 r1 ∞ 1 2 1 1 −4r1 r + r1 + e dr1 2 1 2 4 0 1 120 1 24 1 6 + + = 2 4096 2 1024 4 256 3 3 33 15 + + = . = 1024 256 512 1024

∞

U y3 y2 dτ

= −4 −4

∞

0 ∞

0

∞

+2 0 ∞ +2 0

r12 e−2r1 dr1 r13 e−2r1 dr1

0 ∞

0

r22 e−2r2 dr2 r13 e−2r1 dr1

∞

∞

r2∞ r1

r22 e−2r2 dr2 r2 e−2r2 dr2 r12 e−2r2 dr1 r2 e−2r2 dr2

165

ATOMIC PHYSICS

1 1 3 1 1 9 · −4· · +2· +2· 4 4 8 4 32 512 25 135 5 =− , = − + 8 256 256 = −4 ·

given that:

1 2 1 1 −2r2 = , r + r2 + e 2 2 2 4 r2 ∞ 1 −4r2 1 2 1 r + r2 + e dr2 2 2 2 4 0 1 24 1 6 1 2 1 = + + = , 2 1024 2 256 4 64 32 ∞ 1 1 −2r1 −2r2 r2 e dr2 = , r1 + e 2 4 r1 ∞ 1 24 1 6 9 1 4 1 3 −2r1 r1 + r1 e + = . dr1 = 2 4 2 1024 4 256 512 0

∞

r12 e−2r1 dr1

y3 U y3 dτ

= −4

∞

0 ∞

+2 0

r1 e−2r1 dr1 r12 e−2r1 dr1

∞

0 ∞

r22 e−2r2 dr2 r2 e−2r2 dr2

0

1 1 5 1 5 27 = −4 · · + 2 · =− + =− , 4 4 256 4 128 128 because: 1 1 2r1 r2 e−2r2 dr2 = r1 + e , 2 2 r1 ∞ 1 6 1 2 5 1 3 1 2 −4r1 r1 + r1 e + = . dr1 = 2 4 2 256 4 64 256 0

∞

y4 U y1 dτ

= =

∞ 2 ∞ 4 −2r1 r e dr1 r22 e−2r2 dr2 3 0 1 r1 2 555 185 · = , 3 8192 4096

166

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

since:

1 2 1 1 −2r1 r1 + r1 + e r22 e−2r2 dr2 = , 2 2 4 r1 ∞ 1 7 1 6 1 5 −4r1 r + r + r e 2 1 2 1 4 1 0 1 720 1 120 1 5040 + + = 2 64 · 1024 2 16 · 1024 4 4 · 1024 45 15 555 315 + + = . = 8192 2048 2048 8192

U y2 y4 dτ

given that:

because:

∞

1 y2 y4 dτ r12 ∞ 2 ∞ 5 −2r1 r1 e dr1 r2 e−2r2 dr2 = 3 0 r1 ∞ 2 ∞ 4 −2r2 + r e dr2 r12 e−2r1 dr1 3 0 2 r1 2 15 2 87 5 29 49 = · + · = + = , 3 512 3 2048 256 1024 1024

=

1 6 1 5 −4r1 dr1 r + r e 2 1 4 1 0 1 720 1 120 45 15 15 + = + = , = 2 16384 4 4096 2048 2048 512 ∞ 1 6 1 5 1 4 −4r1 dr1 r + r + r e 2 1 2 1 4 1 0 1 120 1 24 1 720 + + = 2 16384 2 4096 4 1024 45 15 3 87 = + + = . 2048 1024 512 2048

∞

2 1 y3 y4 dτ = r12 3

∞ 0

r14 e−2r1 dr1

∞

r2 e−2r2 dr2 =

r1

1 5 1 4 −4r1 r + r e r1 2 1 4 1 0 1 120 1 24 15 3 21 = + = + = . 2 4096 4 1024 1024 512 1024

∞

7 , 512

167

ATOMIC PHYSICS

U

y42 dτ

since:

∞ 4 ∞ 3 −2r1 = − r e dr1 r24 e−2r2 dr2 3 0 1 0 ∞ 2 ∞ 4 −2r1 + r e dr1 r23 e−2r2 dr2 3 0 1 r1 ∞ ∞ 4 + r16 e−2r1 dr1 r2 e−2r2 dr2 15 0 r1 4 405 4 3 3 2 837 + = − · · + · 3 8 4 3 8192 15 8192 3 879 27 3 333 1203 = − − + =− + =− , 8 4096 2048 8 4096 4096

1 7 3 6 3 5 3 4 −4r1 r + r + r + r e dr1 2 1 4 1 4 1 8 1 0 3 720 120 3 24 1 5040 + + + = 2 64 · 1024 4 16 · 1024 4 · 1024 8 1024 135 45 9 837 315 + + + = , = 8192 4096 2048 1024 8192 ∞ 1 7 1 6 −4r1 r1 + r1 e dr1 2 4 0 1 5040 1 720 = + 2 64 · 1024 4 16 · 1024 45 405 315 + = . = 8192 4096 8192

∞

Normalization matrix y1

y2

y3

y4

y1

9 16

9 16

9 64

0

y2

9 16

21 32

3 16

y3

9 64

3 16

y4

0

0

Kinetic energy y1

y2

y3

y4

y1

3 16

3 16

3 64

0

0

y2

3 16

11 32

1 8

0

1 16

0

y3

3 64

1 8

1 16

0

0

3 16

y4

0

0

0

3 16

168

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

1 matrix r12

Potential energy y1

y2

y3

y4

y1

y2

y3

y4

y1 −

3771 1113 159 − − 4096 1024 512

185 4096

y1

837 231 4096 1024

33 512

185 4096

y2 −

1113 405 135 − − 1024 256 256

49 1024

y2

231 75 1024 256

25 256

49 1024

159 512

−

7 512

y3

33 512

5 128

7 512

185 4096

49 1024

y4

185 49 7 333 − 4096 1024 512 4096

y3

−

y4 −

135 27 − 256 128

7 1203 − 512 4096

Potential energy without interaction y1 y2 y3 y4

Energy without interaction y1 y2 y3

9 8

−

21 16

−

3 8

0

y1

−

21 16

−

15 8

−

5 8

0

y2

−

3 8

−

5 8

−

1 4

0

y3

−

y1

−

y2

−

y3

−

y4

0

0

0

−

3 8

25 256

y4

15 16

−

9 8

−

21 64

−

0

9 8

−

49 32

−

1 2

−

0

y4

21 64

0

1 2

0

3 16

0

0

−

3 16

169

ATOMIC PHYSICS

y1

Total energy y2 y3

y4

921 1024

−

135 512

185 4096

−

317 256

−

103 256

49 1024

135 512

−

103 256

−

19 128

7 512

185 4096

−

y1

−

3003 4096

−

y2

−

921 1024

y3

−

y4

49 1024

7 512

−

435 4096

2s2s 1 S And 2p2p 1 S Terms

3.6.7 [12 ]

2s2s 1 S : 2p2p 1 S :

y1 − y2 + y3 = q, y4 .

H = T + U = T + U0 +

1 r12

(y1 − y2 + y3 )L(y1 − y2 + y3 )dτ = L11 + L22 + L33 − 2L12 + 2L23 = qLq dτ q 2 dτ qU0 q dτ

9 21 1 9 9 3 1 + + − + − = ; 16 32 16 8 32 8 16 9 15 1 21 3 5 1 = − − − + − + =− ; 8 8 4 8 4 4 8 =

12 @ Remember that the 2s2s 1 S and 2p2p 1 S terms are superpositions of the terms called X and Y by the author. The notation used here is the same as in the previous subsection.

170

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

qT q dτ q

1 q dτ r12 qHq dτ

3 11 1 3 3 1 1 + + − + − = ; 16 32 16 8 32 4 16 75 5 231 33 25 77 837 + + − + − = ; = 4096 256 128 512 256 128 4096 3003 317 19 921 135 103 179 = − − − + − + =− . 4096 256 128 512 256 128 4096 =

y42 dτ

=

y4 U0 y4 dτ

=

y4 T y4 dτ

=

y4 r12 y4 dτ

=

y4 Hy4 dτ

=

3 ; 16 3 − ; 8 3 ; 16 333 ; 4096 435 − . 4096

9 9 9 9 − + = , 16 16 64 64 21 3 3 9 − + = , y1 (y1 − y2 + y3 )dτ = 16 32 16 32 9 3 1 1 − + = . y3 (y1 − y2 + y3 )dτ = 64 16 16 64 y1 (y1 − y2 + y3 )dτ =

3.6.8

1s1s Term

ψ ∼ e−r1 −r2 ,

r12 e−2r1 dr1

r22 e−2r2 dr2 =

ψ 2 = 16 e−2r1 −2r2 .

R − r < < R + r, dp =

d. 2Rr

1 , 16

171

ATOMIC PHYSICS

R+r 1 dl 1 dp = = , 2Rr R R−r R+r 1 1 R+r d 1 dp = = log , 2 2Rr 2−r 2Rr R−r 1 −2p 1 1 1 1 −2p e + r1 + . (p + r1 )e dp = − p + r1 + 2 2 4 4 4

p + 2r1 1 −2p 1 1 1 1 e log − + r1 + e p + r1 + 2 2 4 2 4 p 1 −2p 1 1 1 1 1 e dp. + r1 + + − p + r1 + 2 2 4 2 4 p(2r1 + p)

−2r1

1 R+r 1 log = 2 2Rr R−r R

1 2 ψ dr = 32 r12

1+

1 r2 1 r4 1 r2n + + . . . + + . . . . 3 R2 5 R4 2n + 1 R2n

∞

r12 e−2r1 dr1

0

∞

e−2r2 dr2

r1

∞ 1 ∞ 4 −2r1 1 −2r2 r1 e dr1 dr2 2e 3 0 r r1 2 ∞ 1 ∞ 6 −2r1 1 −2r2 + r e dr1 dr2 + . . . . 4e 5 0 1 r1 r2 +

r2 = tr1 (t > 1): 1 −2r1 −2r2 e dτ 16 r12

1 −2r1 −2r2 e dτ r2 >r1 r12 1 −(2+2t)r1 = 32 e dτ. t>1 r12

= 32

172

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

r12 r22 dr1 dr2 = t2 r15 dr1 dτ r2 + r1 1 t+1 1 . log = 2 log 2r1 r2 r2 − r1 t−1 2r1 t 1 2 ψ dτ = 32 e−(2+2t)r1 dt r12 t>1 t + 1 −2(1+t)r1 e tr13 log dr1 dt = 16 t−1 t>1 ∞ ∞ t+1 = 16 dt t log r13 e−2(1+t)r1 dr1 t − 1 1 0 ∞ t+1 t dt, = 6 log (t + 1)4 t−1 1 −2er , (er − 1)2 2er t+1= r , e −1 t (er − 1)3 (er + 1) = . (t + 1)4 16e4r

t+1 = er , t−1 er + 1 , t= r e −1 er − 1 1 = , t+1 2er

dt =

er + 1 (er − 1)4 t+1 2er t dt = − log r dr (t + 1)4 t−1 er − 1 16e4r (er − 1)2 (er + 1)(er − 1) dr. = − 8e2r 1 2 3 ∞ −r 3 1 2 −3r ψ dτ = (e + e )rdr = 1− = . r12 4 0 4 9 3 The probability curve p() (r1 + r2 > , |r1 − r2 | < ) for the mutual distance r12 is obtained as follows. ψ = 4 e−r1 −r2 ,

∞

p() = 8

ψ 2 = 16 e−2r1 −2r2 .

r1 e−2r1 dr1

0

= 8 0 ∞

+

r1 e−2r1 dr1 −2r1

r1 e

r2 e−2r2 dr2

|−r1 | +r1

dr1

r1 +

−r1 r1 +

r1 −

r2 e−2r2 dr2 −2r2

r2 e

dr2 .

173

ATOMIC PHYSICS

−2r2

r2 e

+r1

dr2 = −

r1 +

r1 −

e−2r2 dr2

e−2r2 ,

1 1 1 −2+2r1 − r1 + e 2 2 4 1 1 1 −2−2r1 + r1 + e − , 2 2 4 1 −2r1 +2 1 1 r1 − + e = 2 2 4 1 −2r1 −2 1 1 r1 + + − . e 2 2 4

r2 e−2r2 dr2 =

−r1

1 1 r1 + 2 4

1 1 2 1 −2 p() = 8 e − r1 + r1 + r1 dr1 2 2 4 0 ∞ 1 1 2 1 r1 − r1 + r1 e−4r1 dr1 +e2l 2 2 4 l ∞ 1 1 2 1 −2 −4r1 −e dr1 r + r1 + r1 e 2 1 2 4 0 1 1 3 1 2 = 8 e−2 + + . 12 8 16 p() =

1 2 2 1 2 + 3 + 4 e−2 = + + 2 2 e−2 . 2 5 2 3

−2x

e

1 dx = , 2

2 −2x

1 dx = , 4

4 −2x

3 dx = , 4

x e

x e

1 xe−2x dx = , 4 3 x3 e−2x dx = , 8 x5 e−2x dx =

15 . 8

174

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

2 r12

∞

=

= ...,

p() d

=

3 5 35 3 + + = , 16 4 4 16

p() d

=

1 3 1 + + = 1, 8 8 2

1 p() d

=

1 1 1 5 + + = , 8 4 4 8

1 p() d = 2

1 1 1 2 + + = . 4 4 6 3

∞

r12 =

2 p() d

0

0 ∞

1 = 0

1 r12 1 2 r12

∞

= 0

∞

= 0

——————– 2 3 4 4 −2 2 p () = + 2 + − e . 3 3

3.6.9

1s2s Term

The states are now given by: e−r1 −r2 ,

(r2 − 2)e−r1 − 2 r2 , 1

where the normalization factors are: N1 = 16,

1 N1 N 2 = , 8

N2 = 2,

√

√ 1 = 2 2, N1 N2

so that: 1 1 √ (r2 − 2)e−r1 − 2 r2 . 2

4 e−r1 −r2 ,

1 2 1 1 −2r1 r + r1 + =− , e 2 1 2 4 1 1 1 1 1 r2 2 −2r1 r e dr1 = − + + r2 e−2r2 , r2 0 1 4r2 4r2 2 2 1 −2r1 1 −2r1 r1 + e dr1 = − , r1 e 2 4 ∞ 1 1 −2r1 r1 e dr1 = + r2 e−2r2 . 4 2 r2

r12 e−2r1 dr1

175

ATOMIC PHYSICS

∞

∞

(r22 − 2r2 )e− 2 r2 dr2 0 r1 ∞ ∞ 3 2 − 32 r2 + (r2 − 2r2 )e r1 e−2r1 dr1 0 r3 ∞ 7 8 2 4 4 3 r1 − r1 − r12 e− 2 r1 dr1 = 3 9 27 0 ∞ 1 4 3 3 1 2 − 7 r2 + r − r − r e 2 dr2 2 2 4 2 2 2 0 32 4 16 8 2 8 · 24 · 5 − · 6 · 4 − ·2· 3 = 3 7 9 7 27 7 1 32 3 16 1 8 + · 24 · 5 − · 6 · 4 − · 2 · 3 2 7 4 7 2 7 128 128 384 72 1 512 − − + − − 8 . = 73 49 3·7 27 49 7

3.6.10

r12 e−2r1 dr1

3

Continuation

e−Z(r1 +r2 ) :

1 2 Hψ = −Z + ψ, r12 ¯ = Z 2 − 5 Z, H 8

Hψ · Hψdτ =

2r2 1 4 + 2 ψ2. Hψ · Hψ = Z − r12 r12 ¯ 2 = Z 4 − 5 Z 3 + 25 Z 2 ; (H) 4 64

¯ 2 = Z 4 − 5 Z 3 + 2 Z 2 = (H) ¯ 2 + 53 Z 2 . ψH 2 ψdτ = H 4 3 192

5 5 e(Z− 16 )(r1 +r2 ) , Z ∗ = Z − : 16 5 1 5 1 1 ∗2 − + Hψ = −Z − ψ, 16 r1 16 r2 r12

5 Z ∗2 5 Z ∗3 2Z ∗ 25 1 − + + Hψ · Hψ = Z ∗4 − 8 r1 8 r2 r12 256 r12 25 1 25 1 5 5 1 + + − − + 2 . 256 r22 128 r1 r2 8r1 r12 8r2 r12 r12 ——————–

176

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

1 method: λ 1s2s 1 S : 1s2s 3 S :

5 2 5 (Z − 0.1855)2 = Z 2 − 0.4637Z + 0.0430, 4 4 5 2 5 2 (Z − 0.1503) = Z 2 − 0.3758Z + 0.0282. 4 4

1s2s 1 S − Z 2 : 1s2s 3 S − Z 2 :

Z 2 3 4

Z 2 3 4

3.6.11

1 2 Z − 0.4637Z + 0.0430, 4 1 2 Z − 0.37582Z + 0.0282. 4

1s2s 1 S 1 2 1 2 Z − 0.4637Z Z − 0.4637Z + 0.0430 4 4 0.0726 0.1156 0.8589 0.9019 2.1452 2.1882 1s2s 3 S 1 2 1 2 Z − 0.3758Z Z − 0.3758Z + 0.0282 4 4 0.2484 0.2766 1.1226 1.1508 2.4968 2.5250

Other Terms Normalization matrix

p1 p2 p3 p4

= y1 − 3y2 + 9y3 , = y1 − 2y2 + 3y3 , = y 1 − y2 + y3 , = y4 .

p1

p2

p3

p4

p1

9 16

0

0

0

p2

0

3 32

0

0

p3

0

0

1 16

0

p4

0

0

0

3 16

177

ATOMIC PHYSICS Kinetic energy p1 p2 p3 p4

Potential energy without interaction p1 p2 p3 p4

p1

21 16

3 16

0

0

p1

−

27 8

−

p2

3 16

5 32

1 16

0

p2

−

3 16

−

p3

0

1 16

1 16

0

p3

0

−

p4

0

0

0

3 16

p4

0

Energy without interaction p1 p2 p3 p4 p1

−

33 16

p1

3 16

0

0

1 16

0

1 8

0

3 8

−

1 16

−

0

0

−

3 8

Interaction (1/r12 ) p2 p3

p4

0

0

0

p1

2205 4096

105 4096

21 4096

7 32

0

0

p2

105 4096

165 4096

1 4096

1 16

0

p3

21 4096

1 4096

77 4096

45 4096

3 16

p4

101 4096

45 4096

333 4096

−

p2

0

p3

0

0

p4

0

0

−

−

0

p1 p1

−

Total energy λ = 1 p2 p3

6243 4096

p2

105 4096

p3

21 4096

p4

101 4096

−

105 4096 −

731 4096

1 4096 −

39 4096

21 4096 1 4096 −

179 4096

45 4096

39 4096

p4 101 4096 −

39 4096

45 4096 −

435 4096

101 4096 −

39 4096

178

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS Total potential energy p1 p2 p3 p1

11619 4096

−

663 4096

p2

−

663 4096

−

1371 4096

−

255 4096

−

435 4096

p3

21 4096

−

255 4096

p4

101 4096

−

39 4096

21 4096

45 4096

p4 101 4096 −

39 4096

45 4096 −

1203 4096

[13 ]

p1 Lp1 dτ = L11 + 9L22 + 81L33 + 18L13 − 6L12 − 54L23 , p2 Lp1 dτ = L11 + 6L22 + 27L33 + 12L13 − 5L12 − 27L23 , p3 Lp1 dτ = L11 + 3L22 + 9L33 + 10L13 − 4L12 − 12L23 , p4 Lp1 dτ = L14 − 3L24 + 9L24 ,

p2 Lp2 dτ = L11 + 4L22 + 9L33 + 6L13 − 4L12 − 12L23 , p3 Lp2 dτ = L11 + 2L22 + 3L33 + 4L13 − 3L12 − 5L23 , p4 Lp2 dτ = L14 − 2L24 + 3L34 ,

p3 Lp3 dτ = L11 + L22 + L33 + 2L13 − 2L12 − 2L23 , p4 Lp3 dτ = L14 − L24 + L34 ,

13 @ The author evaluates the matrix elements of operators L, between p states, in terms of those between y states, already considered on the previous pages. In the following, we do not report the mere arithmetic calculations aimed at obtaining the numbers given in the tables.

179

ATOMIC PHYSICS

p4 Lp4 dτ = L44 .

——————–

4 p1 , 3 2 q2 = 4 p2 , 3 4 1 1 17 17 + √ − √ p3 − √ p4 , X = 4 2 4 241 2 4 241 3 4 17 17 1 1 Y = 4 p3 + √ p4 ; + √ + √ 2 4 241 3 2 4 241 q1 =

3 q1 , 4 1 3 = q2 , 4 2 17 17 1 1 1 1 + √ − √ X+ Y , = 4 2 4 241 4 2 4 241 √ √ 17 17 3 1 3 1 X+ Y . = − − √ + √ 4 2 4 241 4 2 4 241

p1 = p2 p3 p4

[14 ]

16 11 A , 9 16 2 12 q2 Aq1 dτ = A , 3 3 16 1 16 17 17 1 13 A − √ A14 , XAq1 dτ = + √ − √ 3 2 4 241 2 3 3 4 241 q1 Aq1 dτ =

14 @

For the new states considered by the author, see the previous footnote.

180

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

16 Y Aq1 dτ = 9 1

16 17 1 A13 + √ − √ 2 4 241 3 3

17 1 A14 , + √ 2 4 241

32 22 A , 3 √ 16 17 17 2 1 2 1 A23 − A24 , XAq2 dτ = 16 + √ − √ 3 2 4 241 3 2 4 241 √ 16 17 17 2 1 2 1 Y Aq2 dτ = 16 A23 + A24 , − √ + √ 3 2 4 241 3 2 4 241 q2 Aq2 dτ =

16 1 1 17 17 33 XAX dτ = 16 A + A44 + √ − √ 2 4 241 3 2 4 241 16 675 34 −√ A , 3 964 16 17 675 33 8 675 44 A34 , Y AX dτ = 668 A − A +√ √ 964 3 964 3 2 241 16 1 17 17 1 33 Y AY dτ = 16 A + A44 − √ + √ 2 4 241 3 2 4 241 16 675 34 +√ A . 3 965

[15 ] XX :

12.38026 A33 + 1.20658 A44 − 7.72988 A34 ,

Y Y :

3.61974 A33 + 4.12675 A44 + 7.72988 A34 ,

XY :

6.69427 A33 − 2.23142 A44 + 5.05789 A34 ,

Xq1 :

4.691 A13 − 1.465 A14 ,

Xq2 :

11.492 A23 − 3.588 A24 ,

Y q1 :

2.5368 A13 + 2.7086 A14 ,

Y q2 :

6.214 A23 + 6.635 A34 ,

15 @ In the original manuscript some numerical (arithmetic) calculations are given (not reported here), leading to the following expressions for the matrix elements.

181

ATOMIC PHYSICS

Normalization matrix q1 q2 X Y

q1

Total potential energy q2 X

Y

q1

1

0

0

0

q1

−5.063

−0.703

−0.012

0.0798

q2

0

1

0

0

q2

−0.708

−3.570

−0.8813

−0.4500

X

0

0

1

0

X

−0.012

−0.6813

−1.75410

0

Y

0

0

0

1

Y

0.0798

−0.4500

0

1.57753

Kinetic energy q2 X

q1

Y

q1

2.333

0.816

0

0

q2

0.816

1.687

0.7182

0.3884

X

0

0.7182

1.00000

0

Y

0

0.3884

0

1.00000

q1

Total energy λ = 1 q2 q3

q4

q1

−2.710

0.112

−0.012

0.0798

q2

0.112

−1.904

0.0370

−0.0617

q3

−0.012

0.0370

0.78410

0

q4

0.0798

−0.0617

0

0.51153

——————–

182

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

Total energy λ = 0.90 q1 q2 X q1

−2.649

q2

X

−0.0109

−0.0109 −1.863

−0.0315

−0.0315

−0.76869

Y

−2.631

q2

X

−0.0106

−1.851

−0.0434

−0.0434

0.76921

−2.611

q2

Y

0

0

Total energy λ = 0.86 q1 q2 X

X

Y

−0.0106

Y

q1

0

0

Total energy λ = 0.89 q1 q2 X q1

Y

−0.0104

Y

−0.0104 −1.837

−0.0547

−0.0547

0.76893

0

0

183

ATOMIC PHYSICS Total energy λ = 0.92 q1 q2 X q1

−2.665

q2

X

−0.0111

Y

3.7.

Y

−0.0111 −1.874

−0.0189

−0.0189

−0.76737

0

GROUND STATE OF THREE-ELECTRON ATOMS

An approximate expression for the energy (in rydbergs) W (which is equal to half the mean value of the potential energy) of the ground state of three-electron atoms with charge Z is here obtained, starting from particular forms for the wavefunctions ψ (or radial wavefunctions χ) of the three electrons. For further details, see Sect. 15 of Volumetto III, referring to the case of two-electron atoms. For Z → ∞ (ρ = Zr): ψ1 = ψ2 = a e−ρ ,

χ1 = χ2 = a ρ e−ρ ,

a ψ3 = √ (2 − ρ)e−ρ/2 . 4 2 a χ3 = √ ρ(2 − ρ)e−ρ/2 . 4 2

5 1 2 ψ1 (q1 )ψ12 (q2 ) dq1 dq2 = , r 4 12 1 2 1 13 1 2 = − 0.0802 = 0.4198, ψ1 (q1 )ψ32 (q3 ) dq1 dq3 = − r 2 162 2 13 32 1 = 0.0439. ψ1 (q1 )ψ3 (q1 )ψ1 (q3 )ψ3 (q1 ) dq1 dq3 = 2 r13 729 2

184

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

−W

32 q 2 5 13 1 Z+ Z − Z −2 − Z 4 4 2 162 729 9 2 5 580 Z Z − + 4 4 729 5 580 1 2Z 2 − Z + Z 2 − Z 4 4 729 5 1 2Z 2 − Z + Z 2 − 0.7956Z 4 4 5 1 2Z 2 − Z + (Z 2 − 3.1824Z) 4 4 5 1 2 2Z − Z + (Z 2 − 4Z + 0.8176Z). 4 4

= = = = = =

[16 ]

3.8. 3.8.1

GROUND STATE OF THE LITHIUM ATOM Electrostatic Potential

An expression for the electrostatic potential energy V of the lithium atom is obtained as a function of the distance r from the nucleus, by means of a semiclassical approach (a Poisson equation for V with an eﬀective charge density). A table with numerical values for this potential is given as well. See also Sect. 3.11. 2 ϕ2 (q1 , q2 )dq2 = k e−43r1 /8 . −

2 dV d2 V + 2 dr r dr

= k e−43r/8 ,

d2 (rV ) = k r e−43r/8 = k r e−αr , dr2 1 k d(rV ) r+ e−αk , − =− dr α α 2 k −rV = 2 r + e−αr + 1. α α

−

16 @ In the original manuscript, in the last line of the previous expression, the ﬁrst two terms are missing.

185

ATOMIC PHYSICS

k = α3 :

17

2 43 −43r/8 , + e r 8 4 43 −43r/8 2 . −2V = + + e r r 4 1 −V = + r

[18 ]

r

−2V

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8

∞ 109.363 49.649 30.041 20.495 14.978 11.469 9.0943 7.4170 6.1930 5.2760 4.5738 4.0257 3.5906 3.2395 2.9522 2.7137

3.8.2

„

3 2 V + r

«

10.750 10.637 10.351 9.959 9.505 9.022 8.531 8.0485 7.5830 7.1404 6.7240 6.3353 5.9743 5.6402 5.3319 5.0478 4.7863

r

−2V

« „ 3 2 V + r

0.85 0.9 0.95 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3

2.5132 2.3427 2.1959 2.0683 1.8571 1.6889 1.5512 1.4359 1.3376 1.2524 1.1779 1.1119 1.0531 1.0003 0.9525 0.9092 0.8696

4.5456 4.3240 4.1199 3.9317 3.5974 3.3111 3.0642 2.8498 2.6624 2.4976 2.3515 2.2214 2.1048 1.9997 1.9046 1.8181 1.7391

r

−2V

2.4 2.5 2.6 2.7 2.8 2.9 3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4

0.8334 0.8000 0.7692 0.7407 0.7143 0.6897 0.6667 0.6452 0.6250 0.6061 0.5882 0.5714 0.5556 0.5405 0.5263 0.5128 0.5000

Ground State

The electrostatic potential inside the lithium atom considered above is now used in order to determine (mainly, numerically) the Schr¨ odinger radial wavefunction for the ground state of this atom. χ + 2(E − V )X = 0; 17 @

The following expression for the electrostatic potential holds for the 2s term of lithium. The numerical values reported in the following table are obtained from the expression of V given just above. In the original manuscript the value in the sixth column corresponding to r = 2 is erroneously written as 2.9997 (instead of 1.9997). 18 @

186

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

W = −2E:

19

χ = (2V + W )χ. 6 2V = − + 10.75 + . . . , r 6 χ = − + 10.75 + W + . . . χ. r χ = x + ax2 + bx3 + . . . , χ = 2a + 6bx + . . . ;

(x = r)

6 2a + 6br + . . . = − + 10.75 + W + . . . (r + ar2 + . . .), r 2a + 6br = −6 + (10.75 + W − 6a)r; a = −3,

b=

28.75 + W . 6

[20 ] 19 @

The energy W is measured in rydbergs. In the following tables, Majorana gave the numerical values of the Schr¨ odinger radial wavefunction χ (and its derivatives) for some values of r. They should have been obtained by solving the diﬀerential equation reported above. However, it is interesting to note that the quoted numerical values do not come out neither by using the series expansion method outlined in the notes (method I), nor by solving numerically the equation with the approximate expression for the potential quoted just above (method II). Probably, the numerical values given by the author were obtained by considering the complete potential considered at the end of the previous subsection (method III). To give an idea of the departure from the Majorana tables, in the following we give some values of χ and its derivatives for W = 0.32, obtained by using the mentioned three methods. Notice that the series solution can be applied only for r 1: 20 @

r 0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00

Series solution (method χ χ 0.00000 1.0000 0.04311 0.7363 0.07484 0.5453 0.09885 0.4270 0.11876 0.3814 0.13820 0.4084 0.16081 0.5081 0.19023 0.6805 0.23008 0.9256 0.28400 1.2433 0.35562 1.6337 0.44859 2.0968 0.56652 2.6326 0.71306 3.2410 0.89184 3.9222 1.10648 4.6759 1.36064 5.5024 1.65794 6.4015 2.00201 7.3734 2.39648 8.4178 2.84500 9.5350

I)

χ −6.0000 −4.5465 −3.0930 −1.6395 −0.1860 1.2675 2.7210 4.1745 5.6280 7.0815 8.5350 9.9885 11.4420 12.8955 14.3490 15.8025 17.2560 18.7095 20.1630 21.6165 23.0700

r 0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00

Numerical solution (method χ χ 0.00000 1.00000 0.04307 0.73387 0.07437 0.52656 0.09652 0.36656 0.11165 0.24454 0.12148 0.15294 0.12735 0.08565 0.13037 0.03775 0.13138 0.00531 0.13110 −0.01481 0.13007 −0.02509 0.12872 −0.02748 0.12743 −0.02345 0.12647 −0.01416 0.12608 −0.00042 0.12649 0.01719 0.12786 0.03832 0.13038 0.06281 0.13420 0.09064 0.13950 0.12197 0.14646 0.15708

II)

χ −6.0000 −4.6851 −3.6392 −2.7940 −2.1145 −1.5716 −1.1381 −0.7923 −0.5163 −0.2968 −0.1210 0.0207 0.1361 0.2324 0.3150 0.3883 0.4565 0.5230 0.5910 0.6632 0.7425

187

ATOMIC PHYSICS

r 0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3

χ 0.00000 0.04307 0.07435 0.09641 0.11126 0.12047 0.12525 0.12652 0.12500 0.12126 0.11573 0.10876 0.10063 0.09156 0.08174

W = 0.32 χ 1.00000 0.73389 0.52574 0.36328 0.23620 0.13645 0.05780 −0.00456 −0.05426 −0.09407 −0.12608 −0.15188 −0.17269 −0.18944 −0.20285

χ −6.0000 −4.6969 −3.6675 −2.6853 −2.2448 −1.7668 −1.3964 −1.1201 −0.8871 −0.7122 −0.5736 −0.4626 −0.3729 −0.2995 −0.23864

χ 0.00000 0.04307 0.07435 0.09641 0.11128 0.12051 0.12530 0.12659 0.12510 0.12139 0.11589 0.10896 0.10087 0.09185 0.08208

W = 0.34 χ 1.00000 0.73392 0.52581 0.36341 0.23643 0.13678 0.05822 −0.00404 −0.05365 −0.09337 −0.12530 −0.15103 −0.17178 −0.18848 −0.20185

χ −6.0000 −4.6961 −3.6661 −2.8636 −2.2429 −1.7640 −1.3945 −1.1082 −0.8853 −0.7105 −0.5720 −0.4613 −0.3718 −0.2986 −0.23898

0.06042

−0.22174

−0.14463

0.06087

−0.22070

−0.14449

0.03764

−0.23261

−0.07613

0.03820

−0.23158

−0.07650

0.01409 −0.00972 −0.03359

−0.23753 −0.23793 −0.23483

−0.02463 0.01494 0.04571

0.01475 −0.00897 −0.03256

−0.23656 −0.23707 −0.23413

−0.02549 0.01561 0.04392

−0.07912

−0.22107

0.08829

−0.07819

−0.22081

0.08569

−0.12138

−0.20068

0.11317

−0.12045

−0.20101

0.10990

−0.15914

−0.17660

0.12602

−0.15835

−0.17763

0.12223

−0.19190

−0.18082

0.13055

−0.19139

−0.15265

0.12637

r 0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00

Numerical solution (method χ χ 0.00000 1.00000 0.04307 0.73381 0.07435 0.52575 0.09641 0.36328 0.11126 0.23620 0.12048 0.13644 0.12526 0.05778 0.12653 −0.00459 0.12501 −0.05430 0.12126 −0.09411 0.11573 −0.12613 0.10875 −0.15193 0.10062 −0.17274 0.09155 −0.18949 0.08173 −0.20289 0.07131 −0.21351 0.06041 −0.22179 0.04916 −0.22808 0.03764 −0.23266 0.02592 −0.23576 0.01408 −0.23758

III)

χ −6.0000 −4.6890 −3.6679 −2.8628 −2.2437 −1.7649 −1.3957 −1.1094 −0.8872 −0.7122 −0.5745 −0.4624 −0.3729 −0.2995 −0.2385 −0.1878 −0.1446 −0.1078 −0.0761 −0.0486 −0.0246

188

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

r 0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3

χ 0.00000 0.04307 0.07435 0.09642 0.11130 0.12054 0.12535 0.12667 0.12521 0.12153 0.11607 0.10918 0.10114 0.09216 0.08244

W = 0.36 χ 1.00000 0.73385 0.52590 0.36358 0.23667 0.13711 0.05865 −0.00352 −0.05304 −0.09268 −0.12453 −0.15019 −0.17088 −0.18753 −0.20086

χ −6.0000 −4.6952 −3.6648 −2.8619 −2.2410 −1.7621 −1.3925 −1.1064 −0.8835 0.7089 −0.5706 −0.4601 −0.3707 −0.2977 −0.23737

χ 0.00000 0.04307 0.07436 0.09643 0.11132 0.12058 0.12541 0.12675 0.12532 0.12167 0.11625 0.10940 0.10140 0.09247 0.08280

W = 0.38 χ 1.00000 0.73387 0.52597 0.36373 0.23691 0.13745 0.05908 −0.00300 −0.05243 −0.09199 −0.12377 −0.14937 −0.17000 −0.18660 −0.19989

χ −6.00000 −4.6944 −3.6635 −2.8602 −2.2392 −1.7602 −1.3907 −1.1045 −0.8819 −0.7073 −0.5692 −0.4588 −0.3697 −0.2969 −0.23677

0.06133

−0.21967

−0.14435

0.06179

−0.21867

−0.14419

0.03876

−0.23056

−0.07685

0.03932

−0.22957

−0.07717

0.01541 −0.00821 −0.03172

−0.23560 −0.23622 −0.23343

−0.02633 0.01229 0.04215

0.01607 −0.00746 −0.03089

−0.23467 −0.23539 −0.23275

−0.02713 0.01102 0.04043

−0.07725

−0.22054

0.08311

−0.07633

−0.22029

0.08060

−0.11951

−0.20132

0.10665

−0.11860

−0.20164

0.10347

−0.15754

−0.17865

0.11845

−0.15676

−0.17966

0.11473

−0.19086

−0.15447

0.12221

−0.19036

−0.15627

0.11808

189

ATOMIC PHYSICS

r 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.20 2.0 2.2 2.4 2.6 2.8 3.0

χ 0.0000000 0.0097048 0.0188379 0.0274266 0.0354970 0.0430739 0.050181 0.056841 0.063075 0.068904 0.074348 0.079425 0.084153 0.088549 0.092628 0.096406 0.099898 0.103117 0.106076 0.108788 0.111264 −0.19190 −0.21945 −0.24184 −0.25927 −0.27205 −0.28054

W = 0.32 χ 1.00000 0.94143 0.88565 0.83253 0.78196 0.73381 0.68798 0.64436 0.60285 0.56334 0.52574 0.48996 0.45591 0.42350 0.39625 0.36328 0.33532 0.30870 0.28335 0.25920 0.23620 −0.15082 −0.12476 −0.09936 −0.07526 −0.05287 −0.03241

χ −6.0000 −5.7155 −5.4432 −5.1828 −4.9341 −4.6969 −4.4706 −4.2548 −4.0493 −3.8537 −3.6695 −3.4903 −3.3220 −3.1620 −3.0099 −2.8653 −2.7280 −2.5976 −2.4738 −2.3563 −2.2448 0.13055 0.12930 0.12416 0.11646 0.10727 0.09726

χ 0.0000000 0.0097048 0.0188379 0.0274267 0.0354971 0.0430740 0.050181 0.056841 0.063076 0.068906 0.074350 0.079428 0.084157 0.088554 0.092635 0.096415 0.099908 0.103128 0.106089 0.108803 0.111281 −0.19139 −0.21940 −0.24242 −0.26066 −0.27444 −0.28411

W = 0.34 χ 1.00000 0.94143 0.88565 0.83253 0.78196 0.73382 0.68800 0.64439 0.60289 0.56340 0.52581 0.49004 0.45600 0.42360 0.39276 0.36341 0.33547 0.30887 0.28354 0.25941 0.23643 −0.15265 −0.12745 −0.10295 −0.07977 −0.05829 −0.03873

χ −6.0000 −5.7153 −5.4429 −5.1823 −4.9334 −4.6961 −4.4696 −4.2537 −4.0481 −3.8524 −3.6661 −3.4889 −3.3205 −3.1604 −3.0082 −2.8636 −2.7263 −2.5958 −2.4720 −2.3545 −2.2429 0.12637 0.12488 0.11961 0.11188 0.10272 0.09282

190

r 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.20 2.0 2.2 2.4 2.6 2.8 3.0

3.9.

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

χ 0.0000000 0.0097048 0.0188379 0.0274267 0.0354972 0.0430744 0.050182 0.056843 0.063078 0.068908 0.074353 0.079432 0.084162 0.088560 0.092642 0.096423 0.099918 0.103140 0.106103 0.108819 0.111300 −0.19086 −0.21931 −0.24296 −0.26202 −0.27679 −0.28764

W = 0.36 χ 1.00000 0.94143 0.88565 0.83254 0.78198 0.73385 0.68804 0.64444 0.60295 0.56347 0.52590 0.49015 0.45612 0.42374 0.39292 0.36358 0.33565 0.30906 0.28374 0.25963 0.23667 −0.15447 −0.13013 −0.10654 −0.08428 −0.06373 −0.04508

χ −6.0000 −5.7151 −5.4425 −5.1817 −4.9327 −4.6952 −4.4687 −4.2527 −4.0470 −3.8512 −3.6648 −3.4875 −3.3190 −3.3190 −3.0066 −2.8619 −2.7246 −2.5941 −2.4702 −2.3527 −2.2410 0.12221 0.12044 0.11502 0.10722 0.09807 0.08822

χ 0.0000000 0.0097048 0.0188381 0.0274270 0.0354976 0.0430749 0.050183 0.056844 0.063080 0.068911 0.074357 0.079436 0.084167 0.088566 0.092649 0.096431 0.099928 0.103152 0.106117 0.108835 0.111318 −0.19036 −0.21926 −0.24353 −0.26339 −0.27915 −0.29118

W = 0.38 χ 1.00000 0.94144 0.88566 0.83255 0.78199 0.73387 0.68807 0.64448 0.60300 0.56353 0.52597 0.49023 0.45622 0.42385 0.39305 0.36393 0.33582 0.30925 0.28395 0.25986 0.23691 −0.15627 −0.13278 −0.11009 −0.08876 −0.06916 −0.05147

χ −6.0000 −5.7149 −5.4421 −5.1812 −4.9320 −4.6944 −4.4678 −4.2516 −4.0459 −3.8500 −3.6635 −3.4860 −3.3175 −3.1573 −3.0050 −2.8602 −2.7228 −2.5923 −2.4684 −2.3508 −2.2392 0.11808 0.11603 0.11042 0.10251 0.09332 0.08348

−2V ∞ 589.26 289.27 189.29 139.32 109.363 89.409 75.175 64.519 56.249 49.649 44.265 39.796 36.029 32.814 30.041 27.628 25.511 23.641 21.980 20.495

ASYMPTOTIC BEHAVIOR FOR THE s TERMS IN ALKALI

The author looked for a solution of the Schr¨ odinger equation for alkali metals, at large distances from the nucleus. In such an asymptotic limit the potential energy experienced by the external electron is approximatively coulombian. Two diﬀerent methods were considered: in the ﬁrst one, the eigenfunction is written in the form of a polynomial times an exponential decreasing factor, while the second one is that typical of homogeneous diﬀerential equations (for lowering the order of the equation by one unit).

191

ATOMIC PHYSICS

3.9.1

First Method

E = −2W :

21

y =

y = P e− y

2 − + E y. r

√ Ex

, √ √ = (P − EP ) e− Ex , √ √ = (P − 2 EP + EP ) e− Ex ;

y

√ P − 2 EP + EP =

2 − + E P, r

√ 2 P − 2 EP + P = 0. r P = αn xn + αn−1 xn−1 + . . . , P = nαn xn−1 + (n − 1)αn−1 xn−2 + . . . , P = n(n − 1)αn xn−2 + (n − 1)(n − 2)αn−1 xn−3 + . . . . √ (r + 1) r αr+1 − 2r Eαr + 2αr = 0;

αr+1

√ 2(r E − 1) αr , = r(r + 1) 1 n= √ , E

αr =

r(r + 1) √ αr+1 . 2(r E − 1)

E=

1 . n2

For n → ∞, αn = 1 and αn−1 =

21 @

−(n − 1)n (n − 1)n2 √ . =− 2 2(1 − (n − 1) E)

Observe that the author apparently uses x or r to denote the same quantity. However, below, it is r = k + x, quantity k being the distance from the last node of the eigenfunction.

192

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

Denoting with D the distance from the last node: E

n

22

P

1

1

x

0.444

1.5

x2 −

0.25

2

x2 − 2x

0.16

2.5

0.111

3

D 0

3

9 1 27 − 1 x2 + x 2 + ... 16 512

x3 − 9x2 +

2

27 x 2

7.1

3.5 0.0625 4 ——————– Denoting with k the distance from the last node: r = k + x, √ 2 P − 2 EP + P = 0, r √ 2 P − 2 EP + P = 0. k+x P = a1 x + a2 x2 + a3 x3 + . . . , P = a1 + 2a2 x + 3a3 x2 + . . . , P = 2a2 + 6a3 x + 12a4 x2 + . . . . x2 x3 1 x 1 = − 2 + 3 − 4 + .... k+x k k k k 22 @ In the following table the author puts for some approximated expressions for the polynomial P for some maximum values n of the index r. For a given n, the ﬁrst one of the coeﬃcient αn is equal to 1, while the other non-vanishing coeﬃcients (with decreasing r) are obtained from the formula r(r + 1) √ αr+1 αr = 2(r E − 1) √ on setting E = 1/n. In the last column of the table, Majorana reports the distance from x = 0 of the greatest root of the considered polynomial. In the following, such a distance will be indicated by k.

193

ATOMIC PHYSICS

[23 ] P =

2a2

+ 6a3 x

+ 12a4 x2

+ 20a5 x3

...

√ √ √ √ √ −2 EP = − 2 Ea1 − 4 Ea2 x − 6 Ea3 x2 − 8 8a4 x3 . . . 2 P k+x

=

2 a1 x k

+

2 a2 x2 k

+

2 a3 x3 k

−

2 a1 x2 k2

−

2 a2 x3 k2 2 a1 x3 k3

...

√ 2a2 − 2 Ea1 = 0, √ 2 6a3 − 4 Ea2 + a1 = 0, k √ 2 2 12a4 − 6 Ea3 + a2 − 2 a1 = 0, k k √ 2 2 2 20a5 − 8 Ea4 + a3 − 2 a2 + 3 a1 = 0; k k k √ a2 = Ea1 , √ 1 3a3 = 2 Ea2 − a1 , k √ 1 1 6a4 = 3 Ea3 − a2 + 2 a1 , k k √ 1 1 1 10a5 = 4 Ea4 − a3 + 2 a2 − 3 a1 . k k k √ a2 = Ea1 , 2√ 1 a1 Ea2 − a3 = , 3 3 k 2√ 1 a2 a1 − 2 , Ea3 − a4 = 4 6 k k 2√ 1 a3 a2 a1 a5 = Ea4 − − 2+ 3 , 5 10 k k k ... 2√ 2 an−2 an−3 an−4 Ean−1 − an = − 2 + 2 ... . n n(n − 1) k k k 23 @ The following method is useful in order to determine the coeﬃcients of the series expansion for P which satisﬁes the diﬀerential equation reported above.

194

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

a2 =

√

Ea1 ,

1 1 2 2 1 a1 = a1 E− , a3 = Ea1 − 3 3 k 3 3k √ 1 3 1 a1 √ 1 Ea1 1 a1 E− a4 = e 2 a1 − + 3 6 k 6 k 6 k2 1 1 3 1 E2 1 1 . e2 − + = a1 3 3 k 6 k2 ——————– √

∼ = x + ax1/ E , √ a ∼ = 1 + √ x1/ E−1 , E √ 1 1 ∼ √ − 1 x1/ E−2 . = a√ E E

P P P √ P − 2 EP +

1 2 P ∼ = a√ k+x E

√ √ 1 √ − 1 x1/ E−2 − 2 E E

√ 1/ E−1

−2ax

√

2ax1/ E 2x + . + k+x k+x

——————–

y

= E− = E−

2 2 y= E− y r k+x x2 2 x + 2 2 − 2 3 + . . . y. k k k

Zeroth approximation: 2 y. y = E− k

First approximation: 2 2x y = E − + 2 y. k k 2 2/3 1/3 k 2 2 2x dx. E− + 2 , dx1 = x1 = 2 k k k2

195

ATOMIC PHYSICS

d2 y = dx21

k2 2

2/3 2 2x E − + 2 y = x1 y. k k x1 ∼ = −2.33;

x=0:

2/3 2 E− , k 2/3 2 2 ∼ , E − = −2.33 k k2

−2.33 ∼ =

2 E∼ = − 2.33 k

3.9.2

2 k2

2/3 =

x2 2

2 2.33 · 22/3 ∼ 2 3.70 − = − 4/3 + . . . . k k k k 4/3

Second Method R

y = e− udx , y = −u y, y = (u2 − u )y. 2 u2 − u = − + E, r 2 = 0. x √ a1 a2 a3 a4 u= E− − 2 − 3 − 4, x x x x u2 − u − E +

√ a0 = − E = −1/n.

a1 a3 a2 u = −a0 − − 2 − 3 − ..., x x x a1 a2 a3 1 u = 2 + 2 3 + 3 4 + ..., x x x 1 1 2 2 u = a0 + (a0 a1 + a1 a0 ) + (a0 a2 + a21 + a2 a0 ) 2 + . . . . x x √ 1 a0 = − E = − , n

196

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

2a0 a1 + 2 = 0,

a1 = −

1 1 = √ = n, a0 E

a0 ar + a1 ar−1 + . . . + ar a0 − (r − 1)ar−1 = 0,

r = 0, 1.

For r > 0 it is:

ar+1 = =

a1 ar + a2 ar−1 + . . . + ar a1 − rar √ 2 E n (a1 ar + a2 ar−1 + . . . + ar a1 − rar ). 2 1 a0 = − , n a1 = n, n3 n2 a2 = − , 2 2 n5 n3 − n4 + . a3 = 2 2 ——————– u = u2 − E − t = xE, √ u = p E,

2 . x

x=

t , E

u p= √ ; E

dp 1 du = 3/2 . dt E dx 1 2 dp = √ (p2 − 1) + √ , dt E t E dp 2n = n(p2 − 1) + , dt t p2 − 1 +

2 √ dp = E . t dt

197

ATOMIC PHYSICS

First approximation: p2 − 1 + p=

2 = 0; t

2 1− . t

[24 ]

3.10.

ATOMIC EIGENFUNCTIONS I

In this part, the author searches for solutions of the Schr¨ odinger equation with a screened Coulomb potential, likely to be applied to speciﬁc atomic problems, although it is not very clear what particular atom the author has in mind (probably he refers to the 1s term of lithium). See also the next Section. In the following we give detailed comments of the mathematical passages reported which, otherwise, would result of unclear interpretation. The equation:

k(k + 1) χ +2 E−V − χ=0 x2

can be solved by setting: R

χ = xk+1 e− u dx , R R k+1 − u xk+1 e− u dx , χ = (k + 1)xk − u xk+1 e− u dx = x R χ = k(k + 1)xk−1 − 2(k + 1)u xk − u x[ k + 1] + u2 xk+1 e− u dx R k + 1 2(k + 1)u 2 k+1 − u dx = − + u e . x − u x2 x We then have the following equation for u: u = 2(E − V ) −

24 @

2(k + 1) u + u2 . x

This Section was left incomplete by the author.

198

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

1st application. Let us consider the following form for the potential: V =a−

b . x

We thus have: u = 2E − 2a +

2b 2(k + 1) − u + u2 , x x

and for u = 0 we get: u=

b . k+1

The energy eigenvalue is: b2 1 −a . E=− 2 (k + 1)2 2nd application. For k = 0, let us consider a screened potential of the form V = −ZV /x, with ZV = 9 − 24.3x + 0x2 , and try a solution of the form: u = 9 − ax + bx2 . By this substitution we have: −a + 2bx 81 − 18ax + 2E − 48.6 + 2a − 2bx, so that:

2 a = − E − 10.8, 3 More in general, the equation: u = u2 + 2E + 2

9 b = − a. 2

ZV − u , x

with: ZV ∼ 8.5 − 15x, becomes: u ∼ u2 + 2E + 30 + 2

8.5 − u . x

199

ATOMIC PHYSICS

For u ∼ 8.5 we get E ∼ −21; other detailed results are reported in the following table 25 26 : x

ZV

0 0.05 0.10 0.15

9 7.85 6.92 6.20

E = −20 u u 9.000 −2.533 8.87 −2.1 8.81 −0.3 8.88 3.1

E = −21 u u 9.000 −3.20 8.83 −3.2 8.70 −1.9 8.65 0.1

E = −20 u u 9.000 −3.867 8.79 −4.3 8.59 −3.6 8.43 −2.6

For very small x, we have to push on the approximation; for example, for 0 < x < 0.05 we could use ZV = 9−24.3x+580x3 . We thus consider: ZV

= 9 − 24.3x + kx3 ,

u = 9 − ax − bx2 + cx3 , and substituting these expressions in the above diﬀerential equation for u, we get the unknown coeﬃcients: 2 9 1 2 a − 81a + 2k . a = − E − 10.8, b = − a, c = 3 2 4 In such an approximation, for the function χ deﬁned above and satisfying now (for k = 0) the equation ZV χ = −2 + E χ, x we obtain the values reported in the following tables: ZV

x

9 7.85 7.36 6.92 6.54 6.20 5.90 5.63 5.14 4.70

0 0.05 0.075 0.10 0.125 0.15 0.175 0.20 0.25 0.30 (0.35) 0.40 (0.45) 0.50

3.90

E χ 0 0.0320 0.0385 0.0413 0.0416

= −21.4 χ χ 1.000 −18 0.357 −8.68 0.177 −5.91 0.055 −3.95

In the original table, the author also reported the values of u for x = 0: −22.8, −28.8 and −34.8 for E = −20, E = −21 and E = −22, respectively. 26 @ The table was evaluated by the author by successive iterations, as can be deduced from the numerical calculations reported in the original manuscript. 25 @

200

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS ZV 9 7.85 7.36 6.92 6.54

x

E χ 0 0.0320 0.0385 0.0414 0.0418

0 0.05 0.075 0.10 0.125

= −21.7 χ χ 1.000 −18 0.358 −8.66 0.178 −5.89 0.057 −3.93

χ 0 0.0320 0.0386 0.0415 0.0419

E = −22 χ 1.000 0.359 0.180 0.059

χ −18 −8.64 −5.88 −3.92

In the considered interval 0 < x < 0.05 we could also use a screening factor XV = 9 − 23.2x and try for a solution of the form: cn xn . χ= n

Substituting it in the following equation: 18 χ =− − 46.4 + 2E χ, x we get the following iterative expression for the coeﬃcients: n(n − 1)cn = −18cn−1 + (46.4 − 2E) cn−2 , cn = −

18 46.4 − 2E cn−1 + cn−2 . n(n − 1) n(n − 1)

The ﬁrst coeﬃcients are c0 = 0, c1 = 1, c2 = −9, c3 = 27 +

27 :

46.4 − 2E , 6

81 − (46.4 − 2E) , 2 1 729 9 + (46.4 − 2E) + (46.4 − 2E)2 . 20 4 120

c4 = − c5 =

27 @ The original manuscript features some numerical calculations (whose interpretation seems unclear) that are apparently related to the solution here investigated.

201

ATOMIC PHYSICS

3.11.

ATOMIC EIGENFUNCTIONS II

The author looks for expressions for the atomic wavefunctions, obtained as solutions of the radial Schr¨ odinger equation. An explicit series solution for a lithium wavefunction is reported. χ + 2(E − V )χ = 0. For the 2s term of lithium: 1 −V = + r χ = P e−

√

2 43 + r 8

−2E r

e−43r/8 .

= P e−r/n ,

(n = n∗ ). √ −2E P e− −2E r , √ √ = P − 2 −2E P − 2EP e− −2E r .

χ = χ

P −

√

√ P − 2 −2E P − 2V P = 0. √

−2E =

1 , n

n = n∗ .

4 43 −43r/8 2 2 + + e P − P + P = 0. n r r 4

P =

∞

as rs ,

a1 = 1,

a0 = 0.

s=1

(−43/8) 2 (s − 1)as−1 + 2as−1 + 4 as−1− n ! s−2

s(s − 1)as −

43 (−43/8) + as−2− = 0. 4 ! s−3 =0

=0

202

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

[28 ] 2 (−43/8) s(s − 1)as − (s − 1)as−1 + 2as−1 + (4 − 2) as−1− = 0. n ! s−2 =0

——————– 1 2 4 43 −43r/8 χ + − 2+ + χ = 0, + e n r r 4

χ=

∞

ls r s ,

b0 = 0,

E=−

1 . 2n2

b1 = 1.

s=1

[29 ] 1 (−43/8) bs−1− = 0. s(s − 1)bs + 2bs−1 − 2 bs−2 + (4 − 2) n ! s−2 =0

[30 ] b1 b2 b3

n−2 = 0.34 n−2 = 0.35 n−3 = 0, 36 1.000000 1.000000 1.000000 −3.000000 −3.000000 −3.000000 4.848333 4.850000 4.851667 ——————–

2Z ( + 1) − y + 2E + x x2

y = 0.

λ = −2E.

2Z ( + 1) y + −λ + − x x2

y = e− 28 @

R

udx

,

y = −u y,

y = 0.

y = (u2 − u )y.

In the original manuscript the following expression is not explicitly equated to 0. As in the previous footnote. 30 @ In the original manuscript the author evidently intended to evaluate (from the previous iterative formula) also the coeﬃcients b4 , b5 , b6 , even for diﬀerent values of n−2 . 29 @

203

ATOMIC PHYSICS

u = u2 − λ + u∼−

2Z ( + 1) . − x x2

+1 x

for x → 0.

y(0) = y(x1 ) = y(x2 ) = . . . = y(xn ) = 0. U = x(x − x1 )(x − x2 ) . . . (x − xn )u = P u, P = x(x − x1 ) . . . (x − xn ). u=

U , P

u =

U P − U P ; P2

√ U = λ. x→∞ P lim

U P − U P = U 2 − λP 2 +

2Z 2 ( + 1) 2 P − P . x x2

For n = 0: P = x, P = 1,

√ U = λ x + a, √ U = λ.

U P − U P = −a, √ U 2 = λx2 + 2a λ x + a2 . √ −a = 2a λ x + a2 + 2Zx − ( + 1). √ a λ + Z = 0, λ=

Z2 , ( + 1)2

a2 + a − ( + 1) = 0; a = −( + 1).

——————–

204

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

y = eb/x+a (x + a)n x3 e−x/n . y = e−

R

udx

y = −u y,

,

y = (−u + u2 )y.

6 1 − u2 + u + 2 = −2V . 2 n x

u=

b 3−n 3 1 + − + . (x + α)2 x + a x n

For x → 0: 1 3−n b 3 + 2− u = − + + x n a a u

u2

3 − n 2b + 3 a2 a

ATOMIC ENERGY TABLES

Energy unit:

Ze2 /a0 = 2Z Rh.

Electrostatic energy 1s

2s

5

17

18

81

2p1

Exchange energy

2p0

2p−1

1s

2p−1

1s 1s

17

77

83

83

83

81

512

512

512

512

2s

2p0

x + ...,

3 − n 2b = + 3 + ..., a2 a 1 3−n b b 2 9 6 1 3−n + + 2 + + + 2 = − x2 x n a a n a a 3 − n 2b +6 + 3 + .... a2 a 3 − x2

3.12.

2p1

−

16 2s

83

237

447

237

512

1280

2560

1280

83

447

501

447

512

2560

2560

2560

83

237

447

237

512

1280

2560

1280

729

2s

2p0

2p−1

2p0

2p−1

729

−

15 2p1

2p1

16

512

15

15

15

512

512

512

−

15

27

512

2560

27

27

2560

1280

−

15

27

27

512

1280

2560

27 2560

−

205

ATOMIC PHYSICS number of electrons

conﬁgurations

energy -E/Rh

energy -E/Rh

Z2

Z2

1

1s

2

S

2

(1s)2

1

S

2Z + 2 −

3

(1s)2 s

2

S

9 2 5965 Z − Z 4 2916

4

(1s)2 (2s)2

1

S

5

(1s)2 (2s)2 (2p)2

2

P

6

(1s)2 (2s)2 (2p)2

3

P

6

(1s)2 (2s)2 (2p)2

1

6

(1s)2 (2s)2 (2p)2

1

(1s)2 (2s)2 (2p)3

4

7

5 Z 4

2Z 2 − 1.25Z

2.25Z 2 − 2.04561Z

S

D

S

7

7

7

3.13.

POLARIZATION FORCES IN ALKALIES

The author considered the polarization forces in alkali elements (in particular, in hydrogen and hydrogen-like atoms), obtaining some approximate expressions for the corresponding correction to the atomic energy levels.

206

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

∇12 ψ(q1 ) + 2(E1 − V1 )ψ(q1 ) = 0, ∇22 ϕ(q2 ) + 2(E2 − V2 )ϕ(q2 ) = 0, where ψ describes fast movements (short periods), while ϕ slow ones (large periods), and ψ, ϕ are separated. q1 = (x1 , y1 , r1 ),

q2 = (x2 , y2 , r2 ),

∂2 ∂2 2 , ∇ = . 2 ∂x21 ∂x22 ⎫ 2x1 x2 ⎪ ⎪ ⎪ ⎪ r3 ⎪ ⎪ ⎪ ⎪ ⎬ 2x x − y y − z z −y1 y2 1 2 1 2 1 2 − = V. 3 3 ⎪ r r ⎪ ⎪ ⎪ ⎪ ⎪ −z1 z3 ⎪ ⎪ ⎭ 3 r

∇12 = x1

2x1 r3

y1

y1 r3

z1

z1 r3

For s terms, r → ∞. ψ(q1 ) −→ ψ (q1 , q2 ): ∇12 ψ (q1 , q2 ) + 2(E1 + δE1 − V1 − V )ψ (q1 , q2 ) = 0, δE1 =

V ψψ dτ1 ,

δE1 = δE1 (q2 ).

At ﬁrst approximation: ψ (q1 , q2 ) = −ψ(q1 ) − 2x2 Zx (q1 ) − y2 Zy (q1 ) − z2 Zz (q1 ). Zx (q1 )Zy (q1 )dx1 dy1 dz1 = 0. Zx , Zy , Zz are inﬁnitesimals for r → ∞.

207

ATOMIC PHYSICS

Zx

is symmetric around x,

Zy

is symmetric around y,

Zz

is symmetric around z,

ψ(q1 )Zx (q1 )dτ1 = 0,

Zx (x1 , y12 + z12 ) = −Zx (−x1 , y12 + z12 ), ... f (x1 , y12 + z12 ) = −f (−x1 , y12 + z12 ).

Zx = f (x1 , y12 + z12 ), Zy = f (y1 , z12 + x21 ), Zr = f (z1 , x21 + y12 ),

δE1 ∼ = −(4x22 + y22 + r22 )

x1 1 ψ(q1 )rx dτ1 ∼ = r3 2

V ψ ψ dτ1 .

ϕ(q2 ) −→ ϕ (q2 ): ∇22 ϕ (q2 ) + 2(E2 + δE2 − V2 − δE1 )ϕ (q2 ) = 0, δE2 =

δE1 ϕ(q2 )ϕ (q2 )dτ2 ∼ =

δE1 ϕ2 (q2 )dτ2 .

ψ = ψ (q1 , q2 )ϕ (q2 ).

∇1 =

= = ∼ =

=

∂ ∂ ∂ i + j + k , ∂x1 1 ∂y1 1 ∂r1 1

∇2 =

∂ ∂ ∂ i + j + k . ∂x2 2 ∂y2 2 ∂r2 2

(∇12 + ∇22 )ψ + 2(E1 + E2 + δE2 − V1 − V2 − V )ψ ∇12 ψ + 2(E1 + δE1 − V1 − V )ψ + ∇22 ψ + 2(E2 + δE2 − V2 − δE1 )ψ ϕ (q2 )∇22 ψ (q1 , q2 ) + 2 ∇2 ϕ (q2 ) · ∇2 ψ (q1 , q2 ) ϕ (q2 )∇22 [ψ(q1 ) + 2x2 Zx (q1 ) − y2 Zy (q1 ) − z2 Zz (q1 )] +2 ∇2 ϕ (q2 ) · ∇2 [ψ(q1 ) + 2x2 Zx (q1 ) − y2 Zy (q1 ) − z1 Zz (q1 )] ∂ϕ (q2 ) ∂ϕ (q2 ) ∂ϕ (q2 ) 0+4 Zx (q1 ) − 2 Zy (q1 ) − 2 Zz (q1 ) ∂x2 ∂y2 ∂z2 ∂ϕ(q2 ) ∂ϕ(q2 ) ∂ϕ(q2 ) ∼ Zx (q1 ) − 2 Zy (q2 ) − 2 Zz (q1 ). = 4 ∂x2 ∂y2 ∂z2 ——————–

208

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

δW

∂ϕ (q2 ) ∂ϕ (q2 ) ψ (q1 , q2 )ϕ (q2 ) 2 Zx (q1 ) − Zy (q1 ) ∂x2 ∂y2 ∂ϕ (q2 ) Zz (q1 ) dτ1 dτ2 − ∂z2

∂ϕ (q2 ) 2 dτ2 δE2 − 4 Zx (q1 )dτ1 x2 ϕ (q2 ) ∂x2 ∂ϕ (q2 ) 2 + Zy (q1 )dτ1 y2 ϕ (y2 ) dτ2 ∂y2 ∂ϕ (q2 ) 2 + Zz (q1 )dτ1 z2 ϕ (z2 ) dτ2 ∂z2 ∂(q2 ) dτ2 . δE2 − 6 Zx2 (q1 )dτ1 x2 ϕ(q2 ) ∂x2

∼ = δE2 −

∼ =

∼ = x2 ϕ

∂ϕ dx2 dy2 dz2 = ∂x2 =

1 ∂ϕ2 x2 dx2 dy2 dz2 2 ∂x2 ∂(x2 ϕ2 ) 1 1 1 dτ2 − ϕ2 dτ2 = − . 2 ∂x2 2 2

dW ∼ = dE2 + 3

Zx2 (q1 )dτ1 .

——————–

x1 ψ1 =

∞

ak ψ k ,

1 ∞ 1 ak , −Zx = 3 1 r E1 − E1k 1 a2k = x21 ψ12 dτ1 .

dE2 ∼ = −6

x22 ϕ2 dτ2

a2k 1 , 1 − Ek r6 E 1 1 k

a2k 6 dE2 = − 6 x22 ϕ2 dτ2 . 1 k r E1 − E1

209

ATOMIC PHYSICS

dW

Zx2 (q1 )dτ1 a2k a2k 6 3 2 2 = − 6 ϕ dτ + . x 2 2 r r6 E11 − E1k (E11 − E1k )2

= dE2 + 3

On denoting with αψ the electric susceptivity, a2k , αψ = 2 r3 ψZx dτ1 = 2 E11 − E1k and with α the susceptivity of the ﬁrst atom, we get: 3α x22 ϕ2 dτ2 . dE2 = − 6 r 2 ak = x21 ψ 2 dτ, $ 2 2 αk2 x1 ψ dτ α . = = − 1 k 2 W E1 − E1 For hydrogen, W = 0.444

e2 . α0

αk2 > (E11 − E1k )2

$

x21 ψ 2 dτ = W2

$

x21 ψ 2 dτ W W1

(W1 is slightly lower than W ). At a very approximate level: $ 2 2 x1 ψ dτ1 αk2 ∼ . = 1 k 2 W2 (E1 − E1 ) 6 2 2 x1 ψ dτ1 x22 ϕ2 dτ 2 dE2 = − W r6 13.5 (for hydrogen this equals to 6 ). r dW = dE2 + 3 Zx2 (q1 )dτ1 6 3 2 2 2 2 ∼ x1 ψ dτ1 x2 ϕ dτ2 + 6 x21 ψ 2 dτ1 = − W r6 r W W1 ⎛ ⎞ ⎜ ⎟ 6 1 2 2 ∼ ⎟. 1− ψ dτ x x22 ϕ2 dτ2 ⎜ = − 1 1 ⎝ ⎠ 6 Wr 2 2 2W1 x2 ϕ dτ2 ——————–

210

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

For hydrogen-like atoms (Z1 ≥ Z2 ): ψ,

Z1 ,

1 E1 = Z12 ; 2

ϕ,

Z2 ,

1 E2 = − Z22 . 2

W = 0.444 Z12 , x21 ψdτ1 =

W1 < 0.444 Z12 .

1 , Z12

13.5 δE2 = − 6 4 2 , r Z1 Z2

x22 ϕ2 dτ2 =

13.5 δW = − 6 4 2 r Z1 Z2

1 . Z22

Z2 1− 2 2W1

.

2W1 ∼ = 0.87 Z12 .

2W1 < 2W,

Z22 13.5 , δE2 = − 6 4 2 . 2 0.87 Z1 r Z1 Z2 13.5 Z23 Z2 δW = − 6 3 3 − , r Z1 Z2 Z1 0.87 Z13

13.5 δW ∼ =− 6 4 2 r Z1 Z2

1−

Z2 = p, Z1 13.5 δW = − 6 3 3 r Z1 Z2 q= p+ 1 − p= 2q

1 p+

1 p

1 1 = , p q

p3 . p− 0.87

=

p . p2 + 1

p2 −

1 p + 1 = 0, q

+ 1 − 1 − 4q 2 1 2q 2 + 2q 4 + . . . − 1 = = , 4q 2 2q 2q

p = q + q3 + . . . ,

p3 = q 3 + . . . .

13.5 δW ∼ =− 6 3 3 r Z1 Z2

q3 3 . q+q − 0.87

211

ATOMIC PHYSICS

By extrapolating to any value of p: 1 −1∼ = 0.15, W1 δW = −

13.5 (q − 0.15q 3 ). r6 Z13 Z23

For Z1 = Z2 , q = 1/2: δW = −

3.14.

13.5 r6 Z13 Z23

(0.5 − 0.15 (0.5)3 )) = −

13.5 · 0.481 6.49 = 6 3 3. 3 3 6 r Z1 Z2 r Z1 Z2

COMPLEX SPECTRA AND HYPERFINE STRUCTURES

In this Section, Majorana studied the problem of the hyperﬁne structure of the energy spectra of complex atoms. The starting point was the (non-relativistic) Land´e formula for the hyperﬁne splitting, which is then generalized to the case (which the author calls the “non Coulomb ﬁeld” case) when the complex atom may be regarded as made of an inner part with an average eﬀective nuclear charge Z1 , and an outer one with an eﬀective nuclear charge Ze , and a principal quantum number n∗ [see, for comparison, the papers by E. Fermi and E. Segr`e, Mem. Accad. d’Italia 4 (1933) 131 and S. Goudsmith, Phys. Rev. 43 (1933) 636]. The hyperﬁne separations between a given group of energy levels were considered in the framework introduced by Houston [see W.V. Houston, Phys. Rev. 33 (1929) 297 and especially E.U. Condon and G.H. Shortley, Phys. Rev. 35 (1930) 1342], where X stands for the exchange perturbation energy (which is eﬀective in the Russell-Saunders or L − S conﬁguration) and A is the perturbation integral measuring the spin energy (which is, instead, eﬀective in the j-j coupling. It is interesting to note that Majorana considered also a generalization of the two mentioned couplings, where both X and A play a role.) The Land´e formula for the hyperﬁne structures (without relativistic corrections) is 2( + 1 1 μ20 , i g(i) cos(i, j) δW = 1840 j+1 r3 cos(i, j) =

i(i + 1) + j(j + 1) − ( + 1) . 2ij

212

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

For the s terms:

δW =

1 r3

= 2πψ 2 (0),

μ20 8π 2 i g(i) cos(i, j) ψ (0). 1840 3

In a Coulomb ﬁeld:

1 r3

Z3 1 , 3 3 a0 n + 12 ( + 1)

and for the s terms: ψ 2 (0) =

δW

= =

Z3 1 . a30 πn3

Z3 4 μ20 i g(i) cos(i, j) 3 3 1840 a0 n (j + 1)(2 + 1) 2 2Z 3 α Rh i g(i) cos(i, j) 3 , 1840 n (j + 1)(2 + 1)

which is valid also for s terms. The Rydberg corrections are Z2 2Rh 3 n

i g(i) cos(i, j)

Z (j + 1)(2 + 1)

.

In a non-Coulomb ﬁeld, an expression analogous to Land´e formula holds:

δW =

α2 Rh 2Z1 Ze2 i g(i) cos(i, j) ∗3 . 1840 n (j + 1)(2 + 1)

α2 Rh = 5.83 cm−1 ,

α2 Rh = 3.17 · 10−3 cm−1 . 1840

The values of 1 1 ∗3 n (j + 1)((2 + 1)

213

ATOMIC PHYSICS

are reported in the following table: n

s

p1

p3

d3

d5

f5

f7

2

2

2

2

2

2

1

2 3

2 9

2 15

2 25

2 35

2 49

2 63

2

1 12

1 36

1 60

1 100

1 140

1 196

1 252

3

2 81

2 243

2 405

2 675

2 945

2 1323

2 1701

4

1 96

1 288

1 480

1 800

1 1120

1 1568

1 2016

s, p 1

p3

d3

d5

f5

f7

2

2

2

2

2

2

1 , 3

1 , 5

3/2 = 1, (j + 1)(2 + 1)

3 , 25

3 , 35

3 , 49

1 . 21

[31 ] n

s

p1

p3

d3

d5

f5

f7

2

2

2

2

2

2

1

1

2

1 8

1 24

1 40

3

1 27

1 81

1 135

1 225

1 225

1 315

4

1 64

1 192

1 320

3 1600

3 2240

3 3136

1 1344

By using the Houston formula (Goudsmith method), for the terms 3 p012 , 1 p we have: 1

31 @ The values in the following table were obtained by multiplying those in the previous one by 3/2.

214

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

⎧ ⎨ j=2 s1 p3 2

2

⎩

j=1

⎧ ⎨ j=1 s1 p1 2

2

⎩

j=0

and, in general, for the terms 3 L−1, , +1 , 1 L :

⎧ ⎨ j =L+1 S 1 L+ 1 2

2

⎩

j=L

⎧ ⎨ j=L S 1 L− 1 2

2

⎩

j =L−1

In the Russell-Saunders approximation (A = 0) the energy of the given levels are as follows: ⎧ ⎨ singlet: X, ⎩

triplet: 0;

j = + 1, j = , j = , j = − 1, E =

0,

X,

0,

0.

For the j-j coupling, the energy of the given levels are instead as follows: ⎧ ⎨ S1/2 L+1/2 : A, ⎩

S1/2 L−1/2 : − A( + 1);

215

ATOMIC PHYSICS

j = + 1, j = , E =

A,

j = − 1,

j = ,

−A( + 1), −A( + 1).

A,

E−1 = −A( + 1).

E+1 = A,

E 2 + a1 E + a2 = 0, ⎧ ⎨ a1 = c1 X + c2 A, ⎩

a2 = c3 X 2 + c4 A2 + c5 XA.

A=0

X=0

A=0

X=0

a1 = −X, a2 = 0,

a1 = +A, a2 = −A2 ( + 1);

a1 = c1 X, a2 = c3 X 2 ,

a1 = c2 A, a2 = c4 A2 ;

c1 = −1,

c2 = +1,

c4 = −( + 1).

c3 = 0,

E 2 + (A − X)E + [c5 AX − ( + 1)A2 ] = 0. Adopting A as energy unit, and measuring X in A units (instead of considering X/A): E 2 − (X − 1)E + [c5 X − ( + 1)] = 0. For X → ∞, the two roots of the previous equation are E = X, E E = −X,

E = −1; E E = c5 X,

c5 = −1. E 2 − (X − 1)E − [X + ( + 1)] = 0. X −1 ± E= 2

X +1 2

2 + ( + 1).

216

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

1

3

X +1 2

2 + ( + 1),

L+1 = , 3

3

L =

X −1 + 2

L =

X −1 − 2

X +1 2

2 + ( + 1),

L−1 = −( + 1). ——————–

For the L − S coupling:

f (ri ) si · i = a S · L.

Ψmm

g 1 r r =√ ψm ϕm ; g r=1

m = S, S − 1, . . . , −S.

m = L, L − 1, . . . , −L; For g = 4: ϕ1

ϕ2

ϕ3

ϕ4

ϕ1 a11 S a12 S a13 S a14 S ϕ2 a21 S a22 S a23 S a24 S ϕ3 ϕ4 r r ϕm = Hψm

4

b44 L

Ai B i L S =

i=1

b11 L b12 L b13 L b14 L b21 L b22 L

i t t Lmm1 Sm m1 Airs Brt ψm ϕm ; 1

i,m1 ,m1 ,s,t

[32 ] g H 1 HΨmm = √ Ψmm = √ g g r=1

i t t Lmm1 Sm m1 Airs Brt ψm ϕm ;

i,m1 ,m1 ,r,s,t

1

√ In the original manuscript, the factor 1/ g, appearing before the second sum in the following expression, is omitted.

32 @

217

ATOMIC PHYSICS

HΨmm =

Hmm ,m1 m1 Ψm1 m1 ,

HΨmm |Ψab = Hmm ,ab , ⎛

Hmm ,ab

⎞ 1 i ⎠ =⎝ Airt Brt Lma Sm b . g i,r,t

——————– E 2 − (X − 1)E − [X + ( + 1)] = 0. For an atom in a magnetic ﬁeld H there is an additional contribution to the energy of the form Hμ0 mg; redeﬁning Hμ0 m → H we have:33 E 2 − (X − 1 + pH)E − [X + ( + 1)] + qXH + tH = 0. Since the considered unperturbed energy levels have diﬀerent multiplicities g and g , the contribution of H is twofold, g H and g H: ⎧ ⎪ ⎨ g + g = p, X −1 X +1 2 ⎪ qX + t = p + (g − g ) + ( + 1). ⎩ 2 2 ——————– Transitions between three energy levels A,B,C:

34

33 @ In the following expression, as reported in the original manuscript, the factor E in the second term and the equating to zero is lacking. 34 @ In the following, E denotes the electric ﬁeld, q AC , qBC the electric dipole moments and νAC , νBC the frequencies of the given transitions.

218

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

MAB = qAC MAC =

c

Transition 2 P − 2 D:

E · qAC E · qBC + qBC ; hνAC hνBC

1 + 1 qAC |qBC | = PAC PBC . hνAC νAC c

35

1 2

3 2

3 2

1 2

1 2

3 2

1 2

1 2

√ √ √ √

5·9+

√ 1·1

100

2·5

5

5·1

5

5·5

25

Transition 2 P − 2 F :

35 @ The numbers in the following tables indicate the amplitudes (third column) and intensities (fourth column) of a spectral line associated with a given transition between two energy levels (speciﬁed in the ﬁrst two columns).

219

ATOMIC PHYSICS 2P 2P 2P 2P

3 2 3 2 1 2 1 2

√

—2 F 7 —2 F —2 F —2 F

√

2 5 2

0 √

7 2 5 2

9 · 20

√ 7 · 1 + 1 · 14

180

5 · 14

70

45

Relative intensity between P 3 and P 1 : 225/70=3.2. 2

3.15.

2

CALCULATIONS ABOUT COMPLEX SPECTRA

[36 ] Eigenvalues of η: j(j + 1) − j (j + 1) − 6. j = j + 2 j = j − 2,

j (j + 1) = (j − 2)(j − 1) = j 2 − 3j + 2; η = 4j = 4j − 8,

−η = 8 − 4j.

−4j + 4m A 0 0 0

A −4j + 2m + 6 B 0 0

0 B −4j + 8 C 0

0 0 C −4j − 2m + 6 D

0 0 0 D −4j − 4m

where:37 + A = 2 (j − m)(j + m − 3),

B=

+ 6(j − m − 1)(j + m − 2),

It appears here the reference to an unknown “second appendix of the §10” [see, probably, E. Fermi and E. Segr`e, Mem. Accad. d’Italia 4 (1933) 131]. 37 @ The symbols A, B, C, D do not appear in the original manuscript, but have been introduced here for obvious typographic reasons (the matrix is much too large). 36 @

220 C=

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

+

6(j − m − 2)(j + m − 1),

+ D = 2 (j − m − 3)(j + m).

[38 ]

1 j j− 2

3 (j − 1) j − , 2

2j(2j − 1)(2j − 2)(2j − 3). j = 5: 38 @ The original manuscript continues with some calculations aimed at ﬁnding the four nonvanishing eigenvalues of the matrix above (whose determinant is equal to 0), the product of which, apart from a numerical factor, is set far below in the text (framed expression). Only for the present case, we have chosen to reproduce those calculations, in this footnote, since the method followed by the author is particularly interesting. For the other cases, with diﬀerent matrices, appearing in this Section we do not report the analogous calculations.

−4(j − m − 3)(j + m) + 4(j + m)(4j + 2m − 6) −4j 2 + 4m2 + 12j + 12m + 16j 2 + 24jm + 8m2 − 24j − 24m =

12j 2 + 24jm + 12m2 − 12j − 12m

=

12(j + m − 1)(j + m). p

6(j − m − 1)(j − m)(j + m − 3)(j + m − 2) (j + m − 1)(j + m) p 48 6(j − m − 1)(j + m − 2) (j − m)(j + m − 1)(j + m)

24

144(j − m − 1)(j − m)(j + m − 1)(j + m) p 48 6(j − m − 2)(j + m − 1) (j − m − 1)(j − m)(j + m) p 24 6(j − m − 3)(j − m − 2)(j + m − 1)(j + m) (j − m − 1)(j − m) p

(j + m − 3)(j + m − 2)(j + m − 1)(j + m) p 2 (j − m)(j + m − 2)(j + m − 1)(j + m) p 6(j − m − 1)(j − m)(j + m − 1)(j + m) p 2 (j − m − 2)(j − m − 1)(j − m)(j + m) p (j − m − 3)(j − m − 2)(j − m − 1)(j − m)

m = 0 (this is imposed since the eigenvalues do not depend on m) 2(j − 3)(j − 2)(j − 1)j 8j(j − 2)(j − 1)j 6(j − 1)j(j − 1)j 2j(j − 1)[(j − 3)(j − 2) + 4(j − 2)j + 3(j − 1)j]

221

ATOMIC PHYSICS

10 · 9 · 8 · 7 = 5040. m=0

m=1

m=5

120 1200 2400 1200 120 5040

360 1920 2160 576 24 5040

5040 0 0 0 0 5040

——————– j =j+1 η = j(j + 1) − (j − 1)j − 6 = 2j − 6,

−η = −2j + 6.

−2j + 4m − 2 A 0 0 0

A −2j + 2m + 4 B 0 0

0 B −2j + 6 C 0

0 0 C −2j − 2m + 4 D

0 0 0 D −2j − 4m − 2

where:39

+ + A = 2 (j − m + 1)(j + m − 2), B = 6(j − m)(j + m − 1), + + C = 6(j − m − 1)(j + m), D = 2 (j − m − 2)(j + m + 1).

[40 ] 2j(2j − 1)(2j − 2)

2j + 2 , 4

39 @ The symbols A, B, C, D do not appear in the original manuscript, but, once more, they have been introduced here for obvious typographic reasons (the matrix is much too large). 40 @ The original manuscript continues with some calculations aimed at ﬁnding the four nonvanishing eigenvalues of the matrix above (whose determinant is equal to 0), the product of which, apart from a numerical factor, is given below in the text (framed expressions).

222

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

2j(2j − 1)(2j − 2)

j+1 , 2

2(j − 1)j(j + 1)(2j − 1). j = 5: 2 · 4 · 5 · 6 · 9 = 2160. m=0

m=1

m=5

360 720 0 720 360 2160

600 480 144 768 168 2160

720 1440 0 0 0 2160

j = j η = j(j + 1) − j (j + 1) − 6 = −6.

4m − 2 A 0 0 0

A 2m + 4 B 0 0

0 B 6 C 0

0 0 C −2m + 4 D

0 0 0 D −4m − 2

where:41 + + A = 2 (j − m + 2)(j + m − 1), B = 6(j − m + 1)(j + m), + + D = 2 (j − m − 1)(j + m + 2). C = 6(j − m)(j + m + 1), [42 ] 41 @ The symbols A, B, C, D do not appear in the original manuscript, but, once again, they have been introduced here for obvious typographic reasons (the matrix is much too large). 42 @ The original manuscript continues with some calculations aimed at ﬁnding the four nonvanishing eigenvalues of the matrix above (whose determinant is equal to 0), the product of which, apart from a numerical factor, is given below in the text (framed expressions).

223

ATOMIC PHYSICS

4 j(j + 1)(2j − i)(2j + 3), 6

2j(2j − 1)

(2j + 2)(2j + 3) . 6

m=0

m=1

m=5

840 30 600 30 840 2340

900 30 486 252 672 2340

180 810 1350 0 0 2340

j = 5:

3.16.

RESONANCE BETWEEN A p ( = 1) ELECTRON AND AN ELECTRON WITH AZIMUTHAL QUANTUM NUMBER

Complex spectra are again considered, now evaluating resonance terms between electrons belonging to diﬀerent shells.

Exchange energy: K(n, 1, m ; n , l , m ) = r

Gk = e (4π)

2 0

∞ ∞

bk Gk ,

R(n, 1, r)R(n , l , r)R(n, 1, r )R(n , l , r )

0

× where:

rnk 2 2 r r dr dr , k+1 r

224

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

m m b0

b1

b2

b3 b4

= 1 = 1 ±1 ±1 ±1 0 ±1 ∓1 0 0

1 0 0 1

0 0 0 0

1/25 3/25 6/25 4/25

0 0 0 0

0 0 0 0

= 1 = 2 ±1 ±1 ±1 ±1 ±1 0 0 0

0 2/5 0 1/5 0 1/15 0 0 0 0 0 0 0 1/5 0 4/15

0 0 0 0 0 0 0 0

3/245 9/245 18/245 30/245 90/245 15/245 24/245 27/245

0 0 0 0 0 0 0 0

±2 ±1 0 ∓1 ∓2 ±2 ±1 0

Only the coeﬃcients b −1 and b +1 are non vanishing.

3.16.1

Resonance Between A d Electron And A p Shell I

`

´ ms = 1/2 m ms 1 −1/2 0 −1/2 −1 −1/2 1 1/2 0 1/2 −1 1/2

` m = 2 0 0 0 A F E S

m = 1 0 0 0 B G D S

´ ms = 1/2 m = 0 m 0 0 0 C H C S

= −1 0 0 0 D G B S

m = −2 0 0 0 E F A S

R R R

where:43 43 @ The symbols A, B, C, D, E, F, G, H, R, S do not appear in the original manuscript, but have been introduced here for typographic reasons. Note that in the last row the author gave the sum of all the terms in the corresponding column (for example, S = A + F + E, or S = B + G + D, etc.). He proceeded similarly with respect to the last column (for example, R = A + B + C + D + E, etc.).

225

ATOMIC PHYSICS

2 3 1 9 A = G1 + B = G1 + G3 , G3 , 5 245 5 245 18 30 1 G3 , G3 , D= C = G1 + 15 245 245 15 45 F = G3 , G3 , E= 245 245 1 24 4 27 G3 , G3 , G = G1 + H = G1 + 5 245 15 245 63 2 24 2 G3 , R = G1 + G3 . S = G1 + 5 245 3 49

3.16.2

Eigenfunctions Of d 52 , d 32 , p 32 And p 12 Electrons

The eigenfunctions are expressed by means of the notation (n , , mj , ms ). We replace (n , , m , ms ) simply with (m , ms ). For d 5 : 2

j

m

5 2

5 2

5 2

3 2

5 2

1 2

5 2

1 − 2

5 2

−

3 2

5 2

−

5 2

„ 2, r r r r

4 5 3 5

„

1 2

1 1, 2

„ 0,

1 2

« r

« +

r

« +

r

2 5

„ « 1 −1, 2

+

1 5

„ « 1 −2, 2

+

r

1 5 2 5 3 5 4 5

„ „ „

1 2, − 2 1, −

1 2

0, −

1 2

« « «

„ « 1 −1, − 2 « „ 1 −2, − 2

226

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

For d 3 : 2

j

m

3 2

3 2

3 2

1 2

3 2

1 − 2

3 2

−

3 2

r r r r

1 5 2 5

„

1 1, 2

„ 0,

1 2

« «

−

−

3 5

„ « 1 −1, 2

−

4 5

„ « 1 −2, 2

−

r r

q r

4 5 3 5 2 5

1 5

„ „ „

2, −

1 2

1, −

1 2

0, −

1 2

« « «

„ « 1 −1, − 2

For p 3 : 2

j

m

3 2

3 2

3 2

1 2

3 2

−

1 2

3 2

−

3 2

„ 1, r r

2 3 1 3

„

1 2

1 0, 2

« r

« +

„ « 1 −1, 2

r +

1 3 2 3

„ „

1 1, − 2 0, −

1 2

« «

« „ 1 −1, − 2

For p 1 : 2

j

m

1 2

1 2

1 2

1 − 2

r r

1 3 2 3

„ 0,

1 2

«

„ « 1 −1, 2

−

−

r r

2 3

„ « 1 1, − 2

1 3

„ « 1 0, − 2

227

ATOMIC PHYSICS

3.16.3

Resonance Between A d Electron And A p Shell II

j

m

m = 5/2

m = 3/2

d5 2 m = 1/2

m = −1/2

A E F 0 S1 T1

B

C

D

...

...

p3

2

3/2 3/2 3/2 3/2

3/2 1/2 −1/2 −3/2

mean values p1 2 1/2 1/2 1/2 −1/2 mean values

mean values

G H S2 T2 S1 S2 S T

where:44 2 3 A = G1 + , 5 245

36 4 C = ..., G1 + G3 , 25 1125 15 10 G3 , G3 , F = D = ..., E= 245 245 5 30 G= H= G3 , G3 , 245 245 2 28 1 7 T1 = G1 + S1 = G1 + G3 , G3 , 5 245 10 245 35 35 G3 , G3 , T2 = S2 = 245 490 2 63 1 21 S = G1 + T = G1 + G3 , G3 . 5 245 15 490 B=

44 @ See the previous footnote. Notice also that S = S + S , and analogously for the T 1 2 terms. Here the manuscript is corrupted and we have represented by dots the expressions we cannot easily interpret.

228

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

j

m

m = 5/2

m = 3/2

A

B

d3 2 m = 1/2

m = −1/2

p3 2

3/2 3/2 3/2 3/2

3/2 1/2 −1/2 −3/2

mean values p1 2 1/2 1/2 1/2 −1/2 mean values

C

D

S1 T1

S2 T2

S mean values

where:45 A = ..., B = ..., C = ..., D = ..., 1 63 1 63 G3 , G3 , T1 = G1 + S 1 = G1 + 15 245 60 980 1 1 T2 = G1 , S 2 = G1 , 3 6 63 2 G3 . S = G1 + 5 245 Mean values: d5 p3 : 2

2

d5 p1 : 2

2

d3 p3 : 2

2

d3 p1 : 2

45 @

2

1 21 1 G1 + G3 + 15 490 6 1 21 1 G1 + G3 − 15 490 3 1 21 1 G1 + G3 − 15 490 4 1 21 1 G1 + G3 + 15 490 2

See the previous footnote.

21 1 G1 − G3 5 245 1 21 G1 − G3 5 245 1 21 G1 − G3 5 245 21 1 G1 − G3 5 245

=

7 1 G1 + G3 , 10 245

=

1 G3 , 14

=

63 1 G1 + G3 , 60 980

1 = G1 . 6

229

ATOMIC PHYSICS

1 If G1 = 1 and G3 = : 2 d 5 p 3 : 0.0881 + 2

1 · 0.1571 = 0.1143, 6

d 5 p 1 : 0.0881 −

1 · 0.1571 = 0.0357, 3

d 3 p 3 : 0.0881 −

1 · 0.1571 = 0.0488, 4

d 3 p 1 : 0.0881 +

1 · 0.1571 = 0.1667. 2

2

2

2

3.17.

2

2

2

2

MAGNETIC MOMENT AND DIAMAGNETIC SUSCEPTIBILITY FOR A ONE-ELECTRON ATOM (RELATIVISTIC CALCULATION)

The following notes are aimed at evaluating the magnetic moment of an hydrogen-like atom by starting from the Dirac equation for an electron in an electromagnetic potential ﬁeld (ϕ, C). In the non-relativistic case: e2 3a20 . 6mc2 Z 2 e e W + ϕ + ρ1 σ · p + C + ρ3 mc ψ = 0, c c c σμ = −

ϕ=+

Ze . r A = (ψ1 , ψ2 ), Ze2 W + + mc A + σ c rc W Ze2 + − mc B − σ c rc

B = (ψ3 , ψ4 ),

e p + C B = 0, c e p + C A = 0. c

230

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

1 Cx = − yH, 2

1 Cy = xH, 2

Hx = 0,

W =−

Hy = 0,

Cz = 0; Hz = H.

Ze2 eH − ρ3 mc2 − cρ1 σ · p − ρ1 (xσy − yσx ). r r ∂W = −μz . ∂H

e Ze2 + ρ3 mc2 + cρ1 σ · p + Hρ1 (xσy − yσx ) ψi = 0, W+ r 2

ψ = ψ0 + Hψ1 + H 2 ψ2 + . . . , W

= W0 + HW1 + H 2 W2 + . . . .

Ze2 2 + ρ3 mc + cρ1 σ · p ψ0 = 0, W0 + r Ze2 e 2 + ρ3 mc + cρ1 σ · p ψ1 + W1 + ρ1 (xσy − yσx ) ψ0 = 0, W0 + r 2 Ze2 + ρ3 mc2 + cρ1 σ · p ψ2 W0 + r e + W1 + ρ1 (xσy − yσx ) ψ1 + W2 ψ0 = 0. 2 e W1 = − 2

ψ˜0 ρ1 (xσy − yσx )ψ0 dτ .

231

ATOMIC PHYSICS

W2 =

e 2

ψ˜0 ρ1 (xσy − yσx )ψ1 dτ − W1

ψ˜0 ψ1 dτ

e ψ˜0 W1 + ρ1 (xσy − yσx ) ψ1 dτ 2

e ˜ ψ1 W1 + ρ1 (xσy − yσx ) ψ0 dτ. 2

= − = −

ψ0 = (A0 , B0 ): Ze2 2 + mc A0 + cσ · pB0 = 0, W0 + r Ze2 2 − mc B0 − cσ · pA0 = 0. W0 + r

m , A0 = f0 S−1

B0 = g0 S1m ,

(m = ±1/2, k = 1) ±l/2

Skm = S1

.

Ze2 h d 2 + mc f0 + c g0 = 0, W0 + r 2πi dr d 2 Ze2 h − mc2 g0 − c + f0 = 0. W0 + r 2πi dr r f0 =

u0 iv0 , g0 = : r r

Ze2 W0 + mc + r

2

Ze2 W0 − mc + r 2

h u0 + c 2π h v0 + c 2π

d 1 − dr r 1 d + dr r

v0 = 0, u0 = 0.

232

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

p2 = −W02 + m2 c4 = , −W02 + m2 c4 = α=

4π 2 e4 2 4 2 m c Z , h2 c2 2πc2 2πe2 mcZ = mZ, hc h

2πe2 , hc W0 = mc2

+ 1 − Z 2 α2 ,

√

v0 = r u0 =

W2 − m2 c2 , c2

1 d − dr r

v0 = v0

e

1+

√

1−Z 2 α2 −Zr/a0

√

a0 =

h2 . 4π 2 mc2

,

√ Zα 2 2 r 1−Z α e−Zr/a0 . 2 2 1−Z α

Z 1 − Z 2 α2 1 − − r a0 r

= v0

Z W − mc2 − 2 r mc a0

and substituting in the equation above: v0

Z W − mc2 − 2 r mc a0

2π + hc

Ze2 W0 + mc + r

2

r mc2 Z hc a0 = − v0 2 2 r(W0 + mc ) + Ze 2πmc2 W0 − mc2 −

u0

u0 = 0,

mc2 Z r h a0 u0 . 2 2 Ze + (W0 + mc )r 2πmc2

mc2 − W0 + =

,

233

ATOMIC PHYSICS

3.18.

THEORY OF INCOMPLETE P TRIPLETS

On pages 61-68 and 90-116 of Quaderno 7, the author elaborated the theory of incomplete P triplets, as published by him in E. Majorana, Nuovo Cim. 8 (1931) 107. In the following, we reproduce only few topics that were not included in the published paper (which may be consulted for further reference).

3.18.1

Spin-Orbit Couplings And Energy Levels c s1 · 1 + c s2 · 2

1 = 1 s1 = 1/2

j1

s1 · 1

3/2 1/2

1/2 −1 2 c = δ, 3

3 δ = c. 2

Interaction Diagonal terms of terms s1 · 1 + s2 · 2 1D 3P 2 1S 0

3P 1

2 3P 0

A + B/25 A − B/5 A + 2B/5

0 −1 −1/2 0

1/2

s · = sx x + sy y + sz z 1 1 = (sx + isy ) (x − ily ) + (sx − isy ) (x + iy ) + σz z . 2 2 The quantity · s for = 1, s = 1/2 is as follows: [See the table on page 234.]

234

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

For atoms with one p electron in the inner shells and one s in the outer one (like neon), denoting with I the exchange energy, we have: [See the tables on page 235.] For high Z and I = 1: [See the ﬁgure on page 236.]

z jz

1 2

1

1 2

1

sz

1 2

1 2

0

−

− 32

1 2

−1

−

1 2

0

0

0

0

0

0

0

√ 2 2

0

0

0

0

√ 2 2

− 12

0

0

0

0

0

0

− 12

√ 2 2

0

0

0

0

√ 2 2

0

0

0

0

0

0

0

1 2

1 2

1 2

−

−

− 32

− 12 −1

0

− 12

1 2

−

1 2

− 12

− 12 0

−1

1 2

1 2

−1

1 2

1 2

1 2

1

−

1 2

1 2

0

1

1 2

1 2

235

ATOMIC PHYSICS

m=2 z

s1z

s2z

1

1 2

1 2

1

1 2

1 2

1 c 2

−I +

m=1 z

s1z

s2z

0

1 2

1 2

1 2

1 2

1

1

−

1 2

−

1 2

1 2

0

−

1

1 2

1 2

1 2

1

−

1 2

√ −I

c

2 2

0

−

1 c 2

−I

√ c

1 2

2 2

1 c 2

−I

0

m=0 z

s1z

s2z

−1

1 2

1 2

1 2

1 2

0

−

0

1

−

1 2

1 2

−I −

1 c 2

−1

0

−

1 2

1 2

0

1 2

−

1 2

1

−

1 2

−

√

2 c 2

0

2 c 2

0

−I

0

√

0 √

1 2

−

1 2

0

−I

1 2

−

1 2

0

0

2 c 2

√

2 c 2

−I −

1 c 2

1 2

236

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

237

ATOMIC PHYSICS

3.18.2

Spectral Lines For Mg And Zn

[46 ] Zn 2086.72 (12 ) 2070.11 21 2104.34 11 2087.27 01 2079.10 10 2096.88 (22 )

46 The

Mg 2779.93 2776.80 2783.08 2779.93 2778.38 2781.52

wavelengths of the following spectral lines are expressed in angstroms.

238

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

3.18.3

Spectral Lines For Zn, Cd And Hg

[47 ]

P2 P1 P2

P1 P0 P0

− − − P2 − P1 P1 − P0 P2 − P0

Zn Cd Hg (399) 220 748 1938 (619) 389 1170 4534 189 544 1774 579 1714 6408

[48 ]

P1 P1 P1 P0

47 As

− P2 − P1 − P0 − P1

Zn 2104.34 47521 2087.27 47910 2079.10 48098 2096.88 47690

Cd 2329.27 42932 2267.46 44102 2239.85 44646 2306.61 43354

Hg 2002.7 49933 1832.6 54567 1774.9 56341 1900.1 52629

above, the wavelengths of the following spectral lines are expressed in angstroms. the following table the author reported the wavelength (in angstroms) and the frequency (in cm−1 ) for the spectral lines in the ﬁrst and the second column, respectively, for each element. As pointed out by the author himself, these values do not take into account the correction induced by propagation of light in air. 48 In

239

ATOMIC PHYSICS

3.19.

HYPERFINE STRUCTURE: RELATIVISTIC RYDBERG CORRECTIONS

A relativistic formula for the Rydberg corrections of the hyperﬁne structures was derived in the following calculations. Some particular cases, including s-orbit terms, were considered in detail. Probably, the present calculations were at the basis of what discussed in an appendix of E. Fermi and E. Segr`e, Mem. Accad. d’Italia 4 (1933) 131 on the same topic, as acknowledged by the authors themselves. √ By using electronic units: γ = k 2 − α2 , α = Z/c. μ0 = γ,

α=

A = μ0 + k = k + γ,

+ k2 − γ 2 ,

μ0 + k = k + γ.

B = nr + γ,

E=c

L=

+ (nr + γ)2 + α2 .

B − c2 , L

E B + 2c = c + c. c L β=

αc , L

αβ =

α2 c . L

dE 1 = αc2 . dnr [(nr + γ)2 + α2 ]3/2 a+ μ0 −1 = β(μ0 + k) = αc

A , L

b+ μ0 −1 = −

bμ0 +1 =

1 α . c 1 − 2γ −α

E B = αc − αc . c L

A−1 1 , c 1 − 2γ

αc B αc B − 2αc − − αc . L L(1 + 2γ) A(1 + 2γ) A L(1 + 2γ)

240

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

1 B α2 c α A c+A c + L L c 1 − 2γ A B A−1 − αc + αc − αc L L c(1 − 2γ) AB α B A + 2α + +α −α L L(1 + 2γ) 1 + 2γ L(1 + 2γ) 2 2 AB α A − A AB − B α + − + = L 1 − 2γ 1 − 2γ 1 − 2γ 1 − 2γ A 2AB B A−1 1 −A + + +α − + . 1 + 2γ 1 + 2γ 1 − 2γ 1 − 2γ 1 + 2γ

2γC =

−C =

α 1 + 2 2 (nr + γ) + α 2γ(4γ 2 − 1) + · 4k(nr + γ) + 2 (nr + γ)2 + α2

−

dE dε

=

z2α 1 2 2 2 [(nr + γ) + α ] 2γ(4γ 2 − 1) + uv 2 2 dr. · 4k(nr + γ) + 2 (nr + γ) + α = − r2

For Z → 0 (α2 → 0, γ = k, nr + γ = n):

−

−C =

±α , 2k (k − 1/2)

dE dε

±Z 2 α . 2n3 k (k − 1/2)

=

In particular (2j + 1 = |2k|), for k = + 1, j = + 1/2:

−

−C =

α , 2( + 1) ( + 1/2)

dE dε

Z 2α , 2n3 ( + 1/2) ( + 1)

=

241

ATOMIC PHYSICS

while, for k = −, j = − 1/2:

−

−C =

−α , 2 ( + 1/2)

dE dε

−Z 2 α . 2n3 ( + 1/2)

=

The ratio R between the Rydberg corrections for the hyperﬁne structures in the relativistic form and those in the classical (non-relativistic) form is then given by:

(2j + 1) (k − 1/2) R= γ(4γ 2 − 1)

nr + γ

2k + +1 . (nr + γ)2 + α2

For nr → ∞: R=

(j + 1/2)(4k 2 − 1) . γ(4γ 2 − 1)

For j = 1/2: R=

1 γ(4γ 2 − 1)

nr + γ

2 + +1 , (nr + γ)2 + α2

and, for n = 1, 2, . . .: 2γ + 1 1 = , −γ γ(4γ 2 − 1)

1s :

R=

2s :

√ 1 + 2 + 2γ R= , γ(4γ 2 − 1)

2γ 2

... ∞s :

R=

3 γ(4γ 2

− 1)

.

242

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

The corrections T on the absolute value of the hyperﬁne structures are instead: T =R

n3 . [(nr + γ)2 + α2 ]3/2

For the s terms we have nr = n − 1, γ 2 + α2 = 1: T =R

n3 . [n2 − 2(n − 1)(1 − γ)]3/2

In particular: 1 , −γ √ 8[(2 + 2γ) + 2 + 2γ] , T = γ(2γ − 1)(2γ + 1)(2 + 2γ)2

1s

T =

2s ...

8

2γ 2

(2γ + 1)(2 + 2γ) (2γ + 1)(2 + 2γ)2 T1s √ √ = . = T2s 2 + 2γ + 2 + 2γ 1 + 1/ 2 + 2γ

For γ = 0.74: 8

3.20.

8.63 2.48 · 3.48 T1s √ = = 5.62. = T2s 1.536 1 + 1/ 3.48

NON-RELATIVISTIC APPROXIMATION OF DIRAC EQUATION FOR A TWO-PARTICLE SYSTEM

After having obtained the usual non-relativistic decomposition of the Dirac wavefunction (at a ﬁrst as well as at a second approximation), the author considered a particular expression for of the electromagnetic interaction between a system of two identical charged particle (probably electrons in an atom). Then, he obtains the radial equations for the Dirac components in a central ﬁeld ϕ.

243

ATOMIC PHYSICS

3.20.1

Non-Relativistic Decomposition α = ρ1 σ,

ρ1 ψ = ρ1 (A, B) = (B, A), σψ = (σA, σB),

ψ = (A, B);

ρ3 ψ = ρ3 (A, B) = (A, −B); ρ1 σψ = (σB, σA);

¯ ¯ ¯ ψαψ = AσB + BσA

e W e + ϕ ψ + ρ1 σ, p + U ψ + ρ3 mc ψ = 0. c c c

e W e W + ϕ A, + ϕ B c c c c e e + σ, p + U B, σ, p + U A + (mc A, −mc B) = 0. c c e e W + ϕ A + σ, p + U B + mc A = 0, c c c e e W + ϕ B + σ, p + U A − mc B = 0. c c c

For U = 0:

e W + ϕ A + (σ, p) B + mc A = 0, c c e W + ϕ B + (σ, p) A − mc B = 0. c c

Since (σ, p) (σ, p) = p2 : e 1 W + ϕ B− p2 − mc B = 0, c c 2mc W = mc2 − eϕ +

1 2 p . 2m

244

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

In a ﬁrst approximation: A=−

1 (σ, p) B, 2mc

while, in the second approximation: A=−

3.20.2

1 W + eϕ (σ, p) B. (σ, p) B + 2mc 4m2 c3

Electromagnetic Interaction Between Two Charged Particles

By considering the total interaction: e2 1 − (α, α ) , r12 the magnetic interaction term is: −

e2 e2 (α, α ) = − ρ1 ρ1 (σ, σ ). r12 r12

The 4 components Aij of the wavefunction may be written as: A12 A21 A22 A11 ψ1 ψ2 ψ1 ψ2 ψ3 ψ4 ψ3 ψ4 ψ1 ψ2 ψ3 ψ4 ψ1 ψ2 ψ3 ψ4 The complete expression for the energy is: W

= −e ϕ(q) − e ϕ(q ) − cρ1 (σ, p) − cρ1 σ · p e2 e2 − ρ1 ρ1 σ · σ . −ρ3 mc2 − ρ3 mc2 + r12 r12

In ﬁrst approximation: A12 = −

1 (σ, p) A22 , 2mc

A21 = −

1 σ , p A22 . 2mc

245

ATOMIC PHYSICS

3.20.3

Radial Equations

A = (ψ1 , ψ2 ), B = (ψ1 , ψ4 ): e W + ϕ A + (σ, p) B + mc A = 0, c c e W + ϕ B + (σ, p) A − mc B = 0. c c By introducing the two-valued Pauli spherical function L corresponding to (, j), and L1 = σz L corresponding to (1 , j) (with 1 = 2j − ): B = g(r)L,

A = f (r)σr L = f (r)L1

(it having been put L = σz L1 ). (σ, p) A = (σ, p) f (r)σr L

x y z σx + σy + σz L = (σx px + σy py + σz pz )f (r) r r r x y z = px f (r) L + py f (r) L + pr f (r) L r r r y x σz L +i px f (r) − py f (r) r r z y σx L +i py f (r) − pz f (r) r r x z σy L. +i pz f (r) − px f (r) r r

px f (r) py f (r)

x L = r

z σx L = r =

h r2 − x2 x x2 L p f (r) + f (r)L + f (r) px L, r 2 3 r 2πi r r z y h yr z σx L pr f (r) − f (r)σx L + f (r) σx py L 3 r r 2πi r r h yr f (r) yz σx zpy L; σx L pr f (r) − f (r) σx L + r2 2πi r3 r

z y f (r) (ypz − zpy ) σx L. py f (r) − pz f (r) σx L = − r r r z h f (r) h f (r) − i (σ, ) , (σ, p) A = L pr f (r) + r 2πi r 2π

h z h f (r) h ∂f + f (r) + (k − 1) f (r). (σ, p) A = L 2π ∂r 2πi r 2πi r

246

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

(σ, ) =

= k − 1, − − 1

(σ, 1 ) =

− − 2 = −(k + 1). +1

h d k−1 W + mc + eϕ f (r) + c − g(r) = 0, 2πi dr r

h d k+1 2 W − mc + eϕ g(r) + c + f (r) = 0. 2πi dr r

2

By setting r · g(r) = v, r · f (r) = i u:49

3.21.

h W + mc + eϕ u − c 2π

h W − mc + eϕ v + c 2π

2

2

d k − dr r d k + dr r

v = 0, u = 0.

HYPERFINE STRUCTURES AND MAGNETIC MOMENTS: FORMULAE AND TABLES

In the following the author reported some ﬁnal formulae concerning his studies on hyperﬁne structures and the atomic magnetic moments (as in the previous Section, he set E = W − mc2 , eϕ = −V ). Related calculations are developed in the next Section. h d k v = 0, (E − V + 2mc ) u − c − 2π dr r h d k (E − V ) v + c u = 0, + 2π dr r 2

49 In

the original manuscript, the second equation in the following is written incorrectly as: `

´ h W − mc2 − eϕ v + c 2π

„

d k + dr r

« u = 0.

247

ATOMIC PHYSICS

⎧ ⎪ ⎪ +1 ⎪ ⎪ ⎨ k=

k=

j+

⎪ ⎪ ⎪ ⎪ ⎩ − 1 2

1 j =+ , 2 1 j =− , 2

2

1 |k| = j + , 2

− ( + 1),

k(k − 1) = ( + 1). Atomic magnetic moment: μ0 =

eh , 4πmc

−M = j g(j) μ0 = −e

μ0 g(j) = −

k e j j+1

k j+1

r u v dr,

(1)

r u v dr.

(1 )

uv dr, r2

(2)

Magnetic ﬁeld at the origin: 2k j C=H= e j+1 2k e C= j(j + 1)

uv dr. r2

(2 )

Nuclear magnetic moment: Mn = i g(i)

μ0 = i g(i) μ, 1840

μ=

eh μ0 = . 4πMn c 1840

(2)

Hyperﬁne structure formula: δW = −(Mn , H) = −(i, j) g(i) μ C = −(i, j) g(i) μ

2k e j(j + 1)

[50 ] 50 @

In the following the author introduced the sum f = i + j.

uv dr, r2

(3)

248

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

(i, j) =

f (f + 1) − i(i + 1) − j(j + 1) . 2 ——————–

In ﬁrst approximation: h u= 4πmc

d k − dr r

v,

h 1 1 =− k+ − ru v dr = − k + 2 4πmc 2 1 1 h uv dr = − (k − 1) = − (k − 1) 3 r2 4πmc r3 r

μ0 , e μ0 . e

Atomic magnetic moment: −

M μ0

=

g(j) =

k(k + 1/2) , j+1

⎧ 2 + 2 ⎪ ⎪ ⎪ ⎪ ⎨ 2 + 1

k(k + 1/2) = ⎪ j(j + 1) ⎪ ⎪ ⎪ ⎩

2 2 + 1

1 j =+ 2 1 j =− 2

, .

Magnetic ﬁeld at the origin: ( + 1) 1 1 2k(k − 1) μ0 3 = −2 μ0 3 , j+1 r j+1 r ( + 1) 1 1 2k(k − 1) μ0 3 = −2 μ0 3 . C = − j(j + 1) r j(j + 1) r

H = j C=−

Hyperﬁne structure formula:

δW =

2k(k − 1) 1 μ20 2( + 1) 1 μ20 (i, j) g(i) (i, j) g(i) = . 3 1840 j(j + 1) r 1840 j(j + 1) r3

For s-terms:

h μ0 uv = −2πψ 2 (0) . dr = −2πψ 2 (0) 2 r 4πmc e

249

ATOMIC PHYSICS

H = j C=− C = −

δW =

8π 2 ψ (0) μ0 , 3

16π 2 ψ (0) μ0 . 3

16π 2 8π 2 μ2 μ20 (i, j) g(i) ψ (0) = 0 (2i + 1) g(i) ψ (0). 1840 3 1840 3 ——————–

In ﬁrst approximation, with a Coulomb ﬁeld: Z3 1 1 = 3 3 3 r a0 n ( + 1/2)( + 1) and, for s-terms, ψ 2 (0) = ⎫ 2( + 1) 1 ⎪ ⎪ ⎪ j(j + 1) r3 ⎬ ⎪ ⎪ 16π 2 ⎪ ψ (0) ⎭ s-terms: 3

δW =

Z3 1 . a30 πn3

=

4 Z3 a30 n3 j(j + 1)(2 + 1)

μ20 Z3 4 (i, j) g(i) 3 3 1840 a0 n j(j + 1)(2 + 1)

(which holds also for s-terms).

μ20 /a3 =

1 2 α Rh, 2

δW =

α=

2πe2 , hc

α2 R/c = 5.83 cm−1 .

2α2 Rh Z3 (i, j) g(i) 3 , 1840 n j(j + 1)(2 + 1)

Z3 δW = δn = 0.00634 (i, j) g(i) 3 cm−1 . hc n j(j + 1)(2 + 1) The term δn1 corresponds to the particular case f = i + j, that is, cos i-j = 1 and (i, j) = i j:

250

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

δn1 = 0.00634

Z 3 i g(i) cm−1 . n3 (j + 1)(2 + 1)

——————– a + b = c: (a, b) , cos a.b = ab (a, b) =

c(c + 1) − a(a + 1) − b(b + 1) . 2

a

b

c

(a, b)

cos acb

1/2

1/2

1 0

1/4 −3/4

1 −3

1

1/2

3/2 1/2

1/2 −1

1 −2

2

1/2

5/2 3/2

1 −3/2

1 −3/2

1/2

+ 1/2 − 1/2

/2 −( + 1)/2

1 −( + 1)/2

a

b

c

(a, b)

cos acb

1

1

2 1 0

1 −1 −2

1 −1 −2

3/2

1

5/2 3/2 1/2

3/2 −1 −5/2

1 −7/3 −5/3

2

1

3 2 1

2 −1 −3

1 −1/2 −3/2

3

1

4 3 2

3 −1 −4

1 −1/3 −4/3

1

+1 −1

−1 −( + 1)

1 −1/ −( + 1)/

251

ATOMIC PHYSICS

3.22.

a

b

c

(a, b)

cos acb

3/2

3/2

3 2 1 0

9/4 −3/4 −11/4 −15/4

1 −1/3 −11/9 −5/3

2

3/2

7/2 5/2 3/2 1/2

3 −1/2 −3 −9/2

1 −1/6 −1 −3/2

3/2

+ 3/2 + 1/2 − 1/2 − 3/2

3/2 /2 − 3/2 −/2 − 2 −3/2 − 3/2

1 1/3 − 1/ −1/3 − 4/3 −1 − 1/

HYPERFINE STRUCTURES AND MAGNETIC MOMENTS: CALCULATIONS

Some calculations concerning atomic systems with magnetic moment are presented in the following, by using similar notations as in the previous Section. The Dirac equation for the u and v wavefunctions underlies such study. Explicit iterative formulae for the perturbative calculation of the wavefunctions are given, as well as the relevant self-consistent relations (left unsolved).

3.22.1

First Method

On using electronic units:51 α = Z/c,

μ0 = 1/2c.

k d Z 2 − v = 0, E + + 2c u − c r dr r d k Z v+c + u = 0. E+ r dr r 51 @

In the original manuscript, an unidentiﬁed reference (see pages 15 and 25) appears here.

252

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

Z k d 2 y2 = ε ry2 , E + + 2c y1 − c − r dr r Z k d y2 + c y1 = 0 E + + r dr r

for ε → 0.

y1 = P e−β r , P =

aμ rμ ,

y2 = Q e−β r , Q =

aμ = aμ + ε a∗μ ,

etc.

bμ rμ , 52

Remembering that α = Z/c: E (μ + k)aμ + α bμ = β aμ−1 − bμ−1 , c E E −α aμ + (μ − k)bμ = + 2c aμ−1 + β bμ−1 − bμ−2 . c c Note that it is unnecessary to vary β. E ∗ E∗ bμ−1 + β ∗ aμ−1 − bμ−1 , c c E∗ E = + 2c a∗μ−1 + β b∗μ−1 + aμ−1 c c

(μ + k) a∗μ + α b∗μ = β a∗μ−1 − −α a∗μ + (μ − k) b∗μ

+β ∗ bμ−1 −

1 bμ−2 . c

E ∗ E aμ + α β + (μ − k) b∗μ β (μ + k) − α c c ∗ E E∗ E ∗ E 1 E ∗ αμ−1 + −β = ββ + + β bμ−1 − bμ−2 . c c c c c c Let us set ν = μ0 + nr 52 @

That is: bμ = bμ + ε b∗μ , β = β + ε β ∗ , E = E + ε E ∗ .

253

ATOMIC PHYSICS

and assume that bν = 0

aν = 0 :

but

(b∗ν = 0) E E∗ E ∗ ∗ aν = β β + aν−1 β (ν + k) − α c c c E∗ E ∗ E bν−2 + β bν−1 − . + −β c c c c ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

(ν + k + 1) a∗ν+1 + α b∗ν+1 = β a∗ν + β ∗ aν − −α a∗ν+1

+ (ν + 1 −

k) b∗ν+1

=

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

E∗ bν c

E E∗ + 2c a∗ν + aν c c

+β ∗ bν −

(1)

(2)

1 bν−1 c

E ∗ b c ν+1 E ∗ ∗ + 2c a∗ν+1 −α aν+2 + (ν + 2 − k) bν+2 = ⎪ c ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎩ +β b∗ν+1 − bν c (ν + k + 2) a∗ν+2 + α b∗ν+2 = β a∗ν+1 −

β a∗ν+2 −

E ∗ b = 0. c ν+2

We can set β ∗ = 0 or, rather:

β∗ = β

E∗ . E

(3)

(4)

254

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

It follows that: E E∗ ββ + c c ∗ E E + β∗ −β c c ∗ E bν β ∗ aν − c E∗ aν + β ∗ bν c ∗

E∗ E

=

E2 2 β + 2 = −2E ∗ , c

= 0, = 0, E∗ E

=

E aν + β bν c

= −2c

E∗ aν . E

E E ∗ aν = −2E ∗ aν−1 − 2 bν−2 , β (ν + k) − α c c ⎧ (ν + k + i) a∗ν+1 + α b∗ν+1 = β a∗ν ⎪ ⎪ ⎨ E E∗ 1 ⎪ ∗ ∗ ⎪ + 2c a∗ν − 2c aν − bν−1 ⎩−α aν+1 + (ν + 1 − k) bν+1 = c E c

(1 )

(2 )

Equations (1 ), (2 ), (3) and (4) are six homogeneous equations in a∗ν , a∗ν+1 , b∗ν+1 , a∗ν+2 , b∗ν+2 and −1.

3.22.2

Second Method ⎧ k Z d ⎪ 2 ⎪ + 2c − + − c y2 = ε ry2 , y E ⎪ 1 ⎪ ⎨ r dr r ⎪ ⎪ k Z d ⎪ ⎪ + y2 + c y2 = ε ry1 . ⎩ E + r dr r ∗

E =Z

r u v dr.

255

ATOMIC PHYSICS

With the previous notations: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

(μ + k) a∗μ + α b∗μ = β a∗μ−1 −

E ∗ + β ∗ aμ−1 b c μ−1

E∗ 1 bμ−1 + aμ−2 , c c E E∗ −α a∗μ + (μ − k) b∗μ = + 2c a∗μ−1 + β b∗μ−1 + aμ−1 c c −

+β ∗ bμ−1 −

1 bμ−2 . c

ν = μ0 + νr . aν = 0.

bν = 0, Note that is is unnecessary to vary β.

E E E∗ E∗ ∗ β (ν + k) − α aν = β β + aν−1 − β c c c c β E E ∗ − β bν−1 + aν−2 − 2 bν−2 c c c

(1)

(b∗ν = 0). β a∗ν+2 − ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

E ∗ b = 0. c ν+2

E ∗ 1 bν+1 + aν , c c E 1 −α a∗ν+2 + (ν + 2 − k) b∗ν+2 = + 2c a∗ν+1 + β b∗ν+1 − bν . c c

(4)

(ν + k + 2) a∗ν+2 + α b∗ν+2 = β a∗ν+1 −

(3)

Note that a∗ν+2 /b∗ν+2 is diﬀerent from what obtained by Eq. (4), so that a∗ν+2 = b∗ν+2 = 0: ⎧ ∗ b∗ν+2 = 0, ⎪ ⎨ aν+2 = 0, ⎪ ⎩ β a∗ − E b∗ + 1 aν = 0. ν+1 c ν+1 c

256 ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

(ν + k + 1) a∗ν+1 + α b∗ν+1 = β a∗ν + β ∗ aν − −α a∗ν+1

+ (ν + 1 −

k) b∗ν+1

=

E∗ 1 bν + aν−1 , c c

E E∗ + 2c a∗ν + aν c c

+β ∗ bν −

(2)

1 bν−1 . c

[See the equations on pages 257 and 258.]53 a11

0

0

0

0

a16

−β

a22

α

0

0

0

0

0

0

β

E c

0

a41

−α

a43

0

0

a46

0

−β

E c

a34

α

0

0

a62

−β

−α

a65

a66

−

= 0.

For a suitable value of β ∗ , from (3) and (4) we get: a∗ν+1 = 0,

b∗ν+1 =

aν , E

⎧ α E∗ 1 ⎪ ⎪ aν = β a∗ν + β ∗ aν − bν + aν−1 , ⎪ ⎨ E c c ⎪ E∗ 1 ν+1−k E ⎪ ⎪ ⎩ aν = + 2c a∗ν + aν + β ∗ bν − bν−1 . E c c c

(2 )

Equations (1) and (2 ) are homogeneous equations in a∗ν , β ∗ and 1, so that: [See equation on page 259.]

53 Note that the second determinant diﬀers from the ﬁrst one with respect the ordering of the rows (1,2,3,4,5,6 in the ﬁrst, and 1,2,6,3,4,5 in the second matrix), as pointed out by the author himself in the original manuscript.

−

E c

0

0

0

E + 2c c

−β

β (ν + k) − α

−

0

0

−β β

−α

ν+k+2

E c

−β

E + 2c c

0

0

0

ν+1−k

α

0

−α

(ν + k + 1)

0

0

0

E c −

E bν−2 c2

E∗ 1 aν − bν−1 E 2

1 − bν c

−2c

0

−2E ∗ aν−1 −

ν+2−k

α

0

0

0

= 0

ATOMIC PHYSICS

257

−

0

0

E + 2c c

−

−β

−α

ν+k+2

E c

−β

E + 2c c

0

β

0

0

ν+1−k

0

α

0

−α

0

0

0

(ν + k + 1)

E c

−β

β (ν + k) − α

ν+2−k

α

0

E bν−2 c2

1 − bν c

0

E∗ 1 aν − bν−1 E 2

0

E c

−

0

−2c

−2E ∗ aν−1 −

0

0

= 0

258 E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

E + 2c c

β(ν + k) − α

β

E c

−

E∗ 1 α bν + aν−1 − αν c c E

1 ν+1−k E∗ aν − bν−1 − αν c c E E E E∗ E β − βaν−1 + bν−1 aν−1 − βbν−1 − aν−2 + 2 bν−2 − c c c c c bν

aν

= 0

ATOMIC PHYSICS

259

4 MOLECULAR PHYSICS

4.1.

4.1.1

THE HELIUM MOLECULE

The Equation For σ -electrons In Elliptic Coordinates

We assume the nuclei to be ﬁxed at a distance r one from the other (in electronic units); the nuclei are supposed to have positive charges, of magnitude Z ≤ 2, taking approximatively into account the screening action of the other electrons.

Z Z + ∇ ψ+2 E+ r1 r2 2

ψ = 0.

By measuring the energy (denoted with W ) in Rh we have W = 2E, from which: 1 1 2 + ψ = 0. ∇ ψ + W ψ + 2Z r1 r2 Putting: u=

r1 + r2 , 2

v=

r2 = u − v,

r1 = u + v, r12 = u2 + 2uv + v 2 ,

r1 − r2 , 2

r22 = u2 − 2uv + v 2 ,

r1 r2 = u2 − v 2 ,

we have ∇2 ψ =

∂2ψ ∂2ψ ∂ψ ∂ψ 2 |∇ u| + |∇ v|2 + ∇·u + ∇ · v, ∂u2 ∂v 2 ∂u ∂v

261

262

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

and, since |∇ u|2 = = |∇ v|2 = ∇2 u = ∇2 v =

1 + cos(r1 , r2 ) 1 r2 + r22 − 4 = + 1 2 2 4r1 r2 2 2 2 1 u +v −2 u −1 , + = 2 2 2 2 2(u − v ) u − v2 1 − v2 1 u2 + v 2 − 2 − = , 2 2(u2 − v 2 ) u2 − v 2 1 1 1 1 2u + = 2 + = , r1 r2 u+v u−v u − v2 1 1 1 1 2v − =− 2 − = , r1 r2 u+v u−v u − v2

it follows that ∇2 ψ =

u2 − 1 ∂ 2 ψ 1 − v2 ∂ 2ψ 2u 2v ∂ψ ∂ψ − 2 ; + + 2 2 2 2 2 2 2 2 2 u − v ∂u u − v ∂v u − v ∂u u − v ∂v

1 − v2 ∂ 2ψ 2u 2v ∂u ∂ψ u2 − 1 ∂ 2 ψ + + 2 − 2 2 2 2 2 2 2 2 u − v ∂u u − v ∂v u − v ∂u u2 − v ∂v 2Z2 2Z1 ψ+ ψ = 0, +W ψ + u+v u−v where, for the sake of generality, we have distinguished Z1 from Z2 (while we take the half-distance between the nuclei equal to 1). On multiplying the previous equation by (u2 − v 2 ): 2 ∂2ψ ∂ψ ∂ψ 2 ∂ ψ + 2u + Z )ψ + (1 − v ) − 2v + 2u(Z 1 2 ∂u2 ∂u ∂v 2 ∂v 2 2 (1) −2v(Z1 − Z2 )ψ + u W ψ − v W ψ = 0.

(u2 − 1)

By setting ψ = P1 (u)P2 (v), and again Z1 = Z2 = Z, we have the following separated equations: (u2 − 1)P1 + 2uP1 + 4uZP1 + u2 W P1 − λP1 = 0, (1 − v 2 )P2 − 2vP2 − v 2 W P2 + λP2 = 0.

(2) (3)

These equations have to be solved together in order to determine W and λ. It is useful to deduce ﬁrstly a relation between W and λ from the second equation, which does not depend on Z (but depends on the distance between the nuclei, which we have deﬁnitively chosen to be

263

MOLECULAR PHYSICS

equal to 2; with a similarity transformation we can always turn back to this case). Such a relation between W and λ depends only on the azimuthal quantum number, related to P2 , and not on the radial one, corresponding to P1 .

4.1.2

Evaluation Of P2 For s-electrons: Relation Between W And λ

The quantity P2 does not change sign if we replace v with −v; v varies between −1 and 1; singular points are at v = −1 and v = 1. Let us set P2 (−1) = 1, so that P2 (−1) is determined: 2P2 (−1)W + λ = 0, P2 (−1) =

W −λ . 2

Quantity λ results as determined as the smallest value for which P2 (0) = 0. In Eq. (2) we put, for the moment, v = x − 1 = −1 + x,

x = v + 1;

it follows: (2x − x2 )P2 + (2 − 2x)P2 − (1 − x)2 W P2 + λP2 = 0; and, setting: W −λ x + bx2 + cx3 + . . . , 2 W −λ = + 2bx + 3cx2 + . . . , 2 = 2b + 6cx + . . . ,

P2 = 1 + P2 P2 after some algebra1 b = c =

1@

(W − λ)2 W + λ − , 16 8 b W −λ W W (W − λ) + b+ − . 3 18 18 18

In the original manuscript some scratch calculations are reported, leading to the following expressions for b and c (obtained by substituting the expansions for P2 , P2 , P2 into the diﬀerential equation for P2 written above).

264

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

W −λ (W − λ)2 W + λ 2 x + cx3 + . . . P2 = 1 + x+ − 2 16 8

v −1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0

λ = −0.3 P2 −P2 P2 1.000 0.350 0.396 967 313 0.365 937 277 0.346 911 243 0.33 888 211 0.31 868 181 0.30 852 151 0.29 838 123

W = −1 λ = −0.4 P2 −P2 P2 1.000 0.300 0.395 972 261 0.379 948 224 0.364 927 188 0.35 910 153 0.34 896 119 0.34 886 085 0.33 879 051

λ = 0.348 P2 −P2 P2 1.000 0.326 0.969 0.942 0.919 0.899 0.882 0.868 0.858 0.850 0.846 0.845

[2 ]

2@

The table reported in the original manuscript contains slightly diﬀerent numerical values with respect to those one can evaluate from the formulae given by the author, namely:

v −1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0

λ = −0.3 P2 −P2 P2 1.000 0.350 0.386 0.967 0.313 0.368 0.937 0.277 0.341 0.911 0.244 0.32 0.888 0.213 0.30 0.869 0.185 0.27 0.851 0.159 0.25 0.837 0.135

W = −1 λ = −0.4 P2 −P2 P2 1.000 0.300 0.395 0.972 0.261 0.377 0.948 0.226 0.359 0.927 0.189 0.34 0.910 0.156 0.32 0.896 0.125 0.31 0.885 0.095 0.29 0.877 0.067

λ = 0.348 P2 −P2 P2 1.000 0.326 0.969 0.942 0.919 0.899 0.881 0.867 0.856 0.847 0.840 0.835

Probably, the numerical values for the second derivative of P2 were deduced in some manner from the following formula (which appears in the manuscript): P2 =

2vP2 − (v 2 + λ)P2 . 1 − v2

265

MOLECULAR PHYSICS

Let us now set: R

P2 = e

zdv , R zdv

P2 = z e

,

P2

2

R

= (z + z ) e

zdv

;

(1 − v 2 )z + (1 − v 2 )z 2 − 2vz + λ − v 2 W = 0. By solving Eq. (4) with respect to

(4)

z:

2v λ − v2W z − − z2. 1 − v2 1 − v2 λ and z are inﬁnitesimals with W ; we will put: z =

z = z 1 + z2 + z3 + . . . ,

(5)

λ = λ1 + λ2 + . . . ,

where z1 stands for a ﬁrst-order inﬁnitesimal, z2 for a second-order inﬁnitesimal, etc. We will have: z1 =

2v λ1 − v 2 W z − , 1 1 − v2 1 − v2

from which, by imposing regularity conditions on the boundaries, v 1 z1 = − (λ1 − v 2 W )dv 1 − v 2 −1 1 1 1 1 2 λ1 − W + W v − W v . = − 1−v 3 3 3

(6) 3

We set: 1 q1 = z1 = − W v, 3 1 1 = λ1 = W. 3

(7) (8)

When determining z2 , etc., we will set: q 1 = z1 , 1 = λ 1 ,

q 2 = z 1 + z2 , q 3 = z 1 + z2 + z3 , . . . 2 = λ 1 + λ 2 , 3 = λ 1 + λ 2 + λ 3 , . . .

3 @ In the original manuscript the upper limit of the following integrals is not explicitly indicated.

266

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

In general we will have: = qn+1

2v n+1 − v 2 W q − − qn2 . n+1 1 − v2 1 − v2

From it: qn+1 = = = =

v 2 1 v W − n+1 − (1 − v 2 ) qn2 dv 2 1 − v −1 v 1 1 1 3 2 2 (1 − v ) qn dv v W + W − v n+1 − n+1 − 1 − v2 3 3 −1 v 1 1 v 2 W − vW + W − 3n+1 − (1 − v 2 ) qn2 dv 3 1−v 1 − v 2 −1 v 1 1 1 1 2 2 − Wv + −n+1 + W − (1 − v ) qn dv , 3 1−v 3 1 + v −1

and, by imposing the regularity at the point v = 1, it must be: n+1

1 1 = W− 3 2

1 −1

(1 − v 2 ) qn2 dv.

(9)

By substituting Eq. (9) into previous equation: qn+1

1 1 = − Wv + 3 1−v

1 1 (1 − v 2 ) qn2 dv 2 −1 v 1 2 2 (1 − v ) qn dv , − 1 + v −1

(10)

or, more easily, qn+1

1 1 = − Wv + 3 1 − v2

1 1 (1 − v 2 ) q 2 dv v 2 −1 v 2 2 − (1 − v ) qn dv .

(10 )

0

By taking into account that qn+1 (v) = −qn+1 (−v), we also have: n+1 = W − 2qn+1 (−1) = W + 2qn+1 (1),

(11)

1 which can replace Eq. (9). Let us now evaluate q2 ; since q1 = − W v, 3 by substitution into Eq. (9 ):

267

MOLECULAR PHYSICS

q2

1 1 = − Wv + 3 1 − v2

2 1 (1 − v ) − W v dv 3 −1 2

v 1 2 (1 − v ) − W v dv , − 3 0

1 v 2

2

that is: 1 W2 1 q2 = − vW + 3 9 (1 − v 2 ) or, more simply:

1

2 1 3 1 5 v− v + v , 15 3 5

1 1 3 2 v − v W 2, q2 = − vW − 3 45 135 1 2 l2 = W − W 2. 3 135

(12)

Recalling that z λ qn n

z1 + z2 + z3 + . . . , λ1 + λ2 + λ3 + . . . , z1 + z2 + . . . + zn , λ1 + λ2 + . . . + λn ,

= = = =

and that zn and λn are inﬁnitesimals of order n, with this procedure we can obtain any term in the series expansion of z and λ with increasing powers of W : 1 3 2 1 v − v W2 + ..., (13) z = − vW − 3 45 135 1 2 λ = W− W2 + .... (14) 3 135 From z we can then obtain P2 : R

P2 = e

zdv

,

by choosing a suitable normalization, in such a way that P2 (−1) = P2 (1) = 1: 1

P2 = e 6 (1−v

2 )W − 1 (1−4v 2 +3v 4 )W 2 +... 540

P2 (0) = e

1 1 W − 540 W 2 +... 6

——————–

.

,

(15)

268

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

The expansions (12), (13) and (15) cannot be used for large values of W . Then, we now consider asymptotic expansions with decreasing powers of W for W tending to the (negative) inﬁnity. We will set: z λ pn Ln

= = = =

y1 + y2 + y3 + . . . , m1 + m2 + . . . , y1 + . . . + yn , m1 + . . . + mn ,

where we always assume that mn+1 /mn or yn+1 /yn are inﬁnitesimals for W → −∞ and consider only inﬁnities of higher order. By substitution into Eq. (5): m1 − v 2 W , 1 − v2 so that, by requiring regularity in the singular points, y12 = −

(16)

(17) L1 = m1 = W, √ p1 = y1 = ± −W . 0 Since p1 (v) = p1 (−v) (and −1 p1 (v)dv is certainly negative) and p1 has the same sign as v, √ v < 0; p1 = y1 = − −W , (18) √ v > 0. p1 = y1 = −W , Note that the discontinuity at the point 0 results in a divergence for z in Eq. (5), which cannot be neglected; however, by replacing the jump with a suitable junction line in the interval −ε, +ε, |z | will be of the √ order of −W /ε, while the other inﬁnities are of the same order of W . Then we can neglect z provided that: √ ε −W 1, and since W tends to the inﬁnity, we may take the limit ε = 0. For the successive approximations we have to consider: 2v Ln+1 − v 2 W p − − p2n+1 , n 1 − v2 1 − v2 and, imposing the regularity conditions, pn =

(19)

Ln+1 = W − 2pn (−1) = W + 2pn (1),

(20)

269

MOLECULAR PHYSICS

one gets pn+1 = −

−

pn+1 =

−

Ln+1 − v 2 W 2v + pn − pn 2 1−v 1 − v2

(v < 0),

(21)

Ln+1 − v 2 W 2v + pn − pn 2 1−v 1 − v2

(v > 0).

(22)

The asymptotic expansions of z do not yield a continuous curve and cannot be √ used in any interval around v = 0 whose extension is of the order of −W . We will ﬁnd later an appropriate approximation formula for z. We now focus directly on the asymptotic expansion of λ as a function of W . By integrating Eq. (2) from −1 to 0 we obtain:

0

v 2 P2 dv

λ = W −10

−1

.

(23)

P2 dv

For W → −∞ it suﬃces to integrate over a very small interval, starting at −1 for any order of approximation; this would be an indication of the fact that the asymptotic expansion is never convergent. By setting x = 1 + v, v = x − 1, Equation (2) becomes: (2x − x2 )P2 + (2 − 2x)P2 − (1 − x)2 W P2 + λP2 = 0,

(24)

and, putting P2 = Re−

√ −W x

√

, √

P2 = (R − R −W )e− −W x , √ √ P2 = (R − 2R −W − RW )e− −W x , it follows: √ (2x − x2 )R − [(2x − x2 )2 −W − (2 − 2x)]R − (2x − x2 )RW √ −(1 − x)2 RW − 2R(1 − x) −W + λR = 0,

270

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

that is: √ (2x − x2 )R − 2[(2x − x2 ) − W − (1 − x)]R √ −[W − λ + 2(1 − x) −W ]R = 0. For x = 0 we will take R = 1. It follows: √ 2R (0) = W − λ + 2 −W , where from: √ W − λ + 2 −W x + bx2 + . . . , R = 1+ 2 √ W − λ + 2 −W + 2bx + . . . , R = 2 R = 2b + . . . . By substitution into the above equation, from the vanishing of ﬁrst-order terms, one has √ √ √ √ 4b − 2(W − λ − 2 −W ) −W − (W − λ − 2 −W ) + 2 −W = 0, where from: √ √ W − lλ − 2 −W (W − λ − 2 −W − 1) √ . −W + b= 2 4 On the other hand, asymptotically we have: λ=W −W

∞ 0

(2x − x2 )P2 dx ∞ , P2 dx 0

and, since P2 = (1 + ax + bx2 + . . .) e−

√

−W x

,

(2x − x2 )P2 = [2x + (2a − 1)x2 + (2b − a)x3 + . . .] e−

√

−W x

,

271

MOLECULAR PHYSICS

we deduce: 2a − 1 2b − a 2 + 3 + −W W2 (−W ) 2 , λ = W −W b 1 a √ + + 3 + ... −W −W (−W ) 2 2a − 1 2b − a 1+ √ + √ −W − 2 2 −W λ = W + 2 −W a b 1+ √ + −W −W √ = W + 2 −W − 1 + . . . . Summing up, for the moment we know the behavior of the function λ = λ(W ) for small and large values of W : W → 0, W → −∞,

2 1 λ= W− W2 + ..., 3 135 √ λ = W + 2W 2 + −W − 1 + . . . .

Let us put again

Rv

P2 = e it follows: z =

−1

rdv

(25)

;

2v λ − v2W z − − r2 . 1 − v2 1 − v2

(5)

As an approximate solution, we take: z = a arctan b v.

(26)

Substituting it into Eq. (5): 2va λ − v2W ab = arctan b v − − u2 arctan2 b v + . . . . 1 + b2 v 2 1 − v2 1 − v2

(27)

We require that this equation be satisﬁed for v = 0; it follows: ab = −λ.

(28)

Regularity conditions for v = 1 impose: 2a arctan b = λ − W.

(29)

272

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

We also require that the equation be satisﬁed for v = 1, since: ab 2av arctan b v − λ + v 2 W = −W − a arctan b − . 2 v→1 1−v 1 + b2 lim

It follows: W + a arctan b +

2ab + a2 arctan2 b = 0. 1 + b2

(30)

From Eqs. (28), (29), (30) we can determine a, b and λ. We can then consider the following equations: λ−W 2 λ−W 2λ − W+ + = 0, (31) 2 2 1 + b2 ⎛ ⎞ λ 1− W −λ ⎜ W ⎟, = tan ⎝b b = tan b (32) λ ⎠ 2λ 2 W λ (33) a=− . b By taking a series expansion, for small W we have: λ = λ W

=

1 W + KW 2 + . . . , 3 1 + KW + . . . . 3

Equation (32) becomes: ⎛ ⎞ 2 − KW + . . . 9 ⎜ 3 ⎟ b = tan ⎝b ⎠ = tan b − bKW + . . . . 2 2 + 2KW + . . . 3 On the other hand: 1 9 b − bKW = arctan b = b − b3 + . . . , 2 3 from which: 1 9 − KbW = − b3 + . . . , 2 3 27 2 b = KW + . . . . 2

273

MOLECULAR PHYSICS

Substituting it into Eq. (31): 1 1 1 2 1 + W − + KW − − 2KW + 9KW + . . . = 0, 9 3 2 3 from which: 1 1 + K − 2K + 9K = 0, 9 2 K=−

1 15 + K = 0, 9 2

2 , 135

1 2 W2 + ..., λ= W− 3 135 which agrees with Eq. (25). We have thus an exact result holding in ﬁrst and second approximation: 1 b2 = − W + . . . . 5

(34)

For the asymptotic expansion (W → −∞), we set: √ λ = W + 2 −W + α + . . . . By substituting it into Eq. (31), noting that b is an inﬁnite of order 1/2 and equating to zero higher-order inﬁnities, we have: √ √ α −W + −W = 0, from which α = −1 and:

√ λ = W + 2 −W − 1 + . . . ,

which again agrees with Eq. (25). We can likely presume that for arbitrary W a very good approximation for λ = λ(W ) is obtained. [4 ] −W 0 1 2 3 4

4@

−λ 0 +0.348 +0.731 1.151

b 0 0.47 0.72 0.94

It is not very clear how the author obtained the values reported in the following table. Probably, for a given value of W , λ was obtained from the approximate Eq. (25) for W → 0 (in this case, for −W = 2, 3 we would have −λ = 0.726, 1.133), while b is deduced from Eq. (31).

274

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

[5 ]

v −1 −0.95 −0.9 −0.85 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0

v −1 −0.95 −0.9 −0.85 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0

λ = −0.72 P2 −P2 P2 1.000 0.640 0.885 969.1 596.8 845 940.3 555.5 808 888.7 844.5 807.2 776 751 732 718 709 706

478.0 406.9 340.5 279 221 165 112 60 9

λ = −1.14 P2 −P2 1.000 0.930 955.3 858.9 914.1 792.0 876.1 729.1 841.1 669.7 780 561 729 463 687 373 654 291 629 214 611 141 600 71 597 3

741 686 638 599 568 543 525 513 508

P2 1.467 1.379 1.297 1.223 1.154 1.032 936 855 791 743 708 687 680

W = −2 λ = −0.74 P2 −P2 P2 1.000 0.630 0.883 969.6 586.8 844 941.3 545.5 808 890.7 847.5 811.2 781 757 739 726 719 717

468.0 396.5 329.8 267.5 208.5 152 97 44 −9

744 690 644 606 577 555 540 532 531

W = −3 λ = −1.16 P2 −P2 1.000 0.920 955.8 849.1 915.0 782.3 877.5 719.5 843.0 660.1 783 551 733 453 692 363 660 280 636 202 620 128 611 56 609 −15

P2 1.463 1.377 1.295 1.222 1.154 1.033 940 862 801 756 725 708 706

5@

P2 1.000 969 941

λ = 0.73 −P2 635 592 550

890 846 809 778.5 754 735.5 722 714 711.5

P2 1.000 955 914 876 841 780 730 688 655 630 612 602 599

P2

473 402 335 273 215 158 105 52 0

λ = 1.143 −P2 0.928 857 790 728 668 559 461 371 289 212 139 69 0

P2

The following two tables seem the continuation of the table appearing at page 264, but it is not clear how the author obtained the numerical values reported here. Note that, as above, in some places the author omits the notation “0.” in the reported numbers. Probably, the numerical values for the second derivative of P2 were deduced in some manner from the following formula (which appears in the manuscript): P2 =

2vP2 − (2v 2 + λ)P2 , 1 − v2

P2 =

2vP2 − (3v 2 + λ)P2 1 − v2

for W = −2, and

for W = −3.

275

MOLECULAR PHYSICS

4.1.3

Evaluation Of P1

In the general case Z1 = Z2 , equations (1) and (2) become: (u2 − 1)P1 + 2uP1 + 2u(Z1 + Z2 )P1 + u2 W P1 − λP1 = 0, (1 − v 2 )P2 − 2vP2 − 2v(Z1 − Z2 )P2 − v 2 W P2 + λP2 = 0.

(35) (36)

For the moment we focus only on P1 or, better, on the ﬁrst eigenfunction that P1 can represent. Then the energy W depends on Z1 + Z2 and λ (we suppose that they are given by or depend in a given way on W ). Let us consider the ground state 1sσ; for σ-electrons we know a relation between W and λ due to Eq. (2). We have only to ﬁx Z1 + Z2 . The expansion for large Z = (Z1 + Z2 )/2 is: W = Z2 + Z + . . . .

(37)

[6 ]

4.2.

VIBRATION MODES IN MOLECULES

A particular study of the vibration modes in molecules was carried out in the following notes. The main scope was to diagonalize the quadratic forms of kinetic (T ) and potential energy (V ) of the coupled oscillators, in order to ﬁnd the eigenfrequencies and eigendirections of their vibration modes. Several cases were considered, and a particularly careful study was devoted to the vibration modes of the molecule C2 H2 (acetylene) that, due to its geometry, presents three eigenfrequencies, two of which are equal. A possible diﬀerent (more general) study, suggested by the 6 @ This Section was probably left incomplete. The corresponding page in the original manuscript reported the following table with practically no entry, pointing out the intention of the author to evaluate P1 and its derivatives for some values of W and λ, in analogy with what was already done for P2 :

Z1 + Z2 = 4, u 1.00

W = λ= P1 P1 P1

W = λ= P1 P1 P1

1sσ W = λ= P1 P1 P1

W = λ= P1 P1 P1

276

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

just considered molecule of acetylene, was envisaged at the end of this Section. 1 1 2 x˙ i , (xi − xi−1 )2 ; V = T = 2 2 ∂V = xi − xi−1 − xi+1 + xi = 2xi − xi+1 − xi−1 . ∂xi

∂T = x˙ i , ∂ x˙ i

The equation of motion is then: x ¨i = xi+1 − 2xi + xi−1 . xi = ci η: x ¨i = ci η¨ = xi+1 − 2xi + xi−1 , ci η¨ = (ci+1 − 2ci + ci−1 )η. η¨ = −λη:

−ci λ = (ci+1 − 2ci + ci−1 ).

cr = k r :

1 −λ = k − 2 + , k k 2 − (2 − λ)k + 1 = 0; λ −4λ + λ2 1 = 1 − ± −λ + λ2 k = 2 2 4 λ 1 = 1− ± λ(λ − 4). 2 4 λ iϕ . ϕ = arccos 1 − k=e , 2 2−λ±

k1 = e ϕ1 =

2πi N

2π , N

,

√

k2 = e2 ϕ2 = 2

cos ϕ = 1 −

λ , 2

2πi N

2π , N

,

... ...

kr = er ϕr =

2πi N

,

...

kN = 1;

r 2π, N

...

ϕn = 2π.

λ = 1 − cos ϕ, 2 r λr = 4 sin2 π. N ——————–

λ = 4 sin2

ϕ ; 2

277

MOLECULAR PHYSICS

1 aik qi qk , 2

U=

T =

1 bik q˙i q˙k . 2

[7 ] qi =

aik qi qk = bik q˙i q˙k =

Sir ξr .

aik Sir Sks ξr ξs = bik Sir Sks ξ˙r ξ˙s =

Ars ξr ξs ,

Brs ξ˙r ξ˙s ,

[8 ] Ars = Brs =

aik Sir Sks ,

A = S ∗ aS,

bik Sir Sks ,

B = S ∗ bS.

Brs = δrs ,

λs δrs =

Ars = λr δrs .

aik Sir Sks ,

ik

δrs =

bik Sir Sks .

ik

(aik − λs bik )Sir Sks = 0

· ξr ,

ik

(aik − λr bik )Sir Sks = 0;

ik

(aik − λs bik )qi Sks = 0;

ik

(aik − λs bik )Sks = 0,

i = 1, 2, . . . , n;

k

7 @ In the original manuscript, the potential and kinetic energies are loosely written as P P U = 1/2 aik , T = 1/2 bik . 8 @ In the original manuscript, the dots (diﬀerentiation with respect to time) over the ξ variables in the last expression were omitted.

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E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

(aik − λt bik )Skt = 0.

k

4.2.1

bik Sit Sks = f (t)δrs .

The Acetylene Molecule

V =

1 2 aq1 + bq22 + aq32 . 2 y1 = q2 − y2 , x1 = q1 + q2 + q3 − x2 = q1 + q2 − y2 , x2 = y2 + q3 ,

y1 + y2 = q2 , x1 + x2 = q1 + q2 + q3 , x2 − y2 = q3 ,

y2 − y1 = 2y2 − q2 . x2 − x1 + 12(y2 − y1 ) = 0. 2y2 + q3 − q1 − q2 + 24y2 − 12q2 = 0, 26y2 − q1 − 13q2 + q3 = 0. y2 =

q1 + 13q2 − q3 , 26

y1 =

−q1 + 13q2 + q3 , 26

x2 =

q1 + 13q2 + 25q3 , 26

x1 =

25q1 + 13q2 + q3 . 26

(26)2 (x21 + x22 + 12y12 + 12y22 ) = (q1 + 13q2 + 25q3 )2 + (25q1 + 13q2 + q3 )2 + 12(q1 + 13q2 − q3 )2 + 12(−q1 + 13q2 + q3 )2 .

279

MOLECULAR PHYSICS

[9 ] q12

q22

q32

q1 q2

q2 q3 q3 q1

1 169 625 26 650 50 625 169 1 650 26 50 12 2028 12 312 −312 −24 12 2028 12 −312 312 −24 650 4394 650 676 676 52 26 25

169

25

26

26

2

26 x˙ 21 + x˙ 22 + 12y˙ 12 + 12y˙ 22 = 25q˙12 + 169q˙22 + 25q˙32 + 26q˙1 q˙2 + 26q˙2 q˙3 + 2q3 q1 . a 0 0 U=

25

1 0 b 0 , 2

T = 26T =

1

T =

a−

1 2

1 26

1 2

13 2

1 . 2

1 26

1 2

25 26

25 λ 26

1 − λ 2 −

9@

25 26 1 2

1 λ 26

1 − λ 2 b−

13 λ 2

1 − λ 2

1

1 13 169 13 , 2

0 0 a

U − λT =

13

13

25

1 λ 26 1 − λ 2 a−

.

25 λ 26

The following table was aimed to fully evaluate the expression just reported above. The numbers given in the lines 2 through 4 are just the coeﬃcients of the terms indicated in the ﬁrst line, while those in the sixth line are the corresponding sums. In the last line the author listed these sums divided by 26.

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E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

[10 ] 2V = aq12 + bq22 + aq32 , 2T =

25 2 13 2 25 2 1 q˙1 + q˙2 + q˙3 + q˙1 q˙2 + q˙2 q˙3 + q˙3 q˙1 . 26 2 26 13 u=

q1 + q3 , 2

v=

q3 = u − v.

q1 = u + v,

q12 + q32 = 2u2 + 2v 2 ,

q1 − q3 , 2

q1 q3 = u2 − v 2 ,

q1 + q3 = 2u.

[11 ] 2V 2T

= 2au2 + 2av 2 + bq22 , 1 1 25 2 25 2 13 2 u˙ + v˙ + q˙2 + 2u˙ q˙2 + u˙ 2 − v˙ 2 = 13 13 2 13 13 24 13 = 2u˙ 2 + v˙ 2 + q˙22 + 2u˙ q˙2 . 13 2

[12 ] 2V = 2av 2

+2au2 + bq2 ,

⎫ ⎪ ⎬

24 13 ⎪ 2T = v˙ 2 +2u˙ 2 + q˙22 + 2u˙ q˙2 , ⎭ 13 2

=⇒ λ1 =

13 a, 12

v=

q1 − q 3 . 2

2V = 2au2 + bq22 , 13 2T = 2u˙ 2 + q˙12 + 2u˙ q˙2 . 2 10 @ In the original manuscript the author evidently attempted to evaluate “directly” the values of λ which satisfy the equation det(U − λT ) = 0. The ﬁrst of the three roots was correctly reported, namely λ1 = (26/24)a, while the expressions of the other two roots were left incomplete. 11 @ In the original manuscript, all the variables entering the expressions for the kinetic energy given below appeared undotted. 12 @ In the following the author pointed out that one eigenvalue is λ = (26/24)a, corre1 sponding to the eigenmode v.

281

MOLECULAR PHYSICS

2a 0

U =

, 0

1

1

13 2

T =

b 2a − 2λ

2

−λ

U − λT =

,

−λ

b−

13 λ 2

.

[13 ] 12λ2 − (13a + 2b)λ + 2ab = 0 ——————– 1 T = (aϕ˙ 21 + aϕ˙ 22 − 2bϕ˙ 1 ϕ˙ 2 ), 2 b < a. x=

ϕ1 + ϕ2 , 2

ϕ1 = x + y, ϕ˙ 21 + ϕ˙ 22 = 2x˙ 2 + 2y˙ 2 , T =

y=

ϕ1 − ϕ2 , 2

ϕ2 = x − y. ϕ˙ 1 ϕ˙ 2 = x˙ 2 − y˙ 2 .

1 2(a − b)x˙ 2 + 2(a + b)y˙ 2 ; 2

∂T = 2(a − b)x, ˙ ∂ x˙

∂T = 2(a + b)y. ˙ ∂ y˙

V = −C1 ϕ1 + C2 (t)ϕ2 = [−C1 + C2 (t)] x − [C1 + C2 (t)]y; 13 @ The following expression, equated to zero, is the determinant of the previous characteristic matrix. It can be noted that the author did not report the expressions for the corresponding two eigenvalues, namely: √ 13 a + 2 b ± 169 a2 − 44 a b + 4 b2 , 24

whose physical meaning probably, was not clear.

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E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

∂V = −C1 + C2 (t), ∂x

∂V = −[C1 + C2 (t)]. ∂y

2(a − b)¨ x = −C1 + C2 (t). ——————–

T V

1 2 aϕ˙ 1 − 2bϕ˙ 1 ϕ˙ 2 + cϕ˙ 22 , 2 = −C1 ϕ1 + C2 ϕ2 ;

=

∂T = aϕ˙ 1 − bϕ˙ 2 , ∂ ϕ˙ 1 ∂V = −C1 , ∂ϕ1

∂T = cϕ˙ 2 − bϕ˙ 1 , ∂ ϕ˙ 2 ∂V = C2 . ∂ϕ2

aϕ¨1 − bϕ¨2 = C1 , cϕ¨2 − bϕ¨1 = C2 . ϕ2 = ϕ˙ 2 = ϕ¨2 = 0: aϕ¨1 = C1 , −bϕ¨1 = C2 ; C2 =

4.3.

b C1 . a

REDUCTION OF A THREE-FERMION TO A TWO-PARTICLE SYSTEM

The following calculations are aimed at studying the system formed by three fermions, the ﬁrst two being described by the state Ψ(q1 , q2 ), and the third one by Ψ(q). After some general remarks, the author shows how the study of the system considered may be reduced to that of a suitable two-particle system. Probably, he refers to the H2+ molecule or similar systems.

283

MOLECULAR PHYSICS

Let us consider an antisymmetric function of q1 , . . . , qn , ψ n (q1 , q2 , . . . , qn ): √ n + 1ψ n+1 (q1 , q2 , . . . , qn+1 ) = ψ n (q1 , . . . , qn ) ψ (qn+1 ) ±ψ n (q2 , q3 , . . . , qn , qn+1 ) ψ (q1 ) +ψ n (q3 , q4 , . . . , qn+1 , q1 ) ψ (q2 ) ±... ±ψ n (qn+1 , q1 , . . . , qn−1 ) ψ (qn ), where the upper signs refer to even n, the lower ones to odd n. ——————– Let us take a set of orthogonal functions ϕ1 , ϕ2 , . . .: √ n! gin1 ,i2 ,... (q1 , q2 , . . . , qn ) = |ϕi1 (q1 )ϕi2 (q2 ) . . . ϕin (qn )| (i1 < i2 < i3 < · · · < in ).

ψ n (q1 , . . . , qn ) =

ai gin (q1 , . . . , qn ),

i

ψ (q) =

r

|a2i | = 1,

ψ n+1 (q1 , . . . , qn+1 ) =

cr ϕr (q),

|c2r | = 1.

ai1 ,...,in cr gin+1 (q1 , q2 , . . . , qn+1 ) 1 ,...,in ,r

i1 ,...,in ,r

(r = i1 , . . . , in ). ——————– Let us now consider the states ψ(n1 , n2 , . . . , nS , . . . , nA ) and ψ (n1 , n2 , . . . , nS , . . . , nA ) with 0, n1 , n2 , . . . , nA = 1, and Ψ = ψψ : Ψ(n1 , n2 , . . . , ns , . . . , nA ) =

±ψ(n1 , n2 , . . . , nA )

n1 ,n2 ,...nA

· ψ (n1 − n1 , n2 − n2 , . . . , nA − nA ). ——————–

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E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

Ψ(q1 , q2 ) = −Ψ(q2 , q1 ), Ψ(q1 , q2 ) =

aik ϕi (q1 ) ϕk (q2 ),

aik = −aki , Ψ(q) =

|a2ik | = 1;

ci ϕi (q),

|c2i | = 1.

Ψ(q1 , q2 )ψ(q) + Ψ(q2 , q)ψ(q1 ) + Ψ(q, q1 )ψ(q2 ) √ 3 = −Ψ(q2 , q1 , q) = −Ψ(q1 , q, q2 ) = −Ψ(q, q2 , q1 ) = Ψ(q2 , q, q1 ) = Ψ(q, q1 , q2 ).

Ψ(q1 , q2 , q) =

√

3 Ψ(q1 , q2 , q) =

aik cr [ϕi (q1 )ϕk (q2 )ϕr (q) + ϕi (q2 )ϕk (q)ϕr (q1 )

i,k,r

+ϕi (q)ϕk (q1 )ϕr (q2 )] . ¯ dτ1 dτ2 dτ ΨΨ

=

1 3

i,k,r;,m,s

=

a ¯ik am c¯r cs [δi δkm δrs + δis δk δrm + δim δks δr + . . .] a ¯ik am [δi δkm δrs + δis δk δrm

i,k,r;,m,s

=

+ δim δks δr ] c¯r cs Ars c¯r cs .

r,s

Ars =

a ¯ik aik δrs +

i,k

= δrs +

a ¯sk akr +

k

(¯ ais ari + air a ¯si )

i

= δrs +

i

(¯ asi air + ari a ¯is ),

i

a ¯is ari

285

MOLECULAR PHYSICS

Ars = δrs − 2

a ¯si ari

i

by using aik = −aki . Ars = δrs − Lrs , ¯ L = AA. ——————– Without interaction we have: a˙ ik = − ¯˙ik = a

2πi (Hi alk + Hk ai ), h 2πi ¯ ¯ k a (Hi a ¯k + H ¯i ). h

[14 ] 2πi d ¯ s − a ¯ i ). (¯ asi ari ) = − (¯ asi ai Hr + a ¯si ar Hi − a ¯i ari H ¯s ari H dt h i

i,

Krs =

a ¯si ari ,

i

[15 ] 2πi ∂ (Ks Hr − Kr Hs ), Krs = − ∂t h

2πi K˙ = − (KH − HK). h

14 @ In the following expression appearing in the original manuscript, the author pointed out the cancellation of the second and fourth term in the sum. 15 @ In the original manuscript, some signs in the following expressions were incorrect.

5 STATISTICAL MECHANICS

5.1.

DEGENERATE GAS

A degenerate gas of spinless electrons in a box of length L is considered in the following. The electrostatic interaction between the particles is taken into account in a peculiar way. [1 ] For spinless electrons: ψ,m,n =

1 L3/2

e2πi(x+my+nz)/h =

T =

px =

h , L

1 L3/2

e2π p·q /h ,

h2 (2 + m2 + n2 ), 2L2 m py =

hm , L

pz =

hn . L

1 Ψ= √ ψ1 (q1 ) · · · ψn (qn ), N! ± ψi = ψi ,mi ,ni . Aik = Iik =

1@

e2 2 |ψ (q1 )| |ψk2 (q2 )| dq1 dq2 = A r12 i

(independent of i and k),

e2 ψ (q1 )ψ k (q2 )ψi (q2 )ψk (q1 ) dq1 dq2 . r12 i

In the original manuscript, the unidentiﬁed Ref. 8.47 appears here.

287

288

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

ΨHΨ dτ =

n i=1

Aik = A,

5.2.

Ti +

1 1 Aik − Iik , 2 2 i,k

Iik =

i,k

e2 h2 . π|pi − pk |2 L3

PAULI PARAMAGNETISM

In the following notes the author reported a (preliminary?) study on Pauli paramagnetism. He considered an ensemble of N degenerate fermions (so that N is proportional to the third power of the Fermi momentum, or V 3/2 where V is the electrostatic potential) interacting with a magnetic ﬁeld H by means of the Pauli term μ0 H, and obtained an expression for the magnetic susceptibility χ. The number of spin-up and spin-down fermions was denoted with n and n , respectively. N = kV 3/2 . 2 N = n + n ,

3 N 3 μ0 H N n = k (V + μ0 H)3/2 kV 3/2 + kV 1/2 μ0 H = + · , 2 2 2 V 2 3 μ N H 3 N 0 − · . n = k (V − μ0 H)3/2 kV 3/2 − kV 1/2 μ0 H = 2 2 2 V 2

n − n = μ0 (n − n ) =

χ=

3 μ0 H N, 2 V 3 μ20 H N. 2 V

3 μ2 (n − n )μ0 = N 0. H 2 V

[2 ] 2 @ In the original manuscript some numerical calculations appear here, that probably represent an attempt to evaluate the magnetic susceptibility of sodium N a (considered as an

289

STATISTICAL MECHANICS

5.3.

FERROMAGNETISM

In this Section, Majorana studied the problem of ferromagnetism in the framework of the Heisenberg model with the exchange interaction. However, it is rather evident that the Majorana approach is seemingly original, since he does not follow neither the Heisenberg formulation (see W. Heisenberg, Z. Phys. 49 (1928) 619) nor the subsequent van Vleck formulation (which followed Dirac) in terms of spin Hamiltonian (see J.H. van Vleck, The Theory of Electric and Magnetic Susceptibilities (Oxford University Press, London, 1932). He considered a system of i atoms (located at positions r1 , r2 , etc.) with spin parallel to the applied magnetic ﬁeld on a total of n atoms, and started by writing the Slater determinants A of the atomic wavefunctions ψ with respect to the possible combinations of i spin-up atoms out of the n total atoms. The Heisenberg exchange interaction (which is of electrostatic origin) Vrs among nearest neighbor atoms (the number of nearest neighbors is denoted with a) was then introduced and the energy E of the system evaluated. The subsequent calculations, performed by employing statistical arguments, were aimed to obtain the magnetization of the system (with respect to the saturation value) when a magnetic ﬁeld H acts on the magnetic moment μ of each atom. For further discussion, see S. Esposito, preprint arXiv:0805.3057 [physics:hist-ph]. r1 , r2 , . . . ri ri+1 , . . . rn A(r1 . . . ri |ri+1 . . . rn ) =

↑↑↑ ... ↓↓↓ ...

ψr1 (qn )δ(sn − 1) . . . ψri (qn )δ(sn − 1) ψri+1 (q1 )δ(s1 + 1) . . . ψri+1 (qn )δ(sn + 1) ... ψrn (q1 )δ(s1 + 1) . . . ψrn (qn )δ(sn + 1) ψr1 (q1 )δ(s1 − 1) ... ψri (q1 )δ(s1 − 1)

...

ensemble of 6 · 1023 (Avogadro’s number) nucleons, or 3 · 1022 nuclei): V = p · 1, 59 · 10−12 volt, χ=

3 3 3 · 0, 85 −6 0, 85 · 10−40 = · 3 · 1022 10 . 2 p · 1, 59 · 10−12 2 p · 1, 59

290

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS 1 1 Ai (r11 , r21 . . . r11 |ri+1 , ri+2 . . . rn1 ) ... τ Aτ (r1τ , . . . . . . r1τ |ri+1 , rnτ )

n! τ= i!(n − i)!

the order of r1 . . . ri or ri+1 . . . rn is not important

.

If H is the interaction operator acting on each particle, the electrostatic interaction potential V0 is given by: V0 = Hψ1 (q1 )ψ 1 (q1 )ψ2 (q2 )ψ 2 (q2 ) . . . ψn (qn )ψ n (qn ) dq1 . . . dqn . The exchange energy between r and s orbits Vrs : e2 ψr (q1 )ψ s (q1 )ψ r (q2 )ψs (q2 ) dq1 dq2 ; Vrs = |q1 − q2 | Vrs = Vsr .

Hmm = V0 −

r<s

m

Vrs +

ri

m

rn

Vrs ,

m r=r1m s=ri+1

and for m = n: ⎧ −Vrs , ⎪ ⎪ ⎪ ⎪ ⎨ Hmn =

⎪ ⎪ ⎪ ⎪ ⎩

0,

for a transition from Am to An by exchanging the opposite intrinsic orientation in the orbits ψr and ψs , for the other cases.

In the ferromagnetic case, if each atom has n neighbor atoms:3 ⎧ (neighbor atoms), ⎨ ε, Vrs = ⎩ 0, (distant atoms). na E = H − V0 + Vrs = H − V0 + ε. 2 r<s 3@

In the original manuscript, the upper limits of the second sum in the expression for Emm are both (incorrectly) written as rin .

291

STATISTICAL MECHANICS m

Emm =

m

ri rn

Vrs = Nm ε,

m r1m ri+1

and for m = n: ⎧ ⎨ −Vrs , Emn =

⎩

0.

Can we consider E as diagonal, in a statistical sense? Let us assume that it can be. For any given value of N , y solutions exist: y = y(N ). y

N

In each of the quantities A we exchange randomly an orbit ↑ with a ↓ one; the quantities A change into B: A1 −→ B1 ... Aτ −→ Bτ

♦

Statistically, the set of B’s coincides with that of the A’s. y0 = y(N0 ), that is, we have y0 quantities A corresponding to N0 . If we perform the transformation ♦, the quantities B corresponding to the y0 quantities A will be distributed between N0 − 2a

and

N0 + 2a,

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E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

and let p2a , p2a−2 , ... p2 , p0 , ... p−2 , ... p−2a ,

N0 + 2a, N0 + 2a − 2, N0 + 2, N0 , N0 − 2, N0 − 2a,

be the probabilities that one out of the mentioned B quantities corresponds to N = N0 + 2a, or N0 + 2a − 2, etc. We can evaluate the average increment: ΔN0 =

a

2r p2r .

−a

In fact, on average an electron ↑ has N0 electrons ↓ i

and

a−

N0 electrons ↑ i

a−

N0 electrons ↓ n−i

as neighbors, while an electron ↓ has N0 electrons ↑ n−i

and

as neighbors. By performing the mentioned exchange, we evidently have: 1 1 + . ΔN0 = 2a − 2N0 i n−i Let us assume that the probabilities p obey the following law (which we can call “normal”) p2r =

1 ΔN0 + 2 4a

a+r

1 ΔN0 − 2 4a

a−r

(2a)! . (a − r)!(a + r)!

Assuming that, for a restricted range, y(N0 + 1) = y0 ek , ... y(N0 ± a) = y0 e±ka ,

293

STATISTICAL MECHANICS

the condition that y(N ) does not change while we pass from A’s to B’s can be expressed as: a p2r e−2kr = 1, −a

which is solved by: ΔN0 2a . e = ΔN0 1− 2a k

1+

The trivial solution: k=0 has to be excluded since, although it does not change y = y(N ) for short ranges, it gives rise to a non constant “ﬂux” of “radions”4 through any section N = N0 of the curve y = y(N ) when passing from the A’s to the B’s. It follows that, by considering y as a continuous function of N : 1 N 1 2− + y a i n−i , = log 1 N 1 y + a i n−i and setting 1 α= a

n 1 1 = + , i n−i a i (n − i) y 2 = log −1 , y αN

we have d log y = log

2 − 1 dN. αN

2 − 1, αN 2 t+1= , αN 2 , N= α(t + 1) t=

4@

We ﬁnd the original text quite obscure.

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E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

dN = −

2 dt; α(t + 1)2

d log y = − log t ·

2 2 log t − α (t + 1) α

log y =

2 dt. α(t + 1)2

dt t(t + 1)

2 log t 2 2 − log t + log(t + 1) + k αt+1 α α 2 2 αN 2 2 log − 1− log − 1 + k, α αN α 2 αN

= =

y=c or

y=c

2 αN

2 α

2 2 − αN

− 2 +N

2 −1 αN

2 α

α

,

2 −1 αN

N

or y = c (t + 1) α t− α +N = c (t + 1) α t 2

2

2 = c, y(0) = c = y α

2

y

, 2 2 −α + α(t+1)

2 + ε2 α

;

= 0.

2 1 π π 1 −αN 2 dN = e y = c 2α . y α α α α n Since the number of solutions is , we have: i

y dN ∼ =

c =

n i

=

1 aπ

n 1 1 i(n−i)a π a i (n − i) 22 n

n i(n − i)

a·2i(n−i) 2 n 1 n n 1α α . = i i 2 2 π

295

STATISTICAL MECHANICS

2 αN

y=c y=c

2 α

2 2 − αN

− 2 +N

2 −1 αN

2 α

α

, N

2 −1 αN

.

——————– Numerical example: n = 10,

n i

i = 3,

= 120,

α=

a = 4,

5 , 42

2 = 16.8, α

c = 0.0002046, [5 ]

16.8

−16.8+N 16.8 , y = 0.0002046 −1 N 16.8 N 16.8 16.8 . y = 0.0002046 −1 16.8 − N N 16.8 N

[6 ] 5@

Note that the correct value of c is 0.0002047. The following table lists some values of y for given N (for example, 0 or 16.8, 1 or 15.8, etc.) as calculated from the previous expressions. Note, however, that the (complete) correct numerical values should be as:

6@

N

y

0 − 16.8 1 − 15.8 2 − 14.8 3 − 13.8 4 − 12.8 5 − 11.8 6 − 10.8 7 − 9.8 8 − 8.8 8.4

0.0002 0.0091 0.094 0.54 2.07 5.66 11.65 18.47 22.90 23.35

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E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

N

y

0 − 16.8 1 − 15.8 2 − 14.8 3 − 13.8 4 − 12.8 5 − 11.8 6 − 10.8 7 − 9.8 8 − 8.8 8.4

0.0002 0.0088 0.089

5.69 18.45 23.35

——————– We can also write: y=c2

2 α

1 2 − α

(1 − α N ) 2

1 − αN 1 + αN

N ,

where

1 , α which points out the symmetry property. N = N −

log y = k −

1 1 − αN log(1 − α2 N 2 ) + N log . α 1 + αN

1 1 log(1 − αN ) = −αN − α2 N 2 − α3 N 3 − . . . , 2 3 1 2 2 1 3 3 log(1 + αN ) = αN − α N + α N + . . . ; 2 3 1 log(1 − α2 N 2 ) = −α2 N 2 − α4 N 4 − . . . , 2 2 3 3 1 − αN = −2αN − α N − . . . ; log 1 + αN 3 1 1 1 1 log y = k − αN 2 − α3 N 4 − α5 N 6 − α7 N 8 − α9 N 10 − . . . . 6 15 28 45

e− kT = e− kT = e−LN = Ce−LN , L ε L= , C = e− α . kT W

Nε

297

STATISTICAL MECHANICS

W 1 1 log(y e− kT ) = k + log C − LN − αN 2 − α3 N 4 − α5 N 6 − . . . 6 15 1 1 − αN . = k + log C − LN − log(1 − α2 N 2 ) + N log α 1 + αN

W 1 − αN d = −L + log log y e− kT = −L + log dN 1 + αN

2 −1 , αN

W 2 2α αN 2 d2 − kT =− = ; log y e =− dN 2 αN 2 2 − αN N (2 − αN ) (1 − α2 N 2 )

y e− kT W

W0

max

−1 +

= y0 e− kT ,

2 = eL , αN0

2 = eL + 1, αN0 N0 =

2 , α(eL + 1)

αN =

eL

2 − αN ==

2 , +1 2eL , eL + 1

αN (2 − αN ) = y0 = c

W − kT

−LN0

y0 e

=e

W − kT

ye

c

= c

eL + 1 eL

eL + 1 eL

eL + 1 eL

4eL . (eL + 1)2 α2

α2

α2

eLN0 ,

LN0

e

−

e

=c

eL + 1 eL

(eL +1)2 α(N −N0 )2 4eL

.

α2 .

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E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

W − kT

ye

dN

eL + 1 2eL

α2

4eL (eL + 1)2 2 √ 1 + e−L α 2 eL = y dN · 2 eL + 1 √ 2 1 + e−L α 2 eL n = i 2 eL + 1 2i(n−i) √ a n 1 + e−L 2 eL n . = i 2 eL + 1 =

y dN ·

M= ↑ 1

↑ 2

μH . kT ↑ 3

... ↑ i

↓ ↓ ↓ ... ↓ i+1 i+2 i+3 n The ratio S between the magnetic moment under the inﬂuence of the ﬁeld H and the saturation magnetic moment is: 2i(n−i) 2i n 1 + e−L a n eM 2i e−M n i n 2 − 1. S= 2i(n−i) n 1 + e−L a n eM 2i e−M n i 2

log

=a

e− kT eM (2i−n) y dN W

1 + e−L 2i(n − i) log + 2M i − i log i − (n − i) log(n − i) + const. n 2

By taking the derivative with respect to i and equating the result to 0: a

1 − e−L 2(n − 2i) log + 2M − log i− 1 + log(n − i)+ 1 = 0, n 2 log

2(n − 2i) 1 + e−L i = 2M + a log . n−i n 2

299

STATISTICAL MECHANICS

2i − n 2i = − 1, n n 2i = 1 + S , n i 1 + S = ; n 2 1 − S n−i = . n 2

S =

log

1 + e−L 1 + S = 2M − 2aS log . 1 − S 2

It follows: log

μ 2 1 + S = 2 H + 2aS log ε . 1−S kT 1 + e− kT

For small H and large T : 2S = 2

2 μ H + 2aS log ε . kT 1 + e− kT

For T lower7 than the Curie point: for a given value of H there exist 2 values of S which, for not extremely high H, are practically equal and opposite. From it follows: a2i(n − i) 1 + e−L i n−i log = log − 2M i . n 2 n−i n − 2i Substituting in : −2M i

i2 (n − i)2 i + log i · − log(n − i) · . n − 2i n − 2i n − 2i

Let us set (y > 0): log log

7@

2 1+y = 2ay log ε , 1−y 1 + e− kT

1 + y + Δy μ 2 = 2 H + 2a(y + Δy) log ε , 1 − y − Δy kT 1 + e− kT

We ﬁnd the original text to be quite obscure, and our own interpretation is only a probable one.

300

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

2 μ Δy H + a log = ε Δy, 2 1−y kT 1 + e− kT [8 ]

Δy =

The LHS in

μH kT 2 1 − a log ε 2 1−y 1 − e− kT

.

can also be written as:

1 + S 1 + e−L 1 + S 1 − S 2 n log + M (1 + S )n − n log n 2 2 2 2 1 − S 1 − S −n log n + const. 2 2 1 + S 1 + e−L 1 + S 1 − S 2 log M S − log = a 2 2 2 2 1−S 1−S + const. − log 2 2 a

5.4.

FERROMAGNETISM: APPLICATIONS

In the following, the author gives some examples of ferromagnetic materials with diﬀerent geometries (corresponding to diﬀerent numbers i of oriented spins on a total of n, and to diﬀerent numbers a of nearest neighbors). Three insert also appear, mainly aimed at evaluating some theoretical quantities related to spontaneous magnetization. a = 3,

8@

i = 3, n − i = 3; n = 20. i

In the original manuscript, the following formula is incorrectly written as:

Δy =

μH kT 1 − y 2 − a log

2 ε − kT

1−e

.

301

STATISTICAL MECHANICS

1, 2, 3 2, 3, 4 3, 4, 5 4.5, 6 5, 6, 1 6, 1, 2 1, 2, 4 1, 2, 5 2, 3, 5 2, 3, 6 3, 4, 6 3, 4, 1 4, 5, 1 4, 5, 2 5, 6, 2 5, 6, 3 6, 1, 3 6, 1, 4 1, 3, 5 2, 4, 6

N

5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 9 9

y1

5 3 5 5 3 5 5 5 7 5 5 7 5 5 7 5 5 7 7 7

ﬀ

18 2

5 9

2 12 6

9 3 = 5 5.4 ; 7

5.4

18 · 0.42 + 2 · 3.62 = 28.8 2 · 2.42 + 12 · 0.42 + 6 · 1.62 = 28.8

y2 N y 1 N y 2 N 2 y1 N 2 y2 N 3 y1 N 3 y2

3 2 5 18 12 6 7 9 2 20 20

90 18 108

6 60 42 108

450 162 612

18 300 294 612

——————– Mean value: 1 a i(n − i) = , n−1 α 3·3·3 = 5.4. 5

2250 1458 3708

54 1500 2058 3612

302

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

a n−1

2 ,

a a−1 n−1 n−2 , a n−1 ,

i−1 n−i−1 i n−i 1 n−i−1 i n−i 1 i(n−i) .

=

+

ni−i2 −n+1 i(n−i) ,

i−1 1 i n−i

=

n−2 i(n−i) ,

1)

−

2)

n−1 n−2 n−3 n n n

[i(n−i)−(n−1)]n3 i(n−i)(n−1)(n−2)(n−3)

i(n−i)−(n−1) , i(n−i)

3)

n−1 n−2 4 n n n

[i(n−i)−(n−1)]n3 i(n−i)(n−1)(n−2)(n−3)

[i(n−i)−(n−1)] 4 n−3 , i(n−i)

4)

n−1 2 1 n n n

[i(n−i)−(n−1)]n3 i(n−i)(n−1)(n−2)(n−3)

i(n−i)−(n−1) 2 i(n−i) (n−2)(n−3) .

[i(n − i) − (n − 1)]n i(n − i)(n − 1)(n − 2)(n − 3) n a−1 i(n − i) − (n − 1) 1 n−2−4 + i(n − i) n − 1 n − 2 n−3 a i(n − i) − (n − 1) 1 1−2 + i(n − i) n − 1 (n − 2)(n − 3) 2 1 = a n [i(n − i) − (n − 1)] i(n − i)(n − 1)(n − 2)(n − 3) +a(a − 1)(n − 2)(n − 3) − 4a(a − 1) [i(n − i) − (n − 1)] +a(n − 2)(n − 3) −2a [i(n − i) − (n − 1)]} (a2 n − 4a2 + 2a)[i(n − i) − (n − 1)] + a2 (n − 2)(n − 3) . = i(n − i)(n − 1)(n − 2)(n − 3) a2

Mean value of the square of the terms in the diagonal: (a2 n − 4a2 + 2a)[i(n − i) − (n − 1)] + a2 (n − 2)(n − 3) (n − 1)(n − 2)(n − 3) 1836 612 24 · 4 + 108 = = 30.6 = = 3·3· 60 60 20 a2 a2 4n − 6 2 2 = i (n − i) + i2 (n − i)2 (n − 1)2 (n − 2)(n − 3) (n − 1)2 a2 4n − 4 − i2 (n − i)2 (n − 2)(n − 3) (n − 1)2 2a a2 n 1 2 + i (n − i)2 − i(n − i) (n − 2)(n − 3) n − 1 (n − 2)(n − 3) 4a2 2a a2 + i(n − i) − i(n − i) + i(n − i). (n − 2)(n − 3) (n − 2)(n − 3) n−1 i(n − i)

303

STATISTICAL MECHANICS

[9 ] terms in the diagonal

eigenvalues

ai(n−i) n−1

ai(n−i) n−1

mean value mean value of the square

a2 i2 (n−i)2 (n−1)2

a2 i2 (n−i)2 (n−1)2

+ k2

+ k2 +

ai(n−i) n−1

Statistically:

mean value mean value of the square

terms in the diagonal

eigenvalues

ai(n−i) n

ai(n−i) n

a2 i2 (n−i)2 n2

+

2ai2 (n−i)2 n3

a2 i2 (n−i)2 n2

+

——————–

n = 24,

9@

i = 6, a = 2; n = 134596. i

In the original manuscript, there appears here the matrix: ˛ ˛ ˛ V4 + V 5 −V5 −V4 ˛˛ ˛ ˛ −V5 V5 + V6 −V4 ˛˛ , ˛ ˛ −V4 −V4 V6 + V4 ˛

whose meaning is unclear to us.

2ai2 (n−i)2 n3

+

ai(n−i) n

304

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

[10 ] y

N

N − N0 (N − N0 )2 y(N − N0 )2

yN

20412 12 244944 68040 10 680400 34020 8 272160 9072 6 54432 3024 4 12096 28 0 0 134596 1264032 1

2.61 0.61 −1.39 −3.39 −5.39 −9.39

6.8 0.4 1.9 11.5 29 88

139000 26000 65000 104000 88000 2000 424000

9.3913

3.15

9.3913 =

3.15

2 · 6 · 18 216 = , 23 23

2 · 2 · 36 · 324 = 3.375. 243

——————–

n = 60,

i = 10,

n − i = 50.

a = 1,

[11 ]

10 @ The numbers in the last line of the following table are the mean values of y, yN and y(N − N0 )2 , respectively, which are obtained by dividing the numbers in the previous line by 134596. 11 @ See the previous footnote. The symbols introduced below have the following meaning: according to what is asserted in the original manuscript: ﬃ„ « ﬃ„ « 30 60 y† = y · 210 , yN ‡ = yN · 210 , 10 20 §

y(N − N0 )2 = y(N − N0 )2

ﬃ„

30 10

«

· 210 .

305

STATISTICAL MECHANICS

y†

N

yN ‡

N − N0 (N − N0 )2 y(N − N0 )2

1 10 10 1.071 8 8.57 0.341 6 2.05 0.037 4 0.15 0.001 2 0 0 2.450 20.77 1

1.52 −0.48 −2.48 −4.48 −6.48 −8.48

2.31 0.23 6.15 20 42 72

§

2.31 0.25 2.10 0.74 0.04 5.44

8.48

2.22

1 · 2 · 100 · 2500 = 2.31. 216000 ——————–

n i

∼ =

Pi Qi

k n nn i−i (n − i)−(n−i) √√ √ √ = , i n−i 2π i n−i i (n − i) i n−i n n n . k= 2π solutions with apparent momentum n − 2i, solutions with intrinsic momentum n − 2i.

i ≤ n/2. Pi =

i j=0

Qi = Pi − Pi−1 .

Qi ,

n . pi = i qi = pi − pi−1 ; i , pi−1 = pi n−i+1 n − 2i + 1 i = pi , qi = pi − pi−1 = pi 1 − n−i+1 n−i+1 n + 1 − 2i n . qi = i n+1−i

306

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

n = 4: i pi ni 0 1 2

Piσ

σ

Piσ 2

σ

Qσi

σ

1 3 2

a i(n − i) , n 2 2 a i (n − i)2 2a i2 (n − i)2 a i(n − i) n , = + + i n2 n3 n σ = Piσ − Pi−1 =

n i

σ

1 4 6

s

a i(n − i) a(i − i)(n − i + 1) n = − i−1 n n n a i(n − 2i + 1) , = i n Qi

qi

=

n i

a i(n + 1 − i) . n ——————–

Curves of the eigenvalues corresponding to apparent momentum N − 2i: yi = yi (N ). For large n, yi tend to the limiting form: 2 2 yi N = n + nfi (x), N. x= log = n + n fi n an an fi (x)max = fi (xi0 ), fi (xi0 ) = 0, fi (xi0 )

N: χ:

xi0 =

= −a

i n−i 2 a i(n − i) =2 , an n n n

1 n−i i2 (n − i)2 +8 2 4 n n n n2

2a i2 (n − i)2 a i(n − i) , + n3 n 8a i2 (n − i)2 4i(n − i) μ2 = + . a n5 a n3 μ2 =

.

307

STATISTICAL MECHANICS

5.5.

AGAIN ON FERROMAGNETISM

In the following pages, the author probably comes back again to ferromagnetism, but the meaning is quite obscure to us. See also E. Majorana, Nuovo Cim. 8 (1931) 78. ψ1 (q1 ) ψ1 (q2 ) . . . ψ1 (qn ) ψ2 (q1 ) . . . ψ2 (qn ) ψ (q ) ψ (q ) . . . ψ (q ) 2 2 2 n 2 1 = ψ1 (q1 ) . . . ... ψn (q1 ) . . . ψn (qn ) ψ (q ) ψ (q ) . . . ψ (q ) n 1 n 2 n n ψ2 (q3 ) . . . ψ2 (q1 ) ... ... ± ψ1 (q2 ) . . . ψn (q3 ) . . . ψn (q1 ) + ... n ϕ(qr+1 , qr+2 , . . . , qn , q1 , . . . , qr−1 )ψ(qr ), n = 2p + 1. r=1

[12 ] 1 ↑↑↑

0

2 ↑↑↓

ϕ1 (ψ2 ψ3 ) − ϕ2 (ψ1 ψ3 )

(123)

3 ↑↓↑

ϕ3 (ψ1 ψ2 ) − ϕ1 (ψ3 ψ2 )

(132)

4 ↑↓↓

0

5 ↓↑↑

ϕ2 (ψ3 ψ1 ) − ϕ3 (ψ2 ψ1 )

6 ↓↑↓

0

7 ↓↓↑

0

8 ↓↓↓

0

(123)

ψ, ϕ, u, v. ψ1 ψ2 (u1 v2 −u2 v1 )ϕ3 u3 +ψ2 ψ3 (u2 v3 −u3 v2 )ϕ1 u1 +ψ3 ψ1 (u3 v1 −u1 v3 )ϕ2 u2 . 12 @ In the original manuscript, some pages of scratch calculations appear here: they deal with combinations of several objects grouped in diﬀerent ways, probably with an eye on the study of ferromagnetism (see below).

PART III

6 THE THEORY OF SCATTERING

6.1.

SCATTERING FROM A POTENTIAL WELL

The author studied here the problem of the scattering of a plane wave from a one-dimensional square potential well. All the physically interesting cases were treated.

e = h/2π = m = 1. ∇2 ψ + 2(E − V )ψ = 0. V = 0: y + 2Ey = 0. 2E = k 2 , y2 = e−ikx .

y1 = eikx ,

y + 2(E − U )y = 0, U = −V ,

y + 2(E + V )y = 0. 2(E + V ) = μ2 , μ=k

1+

2E = k 2 , V . E

By imposing the matching conditions for the wavefunction and its derivative, one obtains:

311

312

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

⎧ V ⎪ 1 + 1 + 1+ 1 − ⎪ E ik(x+a)−iμa ⎪ ⎪ + e ⎪ ⎪ ⎪ 2 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ 1V iμx e , y= g =1+ ⎪ 2E ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ V ⎪ ⎪ 1 − 1 + 1+ 1 + ⎪ E ik(x−a)+iμa ⎪ ⎪ e + ⎪ ⎪ 2 2 ⎩

V E

e−ik(x+a)−iμa , x < −a, − a < x < a,

V E

e−ik(x+a)+iμa , a < x,

E > 0, E=

1 2 μ − V, 2

μ=k

√

1+

E=

V , E

1 2 k , 2

μ k= 1+

. V E

g gives the ratio of the wave amplitude inside and outside the well.1

E < 0, E > −V :

V =i 1+ E

μ = ik

1@

V − 1 = k1 −E

V − 1, −E

V − 1, −E

k1 = ik.

That is: g is given by the ratio a2 + b2 /c2 where a [b] is the coeﬃcient of the ﬁrst [second] wave term in the ﬁrst or third row, while c is the coeﬃcient of the wave term in the second row (c = 1). Note that the quantity we call g, here and in what follows, is in the original manuscript denoted by y, the same as the symbol there used for the wave function.

313

THE THEORY OF SCATTERING

⎧ V V ⎪ 1 + i −E 1 − i − 1 ⎪ −E − 1 −k1 (x+a)−iμa ⎪ k1 (x+a)−iμa ⎪ + , e e ⎪ ⎪ ⎪ 2 2 ⎪ ⎪ x < −a, ⎪ ⎪ ⎪ ⎪ ⎨ y = eiμx , − a < x < a, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ V V ⎪ ⎪ 1 + i −E 1 − i −E −1 −1 ⎪ ⎪ k1 (x−a)+iμa ⎪ e e−k1 (x−a)+iμa , + ⎪ ⎪ 2 2 ⎩ a < x, −V < E < 0, 1 E = μ2 − V , 2 μ = k1

1 E = − k12 , 2

V − 1, −E

k1 =

μ V −E

−1

.

Stationary states:2

2@

In the original manuscript, there appear here the following calculations: 1+i

q

V −E

2

e

−iμa

−1

eiμa = c in = c eniπ/2 .

"

# V sin k(x + a) , i+ E

r cos k(x + a) + i

" eiμa cos k(x + a) − i

# V sin k(x + a) ; E

r i+

r − sin μa cos k(x − a) + cos μa

1+

r cos μa cos k(x − a) − sin μa

1+

V sin k(x + a), E

V sin k(x + a). E

314

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

⎧ ⎪ V ⎪ ⎪ 1+ cos μa sin k(x + a) − sin μa cos k(x + a), ⎪ ⎪ E ⎪ ⎪ ⎪ x < −a, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ V 2 sin μx, g = 1 + cos μa , y= − a < x < a, ⎪ E ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ V ⎪ ⎪ 1+ cos μa sin k(x − a) + sin μa cos k(x − a), ⎪ ⎪ E ⎪ ⎩ a < x, ⎧ ⎪ V ⎪ ⎪ − 1+ sin μa sin k(x + a) + cos μa cos k(x + a), ⎪ ⎪ E ⎪ ⎪ ⎪ x < −a, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ V 2 cos μx, g = 1 + sin μa , − a < x < a, y= ⎪ E ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ V ⎪ ⎪ 1+ sin μa sin k(x − a) + cos μa cos k(x − a), ⎪ ⎪ E ⎪ ⎩ a < x,

μ

k= 1+

,

μ=k

1+

V E

V , E

1 1 E = k 2 = μ2 − V . 2 2

[3 ]

3@

In the original manuscript, the following calculations appear at this point: ! ! r r V V 1+c 1+ cos μa − c i + i 1 + sin μa = 0, E E ! r r V V cos μa − i sin μa = − cos μa + i 1 + sin μa, 1+ c E E q cos μa − i 1 + V sin μa E c=−q . 1+ V cos μa − i sin μa E

[The footnote continues on the next page].

315

THE THEORY OF SCATTERING

Reﬂection

⎧ ⎪ V V ⎪ ⎪ 2 1 + cos 2μa − i 2 + sin 2μa eik(x+a) ⎪ ⎪ E E ⎪ ⎪ ⎪ ⎪ V ⎪ ⎪ ⎪ + i sin 2μa e−ik(x+a) , x < −a, ⎪ ⎪ E ⎪ ⎪ ⎪ ⎨ y= V V iμ(x−a) ⎪ − 1− 1+ 1+ 1+ e e−iμ(x−a) , ⎪ ⎪ E E ⎪ ⎪ ⎪ ⎪ ⎪ − a < x < a, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 2 1 + V eik(x−a) , a < x, E incident energy:

A=4+4

reﬂected energy:

Ar =

V2 V + 2 sin2 2μa, E E

V2 sin2 2μa, E2 V AR = 4 + 4 , E

refracted energy:

2

V sin2 2μa Ar E2 = ; ρ= 2 A 4 + 4 VE + VE 2 sin2 2μa

reﬂecting power:

π μa = n . 2

minima:

3

r 1+ r

V +1 E

!

r cos μa − i

! V 1+ − 1 cos μa + i E r 1+

V E

!

1+

V +1 E

!

r 1+

sin μa =

! V 1+ − 1 sin μa = c , −

r 1−

1+

V E

!

e−iμa ,

! V 1+ − 1 eiμa ; E

r

e−iμa

V +1 E

r

! eiμa .

316

6.2.

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

SIMPLE PERTURBATION METHOD

In the following few passages, Majorana traced the general lines of a simple perturbation method in order to solve the Schr¨ odinger equation for a particle in a potential ﬁeld V in terms of the known eigenstates ψi . ∇2 ψ + 2(E − V )ψ = 0. ∇2 ψ0 + 2Eψ0 = 0. ψ = ψ0 + χ, ∇2 ψ0 + 2(E − V )ψ0 + ∇2 χ + 2(E − V )χ = 0, ∇2 χ + 2(E − V )χ = 2V ψ0 . ∇2 ψi + 2(Ei − V )ψi = 0. 2V ψ0 = 2

ci ψi ,

∇2 χ + 2(E − V )χ = 2 χ=

ci ψi .

di ψi ,

∇2 ψi + 2(E − V )ψi = 2(E − Ei )ψi ; ∇2 χ + 2(E − V )χ = 2 di =

ci . E − Ei

di (E − Ei )ψ,

317

THE THEORY OF SCATTERING

6.3.

THE DIRAC METHOD

The author applied the perturbation theory to the problem of the scattering of a particle of momentum p = hγ from a potential V ; the freeparticle wavefunction is denoted with φγ . Some approximated expressions for the transition probability were obtained within the framework of the Dirac method, which are subsequently applied to the particular case of Coulomb scattering. 1 2 p +V 2m h2 2 γ + V. 2m

E = =

φγ = e2πi(γx x+γy y+γz z) e−2πi(h/2m)γ t , 2

<γ |V |γ >=

φγ V φγ dxdydz = kγ γ e−2πi(h/2m)(γ = kγ γ e2πi(h/2m)(γ

2 −γ 2

2 −γ 2

)t

)t .

ψ=

α˙ γ = −

2πi h

αγ φγ dγ,

kγ γ e2πi(h/2m)(γ

2 −γ 2

)t α dγ . γ

For t = 0 4 : αγ = δ (γ − γ0 ) . For t > 0: 1st approximation: 2 2 2πi kγγ0 e2πi(h/2m)(γ −γ0 )t , h 2m 2πi(h/2m)(γ 2 −γ02 )t αγ = − 2 2 − 1 + δ (γ − γ0 ) . k e γγ 0 h (γ − γ02 )

α˙ γ = −

4 @ Here the author denotes with γ = p /h the momentum (divided by h) of the free particle, 0 0 while δ(x) signiﬁes the Dirac delta-function.

318

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

2nd approximation: 1 2πi 4πim 2πi(h/2m)(γ 2 −γ02 )t kγγ kγ γ0 2 α˙ γ = − kγγ0 e + 3 h h γ − γ02 2 2 2 2 × e2πi(h/2m)(γ −γ0 )t − e2πi(h/2m)(γ −γ )t dγ , αγ = −

kγγ0 2πi(h/2m)(γ 2 −γ02 )t 2m − 1 e h2 (γ 2 − γ02 ) γ 2 − γ02 2 2 4m2 e2πi(h/2m)(γ −γ0 )t − 1 + 4 kγγ kγ γ0 h (γ 2 − γ02 )(γ 2 − γ02 ) 2 2 e2πi(h/2m)(γ −γ )t − 1 dγ + δ (γ − γ0 ) . − (γ 2 − γ 2 )(γ 2 − γ02 )

In ﬁrst approximation, for γ = γ0 , we have: |αγ |2 =

2 2 16m2 2 2 πh(γ − γ0 )t . |k | sin γγ 0 2m h4 (γ 2 − γ02 )2

Neglecting constant terms, for t → ∞ we get: |αγ |2 =

8π 2 m |kγγ0 |2 t δ γ 2 − γ02 , 3 h

and the transition probability is: Pγ0 γ =

8π 2 m 2 2 2 . |k | δ γ − γ γγ 0 0 h3

In second approximation: 2 2 16m2 2 2 πh(γ − γ0 )t |k | sin γγ0 2m h4 (γ 2 − γ02 )2 3 2 2 πh(γ − γ0 )t 32m sin k γγ0 kγγ kγ γ0 + kγγ0 k γγ k γ γ0 + 6 2 2 2m h (γ − γ0 ) 2 2 sin πh(γ − γ )t/2m sin πh(γ 2 − γ02 )t/2m − dγ . × (γ 2 − γ 2 )(γ 2 − γ02 ) (γ 2 − γ02 )(γ 2 − γ02 )

|αγ |2 =

6.3.1

Coulomb Field

For a Coulomb ﬁeld: C = Vγ e2πi(γx x+γy y+γz z) dx dy dz, V = r

319

THE THEORY OF SCATTERING

Vγ = C

e−2πiγ ·q dq = C r kγ γ = Vγ−γ

∞

C 2 sin 2πγr dr = ; γ πγ 2 0 C = . π|γ − γ |2

In ﬁrst approximation: Pγ0 γ =

2 mC 2 8mC 2 2 = δ γ − γ δ γ 2 − γ02 . 0 4 4 3 4 3 h |γ − γ0 | 2h γ0 sin θ/2

In second approximation, for γ = γ0 : 5 2m 1 C 2πi(h/2m)(γ 2 −γ02 )t αγ = − 2 2 − 1 e h γ − γ02 π(γ − γ0 )2 2 2 4m2 e2πi(h/2m)(γ −γ0 )t − 1 C2 + 4 h π 2 |γ − γ |2 |γ − γ0 |2 (γ 2 − γ02 )(γ 2 − γ02 ) 2 2 e2πi(h/2m)(γ −γ )t − 1 dγ . − (γ 2 − γ 2 )(γ 2 − γ02 )

6.4.

THE BORN METHOD

The scattering from a given center was studied here by means of the Born method, and approximated expressions for the scattered partial waves were obtained. ∇2 ψ + k 2 ψ = F ψ. ψ = ψ 0 + ψ1 + ψ2 + . . . , ∇2 ψ0 + k 2 ψ0 = 0, ∇ 2 ψ1 + k 2 ψ1 = F ψ 0 , ∇2 ψ2 + k 2 ψ2 = F ψ 2 , ..., ∇2 ψn + k 2 ψn = F ψn−1 , .... 5 @ Probably, the author started to evaluate the transition probability for Coulomb scattering in a second approximation, but succeeded only in obtaining an expression for the coeﬃcient αγ .

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E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

1 ψn (q) = − 4π

eik|q −q | F (q ) ψn−1 (q ) dq . |q − q |

ψ0 (q) = eiku0 ·q , ik|q −q | e 1 iku0 ·q F (q ) dq , ψ1 (q) = − e 4π |q − q | ik|q −q | ik|q −q | e e 1 iku0 ·q F (q ) F (q ) dq dq , ψ2 (q) = e 2 16π |q − q | |q − q | |u0 | = 1. |q| = r → ∞: 1 ψ1 (q) = − 4πr

|q| = r,

|u| = 1;

q = r u,

|q | = r , r → ∞:

eik|q −q | eiku0 ·q F (q ) dq .

|u | = 1,

q =r u,

|q − q | = r − r u · u , eikr ψ1 (q) = − 4πr

eikr ψ2 (q) = 16π 2 r

eikr (u0 −u)·u F (q ) dq .

eik|q −q | ikr u0 ·u ikr u ·u e e F (q ) F (q ) dq dq . |q − q |

q = q + : eikr ψ2 (q) = 16π 2 r

ik(u0 −u)·q

e

F (q ) dq

eik|| iku0 · F (q + ) d. e ||

1 Fγ eiγ ·q dγ, F = 2π Fγ = F e−iγ ·q dq,

e−iγ ·q

4π eikr dq = 2 , r γ − k2

321

THE THEORY OF SCATTERING

r → ∞: ψ1 (q) = −

eikr , F 4π k(u−u0 )

eikr 2 ik(u0 −u)·q e F (q ) dq Fγ eiγ ·q dγ 2 16π r |ku0 + γ|2 − k 2 2 eikr Fγ dγ ei(ku0 −ku+γ )·q F (q ) dq , = 2 2 2 16π r |ku0 + γ| − k

ψ2 (q) =

eikr ψ2 (q) = 2 8π r ψ2 (q) =

6.5.

eikr 8π 2 r

Fγ Fk(u−u0 )−γ dγ |ku0 + γ|2 − k 2 Fγ−ku0 Fku−γ dγ. γ 2 − k2

COULOMB SCATTERING

The Schr¨ odinger equation for the scattering of a wave from a Coulomb potential is solved and, in particular, the phase advancement is evaluated. Ze charge of the scatterer; Z e charge of the incident particle; M mass of the incident particle. We adopt units such that M = 1, ZZ e2 = 1, h/2π = 1. It follows that: the length unit is h2 /4π 2 M ZZ e2 = (m/M ) (1/ZZ ) a0 ;

6

the energy unit is 4π 2 M Z 2 Z 2 e4 /h2 = 2(M/m) Z 2 Z 2 Rh;

7

the velocity unit is 2πZZ e2 /h = ZZ /137c, where 1/137 = e2 /(1/2π)hc. The Schr¨ odinger equation is:

1 ∇ ψ+2 E− r 2

ψ = 0.

m denotes the electron mass and a0 0.529 · 10−9 the Bohr radius. Rh = 13.54 V [Remember that the symbol V used by Majorana should more appropriately understood as eV]. 6 Here 71

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E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

ψ=

n

α

=0

X

X (r) P (cos θ), r

2 ( + 1) + k − − r r2

X = 0,

2

k 2 = 2E (the velocity of the ingoing particle in large units is v = (ZZ /137)c k). X = X1 + X2 , X1 X2

i = x e F + 1 + , 2 + 2, −2ikx , k i +1 −ikx = x e F + 1 − , 2 + 2, 2ikx , k +1 ikx

F (α, β, x) = 1 +

α α(α + 1)(α + 2) 3 α(α + 1) 2 x+ x + x + .... β 2!β(β + 1) 3!β(β + 1)(β + 2)

Alternative solution

2 ( + 1) X + k − − r r2

2

X = 0,

takes non-integer values greater than −1/2, X = r+1 u, +1 2 2 u + k − u = 0. u +2 r r [8 ]

δ1 u + δ0 + r

1 u = 0, u + 0 + r

δ0 = 0, δ1 = 2( + 1), 0 = k 2 , 1 = −2: u ∼ eiktr (t − 1)+i/k (t + 1)−i/k dt.

8@

This equation is a particular case of the more general one reported just after it, and is also considered by the author in another place; see Appendix 6.10.

323

THE THEORY OF SCATTERING

[...]9 |Im log(1 − t)| ≤ π, |Im log(1 + t)| ≤ π:

1

u= −1

eiktr (1 − t)+i/k (1 + t)−i/k dt .

For r = 0, on setting 1 − t = 2x: 1 1 u(0) = (1 − t)+i/k (1 + t)−i/k dt = (2x)−i/k (2 − 2x)+i/k 2dx −1

= 22+1

0 1

(x)−i/k (1 − x)+i/k dx,

0

u(0) = 22+1

Γ ( + 1 − i/k) Γ ( + 1 + i/k) . Γ (2 + 2)

(1)

|r| > 0:

u1

u = u 1 + u2 , ∞ = e−i(π/2)(+1+i/k) eikx e−krp p+i/k (2 + ip)−ik dp, 0

u2 = ei(π/2)(+1−i/k) e−ikx

∞

e−krp p−+i/k (2 − ip)+ik dp.

0

For real r we have u2 = u1 . u1 = (kr)−(+1) e−i(π/2)(+1+i/k)−(i/k) log kr eikr ∞ × e−p p+i/k (2 + ip/kr)−ik dp. 0

For r → ∞: u1 = (kr)−(+1) eπ/2k e−i(π/2)(+1) eikr−(i/k) log kr 2−i/k Γ( + 1 + i/k) = 2 (kr)−(+1) eπ/2k e−i(π/2)(+1) eikr−(i/k) log 2kr Γ( + 1 + i/k). Now, replace with − ; the phase advancement becomes then: k =

Γ( + 1 + i/k) π − arg . 2 Γ( + 1 − + i/k)

The author then evaluates u and u and veriﬁes that the assumed form for u satisﬁes the previous diﬀerential equation.

9@

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E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

1/4 > a > 0: ( − ) ( + 1 − ) = ( + 1) − a, 2 1 1 2 + − = + − a, 2 2 1

= + − 2

6.6.

1 + 2

2 − a.

QUASI COULOMBIAN SCATTERING OF PARTICLES

Let us assume a scattering potential of the form: √

k , + a2

(1)

r2

a being the magnitude of the radius of the scatterer. By denoting with T the kinetic energy of the incident particles, let us deﬁne the minimum approach distance10 b in the limit Coulomb ﬁeld (a = 0) as: k = T; b

b=

k . T

(2)

The scattering intensity under an angle θ will be obtained on multiplying that appearing in the Rutherford formula by a numerical factor depending on the mutual ratios of a, b, λ/2π (λ being the wavelength of the free particle) and θ. Let us set: i = f (α, β, θ) iR ,

(3)

where iR is the intensity calculated from the Rutherford formula (a = 0) and a b α= (4) , β= . λ/2π λ/2π Since for a = 0 the Rutherford formula is exact, we have: f (0, β, θ) = 1. 10 @

That is, the scattering parameter.

(5)

325

THE THEORY OF SCATTERING

Let us now consider a ﬁxed α and take the limit β → 0. At zeroth order approximation, i.e., exactly for β = 0, we can use the Wentzel method. By choosing as mass unit M , wavelength unit λ/2π and velocity unit v for the incident particles, from λ = h/M v it follows that h = 2π in our units. Moreover, the kinetic energy of the incident particle is 1/2. From Eqs. (4) and (2) it follows that b = β, k = β/2 and a = α. By substituting these into Eq. (1), we get the expression for the potential energy, and the Schr¨ odinger equation corresponding to the eigenvalue 1/2 will be: β 2 ∇ ψ+ 1− √ (6) ψ = 0. r 2 + α2 Let us set: ψ = ψ0 + ψ1 + ψ2 + . . . , where: ∇2 ψn + ψn = √

r2

β ψn−1 . + α2

(7)

√ 2 2 In order to avoid convergence of β/ r + α let us √problems, instead consider the expression β 1/ r2 + α2 − 1/R for r < R and 0 for r > R; in the ﬁnal results we will take the limit R → ∞. Eq. (7) is then replaced by: 11 ∇2 ψn + ψn = P ψn−1 ; ⎧ 1 1 ⎪ ⎪ , − ⎨ β √ 2 r + α2 R P = ⎪ ⎪ ⎩ 0,

(8)

for r < R; for r > R.

Setting ψ0 = eiz , we have: ∇2 ψ1 + ψ1 = P eiz .

(9)

For an univocal solution of Eq. (9) we will choose ψ1 to represent a diverging wave. In this case Eq. (9) can be integrated and, putting r12 = (x − x )2 + (y − y )2 + (z − z )2 , 11 @

In the original manuscript, the factor β is lacking.

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E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

we get: ψ1 (x, y, z) = −

1 4π

P (x , y , z )

ei(r12 +z ) dx dy dz . r12

(10)

Assuming the point (x, y, z) to be far from the origin, we have (r is the distance from the origin, θ the angle between the vector radius and the z-axis): r→∞:

1 ψ1 (r, θ) = − 4πr

P (x , y , z ) eir

× eir (cos θ −cos θ cos θ −sin θ sin θ

cos φ )

dx dy dz ,

cos θ (1 − cos θ) − sin θ sin θ cos φ = 2 sin θ/2 sin θ/2 cos θ − cos θ/2 sin θ cos φ = 2 sin θ/2 cos (π/2 − θ/2) cos θ + sin (π/2 − θ/2) sin θ cos φ − π , r→∞:

eir ψ1 (r, θ) = − 2 r sin θ/2

∞

r P (r ) sin 2 sin θ/2 r dr ,

0

(11) whence we easily deduce: 2 f (α, 0, θ) = sin θ/2 β

∞

r P (r ) sin 2 sin θ/2 r dr .

(12)

0

√ In we simply replace P with β/ r2 + α2 , the integral in Eq. (12) does not converge; however, we can circumvent this diﬃculty by keeping indeterminate the upper integration limit and assuming, for the resulting integral, its mean value which for the upper limit tends to inﬁnity. We thus ﬁnd: f (α, 0, θ) = 2 sin θ/2 0

= 0

∞

∞

√

r sin (2 sin θ/2 r) dr r 2 + α2 (13)

x sin x dx

= ϕ (α sin θ/2) . x2 + 4α2 sin2 θ/2

327

THE THEORY OF SCATTERING

6.6.1

Method Of The Particular Solutions

u

β ( + 1) + 1− √ − 2 2 r2 r +α

u = 0.

(14)

For the hydrogen atom we consider the values β = 0.4, 0.5, 0.6, 0.7 and α = 0, 0.2, 0.4, 0.6, 0.8, 1. The solution of Eq. (14) is reported numerically in the following tables for = 0 and β = 0.4. 12

r 0

u 0

0.1

0.1018

α=0 u 1 1.019

u 0.400

u 0

α = 0.2 u u 1 0

u 0

α = 0.4 u u 1 0

0.305 1.049

0.2

0.2067

0.207 1.070

0.3

0.3137

0.109 1.080

0.4

0.4217

0.000 1.080

0.5

0.5297

-0.106 1.069

0.6

0.6366

-0.212

0.7 0.8 0.9 1.0 1.1 1.2 1.3

12 @ The author uses a numerical algorithm (unknown to us) in order to infer the solution u(r) of Eq. (14) from its second (and ﬁrst) derivative, and the ﬁrst few results obtained are displayed in the tables.

328

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

r 0

u 0

α = 0.6 u u 1 0

u 0

α = 0.8 u u 1 0

u 0

α = 1.0 u u 1 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3

6.7.

COULOMB SCATTERING: ANOTHER REGULARIZATION METHOD

Let us assume the potential to be as follows: for r < R, V0 , V = k/r, for r > R.

(1)

V

R V0

r

329

THE THEORY OF SCATTERING

Denoting with T the kinetic energy of the incident particles, the minimum approach distance in the Coulomb ﬁeld will be: b=

k . T

(2)

The scattering intensity under an angle θ will be given by the product of the intensity scattering due to the Coulomb ﬁeld, obtained from the Rutherford formula, times a numeric function depending on θ, R/λ, b/λ, V0 /T : 13 b V0 R f , , ,θ , (3) T λ/2π λ/2π where λ is the wavelength of the free particle. Let us choose as mass unit M , velocity unit v and length unit λ/2π relative to the free particle. In such units, h = λM v 14 is equal to 2π, while T is 1/2. Moreover, let us set: R b V0 , α= ,β = , (4) A= T λ/2π λ/2π so that: V0 R b i =f , , , θ = f (A, α, β, θ). (5) iR T λ/2π λ/2π In our units we have: V0 =

A , 2

R = α,

b = β,

k=

1 β, 2

(6)

and the Schr¨ odinger equation corresponding to the eigenvalue 1/2 takes the form: ∇2 ψ + (1 − A) ψ = 0, for r < R, β ∇2 ψ + 1 − ψ = 0, r

(7) for r > R.

For the hydrogen we have: β = 0.4, 0.5, 0.6, 0.7; α = 0.4, 0.5, 0.6, 0.7, 0.8; A = (2), (1.5), 1, 0.5, 0, − 0.5, − 1, − 1.5, 2, − 2.5, − 3, − 3.5, 4, − 4.5, − 5, − 5.5, 6, − 6.5, − 7, − 7.5, − 8. 13 @

In the original manuscript, the ﬁrst dependent variable in Eq. (3) is V0 /2T rather than V0 /T . However, in the following the author considered the latter parametrization. 14 @ In the original manuscript, the author wrote erroneously h = λ/M v.

330

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

6.8.

TWO-ELECTRON SCATTERING v, v be the velocities of the two beams; n0 , n0 be the rest number densities of the two beams; n = n0 / 1 − v 2 /c2 , n = n0 / 1 − v 2 /c2 be the number densities in the laboratory reference frame; vr be the relative velocity according to the relativistic kinematics; S(vr ) be the cross section.

The number N of collisions for unit volume and time can be written as: vr N = a v, v n n = S(vr ) n0 n0 . 1 − vr2 /c2 In terms of a we thus have:

1 − vr2 /c2 a S= vr 1 − v 2 /c2 1 − v 2 /c2

(classically (that is: non relativistically), we have instead S = a/|v −v |). Without considering the resonance in the scattering cross section, let u (0 ≤ u ≤ vr ) be the velocity of the ﬁrst electron after the collision in its initial reference frame; we have: dS = S (vr , u) du. Let us now denote with u1 the relative velocity between the frame of the ﬁrst electron before (after) the collision and that of the second electron after (before) the collision. By taking into account the resonance between the two electrons, u and u1 are indistinguishable. Putting, conventionally, u ≤ u1 , the maximum value of u is given by 15 : 4 (1 − vr2 /c2 ) y y2 = 2 − umax = umin = c1 − 2 2 c2 1 + 1 − vr2 /c2 1 − 1 − vr2 /c2 y=c . 1 + 1 + vr2 /c2 The relation between u and u1 is the following: 1 1 1 + =1+ . 2 2 2 2 1 − u /c 1 − vr2 /c2 1 − u1 /c 15 In

this case we have u = u1

331

THE THEORY OF SCATTERING

6.9.

COMPTON EFFECT n = n0 / 1 − v 2 /c2 , number of electrons per cm3 ; n0 , rest number densities of the electron beams; N , number of photons

16

per cm3 ;

N0 = N ν0 /ν, number of photons per cm3 in the electron frame; hν, energy of one photon; hν0 , energy of one photon in the electron frame (before or after the collision); u1 , relative velocity between the ingoing electron frame and the outgoing one (according to relativistic kinematics); S(ν0 ), cross section. The number of collisions for unit volume and time is thus: S(ν0 ) n0 N0 c = a n N, so that, in terms of a, S(ν0 ) =

a ν 1 . c 1 − v 2 /c2 ν0

The diﬀerential cross section can be written as: dS = F (ν0 , u) du,

so that

∞

F (ν0 , u) du.

S= 0

Classically, the cross section is given by: Sclass =

8π e4 . 3 m2 c4

16 @ For the sake of clarity, here and in the following we have translated with “photons” what was termed “quanta” in the original manuscript.

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E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

6.10.

QUASI-STATIONARY STATES

The author considered the transition from a discrete (unperturbed) state ψ0 with energy E0 to a continuum (perturbed) state ψ, assuming that the unperturbed system has two continuum spectra φW and ψW with energy E0 + W . What here reported are the scratch calculations which prepared the Sect. 28 of Volumetto IV, to which we refer the reader for notations and further explanations of the arguments treated by the author. However, a further generalization is present here with respect to what considered after Eq. (4.499) of Volumetto IV. ψ =

2 |I|2

|I|2

I ψ + ψ − 0 W Q2 Q4 Q2 φW

IL dW − 4 Q W − W

1 2 2

/Q + π 2 Q2

ψW dW W − W

2 IL φW Q4 e−2πi( −)t/h dW +I ψ0 ( 2 /Q2 + π 2 Q2 )(W − W ) e−2πi( −)t/h ψW |I|2 dW + 2 Q ( 2 /Q2 + π 2 Q2 )(W − W ) e−2πi( −)t/h dW ψW dW 2 −|I| ( 2 /Q2 + π 2 Q2 )(W − W ) W − W e−2πi( −)t/h φW IL dW + 2 Q ( 2 /Q2 + π 2 Q2 )(W − W ) e−2πi( −)t/h dW φW dW −IL ( 2 /Q2 + π 2 Q2 )(W − W ) W − W |L|2 IL + 2 ψW − 2 φW . Q Q +

ψ = A ψ0 + B ψW + CφW + b ψW dW + c φW dW e−2πiEt/h , Quantity A: Q2 1 = , ( 2 /Q2 + π 2 Q2 )( − ) ( + iπQ2 )( − iπQ2 )( − ) 1 2 2 R1 = e2πit/h e−2π Q t/h , 2πi( + iπQ2 )

333

THE THEORY OF SCATTERING

1 R2 = 2 2 ,

/Q + π 2 Q2 1 1 2 2 e2πit −2πi R1 + R2 = 2 2 e−2π Q t/h 2

/Q + π 2 Q2

× − 2 + iπ − iπ , Q 1

I 2πit/h −t/2T A= 2 2 e 1 − e

/Q + π 2 Q2 Q2 −Iπi 1 − e(2πi/h)t e−t/2T , A=

I 2πit/h −t/2T 1 − e . e

+ iπQ2

Quantity B: π 2 |I|2 |L|2 1

2 |I|2 1 + + = 2 |I|2 + |L|2 = 1, 2 2 2 2 4 2 2 2 2 2

/Q + π Q Q

/Q + π Q Q Q

B = 1. Quantity C: π 2 IL IL

2 IL 1 + − 2 = 0, 2 2 2 2 4 2 2 2 2

/Q + π Q Q

/Q + π Q Q C = 0.

334

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

Quantity b:

17

e−2πi( −)t/h d ( 2 /Q2 + π 2 Q2 )( − )( − ) Q2 e−2πi( −)t/h d , = ( + iπQ2 )( − iπQ2 )( − )( − ) 1 1 −2πi R0 + R1 + R2 , 2 2 1 , −2πi( + iπQ2 )( + ipiQ2 ) 1 1 R1 = 2 2 ,

/Q + π 2 Q2 − −1 1 . R2 = e2πi(− t/h 2 2 2 2

/Q + π Q −

R0 = e2πit/h e−2π

2 Q2 t/h

1 1 1 1 −2πi R0 + R1 + R2 = 2 2 2 2 2 2 2 2

/Q + π Q /Q + π 2 Q2

2πit/h −t/2T × e e − iπ − iπ Q2 Q2

2 2 2πi (− )t/h πi πie

, − + π 2 Q2 + + π 2 Q2 Q2

− Q2

−

|I|2 1 1 e2πi(− )t/h

|I|2 1 b = − 2 + Q − 2 /Q2 + π 2 Q2 Q2 − 2 /Q2 + π 2 Q2 2 |I| 1 + 2 2 2 2 2 2

/Q + π Q /Q + π 2 Q2

2πit/h −t/2T × e e − iπ − iπ Q2 Q2

2 2 2πi (− )t/h πi

2 2 2 2 πie − +π Q + +π Q Q2

− Q2

− =

17 @

|I|2 1 1 |I|2 − e2πi (− )t/h 2 2

+ iπQ −

+ iπQ − 2 2πi t/h−t/2T |I| e , + ( + iπQ2 )( + iπQ2 )

Cf. the ﬁgure above.

335

THE THEORY OF SCATTERING

b =

|I|2 −1 + e2πi t/h−t/2T ( + iπQ2 )( + iπQ2 )

+ iπQ2 2πi (− )t/h + 1−e ,

−

|I|2 −e2πi (− )t/h + e2πi t/h−t/2T ( + iπQ2 )( + iπQ2 )

b =

+ iπQ2 2πi (− )t/h + 1−e .

−

Quantity c:

1

IL 1 e2πi(− )t/h

IL 1 + Q2 − 2 /Q2 + π 2 Q2 Q2 − 2 /Q2 + π 2 Q2 IL 1 + 2 2 2 2 2 2

/Q + π Q /Q + π 2 Q2

× e2πit/h e−t/2T − iπ − iπ Q2 Q2

2 2 2πi (− )t/h

πi πie

+ π 2 Q2 + + π 2 Q2 − Q2

− Q2

−

c = −

=

1 1 IL IL − e2πi (− )t/h

+ iπQ2 − + iπQ2 − IL e2πi t/h−t/2T + . ( + iπQ2 )( + iπQ2 )

c =

IL −1 + e2πi t/h−t/2T ( + iπQ2 )( + iπQ2 )

+ iπQ2 2πi (− )t/h + 1−e .

−

336

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

I −2πiEt/h −2πi(E0 −k)t/h −t/2T ψ = e−2πi Et/h ψW + − e e e ψ0

+ iπQ2 I IψW + LφW −2πi Et/h 2πi t/h−t/2T 1 − e d − e

+ iπQ2

+ iπQ2 IψW + LφW −2πi Et/h 1 2πi(− )t/h 1 − e +I e d ,

+ iπQ2

− ψ = e−2πi Et/h ψW + a e−2πi E0 t/h ψ0 + bW IψW + LφW dW · e−2πi E t/h , Hψ = E e−2πi Et/h ψW + IW e−2πi E0 t/h ψ0 + a E0 e−2πi E0 t/h ψ0 −2πi E0 t/h + ae I W ψW dW + a e−2πi E0 t/h LW φW dW + E bW IψW + LφW e−2πi E t/h dW 2 +Q bW dW · ψ0 e−2πi E t/h , IW = I:

2πi −2πi W t/h 2 −2πi W t/h e bW e I +Q dW , a˙ = − h 2πi −2πi W t/h a. bW = − e h ψ = e−2πi Et/h ψW I −2πiEt/h −2πi(E0 −k)t/h −t/2T e ψ0 − e e

+ iπQ2 IψW + LφW −2πi E t/h I − e

+ iπQ2

+ iπQ2 2πi t/h−t/2T × 1−e d +

IψW + LφW −2πi Et/h I + e

+ iπQ2

− × 1 − e2πi(− )t/h d .

337

THE THEORY OF SCATTERING

ψ = ψ + ψ , I e−2πiEt/h ψ0 ψ = e−2πi Et/h ψW +

+ iπQ2 IψW + LφW −2πi Et/h I 2πi ( −)t/h e − 1 − e d ,

+ iπQ2

− I e−2πi(E0 −k)t/h e−t/2T ψ0 ψ = −

+ iπQ2 IψW + LφW −2πi E t/h I 2πi t/h−t/2T − e 1 − e d .

+ iπQ2

+ iπQ2

Appendix: Transforming a diﬀerential equation δ1

1 u + 0 + u = 0, u + δ0 + r r

χ = rk u. u = r−k u, k χ − , u = u χ r k 2 χ2 χ χ k − − 2 + 2 +u u = u χ r χ χ r k χ k(k + 1) χ −2 + . = u χ r χ r2

k χ k(k + 1)

1 χ k δ1 χ kδ1 χ −2 + − δ + − 2 + 0 + = 0, + δ 0 0 χ r χ r2 χ r r χ r r δ1

1 − kδ0 k(k + 1) − kδ1 k χ + δ0 + −2 χ + 0 + + χ = 0. r r r r2 k=

δ1 ; 2

δ1 = 2k,

1 − kδ0 k(k − 1) χ = 0. − χ + δ0 χ + 0 + r r2

7 NUCLEAR PHYSICS

7.1.

WAVE EQUATION FOR THE NEUTRON

Denoting with ε the electric or diamagnetic susceptivity, the Lagrangian describing the electromagnetic ﬁeld is: 1 − ε(E 2 − H 2 ). 2 Using Dirac coordinates,

7.2.

ε W 2 2 + ρ1 σ · p + ρ3 mc + ρ3 (E − H ) ψ = 0. c 2c

RADIOACTIVITY

In the following table the author referred to some radioactive nuclides grouped by their atomic number Z. The number following the (old) name of the given isotope is its mass number. Probably this table was aimed at cataloguing the isotopes existing at the time of Majorana according to Z for further studies. [1 ] Z = 90

1@

U X1 UY Io Rd Ac Th Rd Th

234 231 230 227 232 228

Z = 89

Ac Ms Th2

In the original manuscript, the unidentiﬁed Ref. 9.28 appears here.

339

227 228

340

7.3.

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS Z = 88

Ra Ac X Ms Th1 Th X

Z = 84

Ra A Ra C Ra F Ac A Ac C Th A Th C

Z = 82

Ra B Ra D Ac B Th B

226 223 228 224

Z = 86

Rn An Tn

222 219 220

218 214 210 215 211 216 212

Z = 83

Ra C Ra E Ac C Th C

214 210 211 212

Z = 81

Ra C Ac C Th C

214 210 211 212

210 207 208

NUCLEAR POTENTIAL

In the following pages, the author considered the problem of ﬁnding the nucleon potential inside a given nucleus. In particular, he focused on the interaction between neutrons and protons, assuming that the interaction between protons is approximatively given only by the usual electrostatic repulsion, while that between neutrons is negligible. Many of the results discussed apply to a general nucleus of atomic number Z and mass number A, although particular attention was here given to α particles. What reported in the following is, at the same time, a preliminary study and a generalization of what published by Majorana in Z. Phys. 82 (1933) 137, or in La Ricerca Scientiﬁca 4 (1933) 559, on the nuclear exchange forces.

7.3.1

Mean Nucleon Potential

Some expressions for the matrix elements of the interaction potential between neutrons and protons in a given nucleus were deﬁned in the following. The author considered the case of a nucleus composed of a number a of protons (whose wavefunctions, depending on the coordinates q, were denoted with ψ) and A of neutrons (whose wavefunctions, depending on the coordinates Q, were denoted with ϕ). The state function of the nucleons was written as a Slater determinant. With reference to the published papers quoted above, the given form of the matrix elements of the interaction potential (also considered in the

341

NUCLEAR PHYSICS

following subsections) in terms of Dirac δ-functions corresponds to the hypothesis that the mean energy per nucleon cannot exceed a certain limit, whatever large be the nuclear density. It is also assumed that the density of neutrons is larger than that of protons. In the second part, it seems that the author considered the particular case of a nucleus of helium (with only two protons and two neutrons), probably thought as composed of two deuterium nuclei (denoted, in the original manuscript, as d and D, respectively). However, it is also possible that the author was initially studying the scattering of two nuclei with mass numbers a and A, respectively, and that only later on he turned to the particular case cited above. The interaction potential between the nucleon s in the ﬁrst nucleus and the nucleon S in the second one was denoted with V sS . ψ1 , ψ2 , . . . ψa ; q1 , q2 , . . . qa ; ϕ1 , ϕ2 , . . . ϕA ; Q1 , Q2 , . . . QA (A ≥ a).

ψ (q ) 1 1 1 ψ = √ ... a! ψ (q ) a 1 ϕ1 (Q1 ) 1 ϕ = √ . . . A! ϕ (Q ) 1 A

. . . ψ1 (qa ) , . . . ψa (qa ) . . . ϕ1 (QA ) . . . . ϕA (AA )

q , Q |V |q , Q = δ(q − Q ) δ(Q − q ) f |q − Q |. qs , Qs |V sS |qs , QS = δ(qs − QS ) δ(QS − qs ) f |qs − QS |; VS =

s

V

sS

=

f |qs − QS | = V

s

V sS ,

Vs =

V sS ;

S

ψ(qs )ϕ(Qs )δ(QS − qs ) × δ(qs − QS )ψ(qs )ϕ(QS )dqs dQS dqs dQS ;

f |qs − qs | ψ(qs )ψ(qs )ϕ(qs )ϕ(qs )dqs dqs ; = f |qs − qs | ψ(qs )ψ(qs ) ϕS (qs )ϕS (qs ) dqs dqs . S

342

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

q |V s |q = f |q − q |

ϕS (q )ϕS (q ).

S

7.3.2

Computation Of The Interaction Potential between Nucleons

The following calculations seems to be aimed at obtaining an expression for the interaction potential between nucleons (the primed quantities probably refer to neutrons, while the unprimed ones to protons); see also the beginning of the next subsection. dq dq dq = dq . |q − q |2 (q − q )2 q dq |q − q |2 s s2 2s ds

= = = = = =

(ρ, ϑ, ϕ), ρ2 sin ϑ dϑ dϕ dρ, |q 2 | + ρ2 − 2|q|ρ cos ϑ, |q − q |, |ϕ2 | + |ρ2 | − 2|q|ρ cos ϑ, 2|q|ρ sin ϑ dϑ.

R > q:

R + q dq R2 − q 2 log . = π 2R + |q − q |2 q R −q

R > q: |q| − ρ ≤ s ≤ |q| + ρ, sρ sρ dq = ds dϕ dρ = 2π ds dρ. |q| |q|

2π dq = |q − q |2 |q|

ρdρ

ρ dρ log(q + ρ) = =

ds 2π + ... = s |q|

1 2 ρ log(q − ρ) − 2 1 2 ρ log(q + ρ) − 2

ρdρ log

|q| + ρ + .... |q| − ρ

ρ2 1 dρ 2 q+ρ 1 1 (ρ − q)2 − q 2 log(q + ρ). 4 2

343

NUCLEAR PHYSICS

dq |q − q |2

2π q

=

q

0

q + ρ 2π + ρ dρ log q−ρ q

R

ρ dρ log q

ρ+q ρ−q

1 2π 1 2 R log(R + q) − (R − q)2 q 2 4 1 1 1 − q 2 log(R + q) + q 2 + q 2 log q 2 4 2 1 1 2 R log(R − q) − (R + q)2 − 2 4 1 2 1 1 2 2 − q log(R − q) + q + q log q . 2 4 2

=

dq = dx dy dz :

F (q) =

dq q
|q − q|2

=

⎧

R2 − q 3 R + q ⎪ ⎪ log , π 2R + ⎪ ⎪ ⎨ q R −q

q < R ;

⎪ ⎪ q + R q 2 − R2 ⎪ ⎪ ⎩π 2R − log , q q − R

q > R .

1 F (0) = 4R , π

1 F (R ) = 2R . π

q > R :

1 R3 1 R5 R q + R + =2 + + ... ; log q − R q 3 q3 5 q5

R 1 R3 2 R4 2 R2 q 2 − R2 ·2 + + . . . = 2R 1 − − − ... . q q 3 q3 3 q2 15 q 4 F (q) + F

(q >

R )

(q <

R )

R2 q

= 4πR .

1 R4 1 R2 1 + + ... ; + 3 15 q 2 35 q 4

:

4πR3 F (q) = q2

:

1 q4 1 q6 1 q2 F (q) = 4πR 1 − − − − ... . 3 R2 15 R4 35 R6

344

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

q < R < R , t < R < R :

R

F (q)dq = 4π q
2

R + t 2t R + (tR − t ) log R −t 2

0

2

3

dt.

F (q)dq = 4πt2 F (t)dt.

R2 − t2 R + t log . F (t) = π 2R + t R −t

t log(t + R )dt = =

1 2 t log(t + R ) − 2 1 2 t log(t + R ) − 2

t2 1 dt 2 t + R 1 1 (t − R )2 − R2 log(t + R ). 4 2

1 1 t4 log(t + R ) − dt t log(t + R )dt = 4 4 t + R 1 1 1 1 4 t log(t + R ) − t4 + R t3 − R2 t2 = 4 16 12 8 1 3 1 4 + R t − R log(t + R ). 4 4 R < R : 1 R + R 2 2 3 F (q)dq = 4π R R − (R2 − R2 )R2 log + RR3 3 2 R −R 1 R + R 1 3 1 3 − RR − R R − R4 log 4 R −R 6 2 R + R 1 4 + R log 4 R −R 1 1 2 R + R 2 1 3 3 2 2 R R + RR − (R − R ) log . = 4π 2 2 4 R −R

3

R > R:

q
q
dq dq |q − q|2

R + R 3 3 2 2 2 . = π 2R R + 2RR − (R − R ) log R −R 2

345

NUCLEAR PHYSICS

7.3.3

Nucleon Density

In the following the author worked out some expressions for the nucleon density, starting from the potential and kinetic energy densities V and T of a system of nucleons (the proton and neutron density are denoted Y with ρ(= Z 1 ψi ψ 1 ) and ρ (= 1 ϕi ϕ1 ), respectively). Notice that the potential energy density V is given, up to a factor −π 2 , by the last formula in the previous subsection, with the replacements R, R → ρ, ρ . Potential energy per unit volume: ρ 3 + ρ 3 1

13

−V = 2ρρ

23

1 3

+ 2ρ ρ − (ρ

2 3

− ρ ) log 2

1

|ρ 3 − ρ 3 | 1

1

.

Kinetic energy per unit volume: 5 3 5 T = (ρ 3 + ρ 3 ). 5

ρ = ρ :

2

2500 , 81 6 5 1250 ρ3 = . 5 81 4

−V

= 4ρ 3 =

T

=

1 T = − V. 2 4 6 5 ρ3 = ρ3 , 5

1

ρ3 = V ρ T 2ρ

−

−

∂V ∂ρ ∂T ∂ρ

10 5 = , 6 3 = =

ρ=

125 ; 27

20 , 3 5 . 3

8 1 40 ∂V = ρ3 = , ∂ρ 3 9 2 25 ∂T = ρ3 = ; ∂ρ 9

= − =

2 @ The numerical values 2500/81 and 1250/81 seem to have been written by the author after he deduced the numerical value for ρ (see below).

346

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

−

∂T ∂V + ∂ρ ∂ρ

5 = . 3

ρ = ρ (ρ < ρ ): ∂V − ∂ρ

−

∂V ∂ρ

∂T ∂ρ ∂T ∂ρ

1 1 2 ρ 3 + ρ 3 2 −2 4 −1 2 = 2ρ + ρ 3 ρ + ρ 3 ρ 3 − ρ 3 log 1 1 3 3 ρ 3 − ρ− 3 1 1 2 1 1 1 2 2 − ρ− 3 ρ 3 − ρ 3 (ρ 3 + ρ 3 ) · log ρ 3 + ρ 3 , 3 1 1 1 2 ρ 3 + ρ 3 2 − 2 4 − 1 2 = 2ρ 3 + ρ 3 ρ − ρ 3 ρ 3 − ρ 3 log 1 1 3 3 ρ 3 − ρ− 3 2 1 1 2 2 1 1 1 ρ 3 + ρ 3 · log ρ 3 + ρ 3 ; + ρ− 3 ρ 3 − ρ 3 3

13

2

= ρ3 , = ρ 3 . 2

1 T =− V: 2 5 1 1 2 1 2 ρ 3 + ρ 3 3 5 (ρ 3 + ρ 3 ) = ρρ 3 + ρ ρ 3 − (ρ 3 − ρ 3 )2 log 1 1 . 5 2 ρ 3 − ρ 3 1

1

ρ = kρ: 1

5 1 4 1 4 2 k3 + 1 3 5 , ρ 3 (1 + k 3 ) = (k + k 3 )ρ 3 − ρ 3 (k 3 − 1)2 log 1 5 2 k3 − 1 1 5 1 1 2 k3 + 1 3 1 2 . ρ 3 (1 + k 3 ) = (k + k 3 ) − (k 3 − 1) log 1 5 2 k3 − 1

[3 ] ρ=

⎫3 ⎧ 1 1 2 ⎪ k 3 + k − 12 (k 3 − 1)2 log k 31 +1 ⎪ ⎬ ⎨ 125 27 ⎪ ⎩

k 3 −1

1+k

5 3

2

⎪ ⎭

.

3 @ In the original manuscript, the power 2 of the factor (k 3 − 1) in the following equation is missing.

347

NUCLEAR PHYSICS

7.3.4

Nucleon Interaction I

Explicit expressions for a particular form of the interaction potential between Z protons and Y neutrons are worked out. See also the paper published by Majorana in Z. Phys. 82 (1933) 137, or in La Ricerca Scientiﬁca 4 (1933) 559. Denote with q, Q the center-of-mass coordinates. q Q |V |q u Qu = −δ(q − Q ) δ(Q − q )

λe2 . r

N =Z +Y. 1 ψ1 (q1 ) . . . ψZ/2 (qZ/2 )ψ1 (qZ/2+1 ) . . . ψZ/2 (qZ ), (Z/2)! 1 ϕ1 (Q1 ) . . . ϕY /2 (qY /2 )ϕ1 (QY /2+1 ) . . . ϕY /2 (QY ). (Y /2)! U

= − +

Y Z i=1 =1 Z

ψ i (q )ψi (q )ϕ (q )ϕ (q )

ψ i (q 2 )ψi (q )ψ k (q )ψk (q )

i
λe2 dq dq |q − q |

e2 dq dq |q − q |

+ negligible exchange terms. [4 ] ρ=

Z 1

ψi ψ˜i ,

ρ =

Y

ϕi ϕ˜i .

1

U

λe2 q |ρ|q q |ρ |q dq dq |q − q | 1 e2 q |ρ|q q |ρ|q + dq dq . 2 |q − q |

= −

λe2 q |ρ |q |q − q | + δ(q − q ) q |ρ|q

q |VP |q = −

Q |VN |Q = − 4@

e2 dq , |q − q |

λe2 Q |ρ|Q . |Q − Q |

Notice that here the author refers to the “ordinary” exchange energy depending on the electrostatic interaction among protons.

348 [5 ]

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

1 Aq− v2 ,q+ v2 = e−2πi p·v /h A(p, q) dp, h3 A(p, q) = e2πi p·v /h Aq− v2 ,q+ v2 dv.

In Classical Mechanics: 6 ⎧ p < P, ⎨ 2, ρ= ⎩ 0, p > P;

ρ =

⎧ ⎨ 2,

p P .

0,

1 1 λe2 e2 VP (p, q) = 3 , p ) dq dp − ρ (Q, p ) dp , ρ(q h |q − q| h π|p − p |2 1 λe2 ρ(q, p) dp. VN = − h π|p − p |2 p
VP (p, q) =

P = P (q), 4 dp = πP 3 , 3

P3 8π e2 dq 3 |q − q| h3

2λe2 − h VP (p, q) =

P 2 − p2 P + p 2P + log p P −p

p P .

In the following, the author deals with a semiclassical approach, which is valid when the number of particles is suﬃciently large. The quantities VP and VN considered below are, then, the classical functions corresponding to the quantum matrix elements discussed before. See E. Majorana, Z. Phys. 82 (1933) 137 or La Ricerca Scientiﬁca 4 (1933) 559. 6 @ In the following, the author postulates for simplicity that the one-particle states are either empty or doubly occupied with opposite spins. Moreover, by assuming that at a given position q (or Q) the protons (or neutrons) occupy the states with minimum kinetic energy, it follows that a maximum value P for the proton momentum (and, similarly, P for neutrons) does exist. See the papers quoted in the previous footnote.

349

NUCLEAR PHYSICS

[7 ] 2λe2 VN (p, q) = − h 2λe2 VN (p, q) = − h

P 2 − p2 P +p 2P + log P P −p p2 − P 2 p+P 2P − log P p−P

C(q) =

VP (P, q) =

,

p < P;

,

p > P.

8π P 3 (q ) e2 dq . 3 |q − q| h3

⎧

2λe2 P 2 − P 2 P + P ⎪ ⎪ C− 2P + log , ⎪ ⎪ ⎨ h P P −P

P P ;

VN (P , q) =

⎧

2λe2 P + P P 2 − P 2 ⎪ ⎪ − log 2P − , ⎪ ⎪ ⎨ h P P −P

P P .

T =

P2 . 2M

7@

The original manuscript presents here an insert dealing with the following Fourier transforms: Z Z ϕ(ξ) = e−2πiξx f (x)dx, f (x) = e2πiξx ϕ(ξ)dξ, ϕ (ξ) =

Z

e−2πiξx f (x)dx,

f (x) =

Z

e2πiξx ϕ (ξ)dξ,

where, in particular: ϕ (ξ) =

1 ϕ(ξ), ξ

f (x) =

Z

1 f (x1 )dx1 . π(x − x1 )2

350

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

[8 ]

VP (P, q) +

P2 = −AP , 2M

VN (P , q) +

P 2 = −AN . 2M

By considering a statistical method:

T P (q) =

9

3P 2 , 10M

2λe2 3 3 P 3 3 (P 2 − P 2 )2 P + P V P (q) = C − − log P + , h 2 2 P2 4 P3 |P − P | T N (q) =

3P 2 , 10M

2λe2 3 3 (P 2 − P 2 )2 P + P 3 P3 ; V N (q) = − − log P+ h 2 2 P 2 4 P 3 |P − P |

P 3 (V P − C) = P 3 V N . Limiting condition: −

P3 P 3 V (Q) + T (Q) + T (Q) P P N P 3 + P 3 P 3 + P 3 =

P 3 P3 A + AN . P P 3 + P 3 P 3 + P 3

8 @ In the following, the author probably denotes with A P (or AN ) the energy associated with the proton (or neutron) exchange interaction. 9 @ An application of the theory of nuclear forces introduced above to heavy nuclei, composed of a large number of nucleons, is now apparently investigated, so that statistical methods may apply.

351

NUCLEAR PHYSICS

7.3.4.1 Zeroth approximation. C = 0; P = constant, P = constant. k = P /P : 2λe2 k+1 2 P 2k + (k − 1) log , VP (P, q) = − h |k − 1| 2λe2 k2 − 1 k+1 P 2− log . VN (P , q) = − h k |k − 1|

TP VN

TN VP

3P 2 , 10M 3 3 3 3 2 2λe2 k+1 2 P k + k − (k − 1) log , = − h 2 2 4 |k − 1| =

3k 2 P 2 , 10M 3 2λe2 3 3 (k 2 − 1)2 k+1 P + = − − log . h 2 2k 3 4 k3 |k − 1| =

Particular case: k = 1. VP (P, q) = VN (P , q) = −4 V N (q) = −

−

λe2 P, h

6λe2 P = V P (q), h

T = TN =

P2 ; 2M

3P 2 . 10M

3P 2 4λe2 P2 3λe2 P+ =− P+ , h 10M h 2M P2 λe2 P = , h 5M P = 5M

VP (P, q) = −20M V = −30M

λ2 e4 , h2

λ2 e4 , h2

λe2 . h Tnuc = T =

25 λ2 e4 M 2 ; 2 h

15 λ2 e4 M 2 . 2 h

352

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

AP = AN =

15 λ2 M e4 . 2 h2

[10 ]

7.3.5

Nucleon Interaction II

Explicit expressions for another particular form of the interaction potential between Z protons and Y neutrons are worked out.

q , Q |V |q , Q = −δ(q − Q )δ(Q − q )A e−|q −q

|/ε

.

ψi ψ˜i ; for neutrons: ρ = Y1 ϕi ϕ˜i . e2 −|q −q |/ε q |ρ |q + δ(q − q ) q|ρ|q dq, q |VP |q = −A e |q − q |

For protons: ρ =

Z 1

q |VN |q = −A e−|q −q

|/ε

q |ρ|q .

In Classical Mechanics11 , assuming a degenerate gas of nucleons: ⎧ ⎧ p P . 0, p > P;

A e−|q −q

|/ε

= A e−v/ε = A e−(h/ε)(v/h) = A e−k

VP (p, q) =

1 h3

,

(k = h/ε).

−A VN (p, q) = −A

10 @

v/h

e2 ρ(q , p ) dq dp |q − q | 8πh/ε

(h2 /ε2 + 4π 2 |p − p |2 )2 8πh/ε (h2 /ε2

+

4π 2 |p

−

p |2 )2

ρ (q, p ) dp , ρ(q, p ) dp .

In the original manuscript there appears also the following note:

15 M e4 = 9500 V 2 h2 (V stands for eV), where the nucleon mass value M 938 MeV had been used. 11 @ See footnote 6.

353

NUCLEAR PHYSICS

Let us set: P0 =

h , 2πε

h = 2πP0 . ε

[12 ] 1 e2 VP (p, q) = ρ(q , p ) dq dp h3 |q − q | A P0 − 2 ρ (q, p ) dp , π (P02 + |p − p |2 )2 A P0 ρ (q, p ) dp . VN (p, q) = − 2 2 π (P0 + |p − p |2 )2 P = P (q), P = P (q),

p
dp = 4πP 3 /3.

For a degenerate gas of nucleons: 8π 1 VP (p, q) = 3 h3 VN (p, q) = −

2A π2

2A e2 P 3 (q ) dq − 2 |q − q | π

p
VP (p, q) =

2 p

P0 dp , 2 2 + |p − p | )

P0 dp . (P02 + |p − p |2 )2

8π e2 P 3 (q ) 2A dq − 3 |q − q| h3 π

arctan

P + p P0

P 2 + (P + p)2 P − p 1 P0 log 02 + arctan − P0 2 p P0 + (P − p)2 2A VN (p, q) = − π

P −p P +p + arctan arctan P0 P0

1 P0 P 2 + (P + p)2 log 02 − 2 p P0 + (P − p)2 12 @

.

In the original manuscript, the unidentiﬁed Ref. 5.25 appears here.

,

354

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

[13 ]

VP (P, q) +

P2 = −AP , 2M

VN (P , q) +

VP (P, q) =

P2 = −AN . 2M

8π e2 P 3 (q ) 2A dq − 3 |q − q| h3 π

arctan

P + P P0

P − P 1 P0 P 2 + (P + P )2 log 02 + arctan − P0 2 p P0 + (P − P )2 2A VN (P , q) = − π

arctan

P − P P + P + arctan P0 P0

1 P0 P02 + (P + P )2 − log 2 P P02 + (P − P )2

.

Limiting conditions: − P 3 V P (Q) + P 3 T P (Q) + P 3 T N (Q) = P 3 AP + P 3 AN ;

P 3 V P = P 3 VN + P 3 C, P 3 V P − C = P 3 V N . C=C=

13 @

8π e2 P 3 (q ) dq . 3 |q − q | h3

In the original manuscript, the unidentiﬁed Ref. 11.59 appears here.

,

355

NUCLEAR PHYSICS

3

P VN

2A = − π

P + P P0 P P + (P 3 + P 3 ) arctan P0

−(P 3 − P 3 ) arctan

P − P P0

P 2 + (P + P )2 3(P 2 + P 2 ) + P02 −P0 log 02 4 P0 + (P − P )2

= P 3 (V p − C).

7.3.5.1 Evaluation of some integrals. For p < P : [14 ]

P0 dp = + |p − p |2 )2 p+s P s+p p 2πP0 t dt t dt s ds + s ds = 2 2 2 2 2 2 P p 0 p−s (P0 + t ) s−p (P0 + t ) p 1 1 1 2πP0 s ds · − = p 2 P02 + (p − s)2 P02 + (p + s)2 0 P 1 1 1 s ds · − + 2 P02 + (s − p)2 P02 + (s + p)2 p 1 1 1 2πP0 P s ds · − = p 2 P02 + (p − s)2 P02 + (p + s)2 0 2 p

(the last expression holds also for p > P ).

s ds 2 P0 + (p − s)2

= =

14 @

(p − s) d(p − s) p ds + 2 2 2 P0 + (p − s) P0 + (p − s)2 p 1 s−p arctan , log P02 + (p − s)2 + 2 P0 P0

In the original manuscript, the unidentiﬁed Ref. 2.50 appears here.

356

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

s ds + (p + s)2

=

s ds 2 P0 + (p − s)2

=

s ds 2 P0 + (p + s)2

=

P02

P

0

P

0

p

p 1 p+s arctan ; log P02 + (p + s)2 − 2 P0 P0 1 P 2 + (P − p)2 log 0 2 2 P0 + p2

P − p p p arctan , + + arctan P0 P0 P0 1 P 2 + (P + p)2 log 0 2 2 P0 + p2

P − p p p arctan . − − arctan P0 P0 P0

1 P0 P0 P02 + (P + p)2 log dp = π − 2 p (P02 + (p − p )2 )2 P02 + (P − p)2 P + p P − p + arctan + arctan P0 P0

.

[15 ]

P P + p P0 3 2 dp = 4π p arctan dp 2 2 2 2 P0 p

dp

P + p dp P0 P + p 1 1 3 p3 = p arctan − P0 dp 3 P0 3 P02 + (P + p)2 P + p 1 = p3 arctan 3 P0 1 (p + P )3 − 3P (p + P )2 + 3P 2 (p + P ) + P 3 − P0 dp 3 P02 + (p + P )2 p2 arctan

15 @

In the original manuscript, the unidentiﬁed Ref. 3.43 appears here.

357

NUCLEAR PHYSICS

1 3 p + P 1 = p arctan − P0 (p + P ) dp − 3P dp 3 P0 3 (3P 2 − P02 )(p + P ) P (P 2 − 3P02 ) + dp − dp P02 + (p + P )2 P02 + (p + P )2 p + P 1 1 1 3 − P0 p2 − 2P p = p arctan 3 P0 3 2 +

P (P 2 − 3P02 ) p + P 3P 2 − P02 arctan log P02 + (p + P )2 − . 2 P0 P0

1 P − p 1 3 P − p 1 = p arctan + P0 p2 + 2P p p arctan P0 3 P0 3 2 2

2 P (P 3 − 3P02 )2 P − p 3P 2 − P02 2 log P0 + (p − P ) − arctan + . 2 P0 P0 p log

P02 + (p + P )2 dp P02 + (p − P )2

P 2 + (p + P )2 1 − = p2 log 02 2 P0 + (p − P )2 p2 (p − P ) dp + P02 + (p − P )2

p2 (p + P ) dp P02 + (p + P )2

P 2 + (p + P )2 1 2 p log 02 2 P0 + (p − P )2 (p + P )3 − 2P (p + P )2 + P 2 (p + P ) − dp P02 + (p + P )2 (p − P )3 + 2P (p − P )2 + P 2 (p − P ) dp + P02 + (p − P )2 P02 + (p + P )2 1 2 − (p + P ) dp + 2P dp = p log 2 2 P0 + (p − P )2 2P P02 dp (P 2 − P02 )(p + P ) dp + − P02 + (p + P )2 P12 + (p + P )2 (P 2 − P02 )(p − P ) + (p − P ) dp + 2P dp + dp P02 + (p − P )2 =

358

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

−

P02

2P P02 dp + (p − P )2

P02 + (p + P )2 P 2 + (p + P )2 P 2 − P02 1 2 p log 02 log + 2P p − 2 2 P0 + (p − P )2 P02 + (p − P )2

=

−2P P0 arctan

P + p P − p + 2P P0 arctan . P0 P0

P + p P − p dp + p2 arctan dp p arctan P0 P0 1 P 2 + (p + P )2 dp − P0 p log 02 2 P0 + (p − P )2 2

1 P + p 1 3 P − p 1 + (p − P 3 ) arctan = P0 P p + (p3 + P 3 ) arctan 3 3 P0 3 P0 −

3P0 P 2 + 3P0 P 2 + P03 P 2 + (p + P )2 . log 02 12 P0 + (p − P )2

p
p

P0 dp dp (P 2 + |p − p |2 )2

4 2 P + P = π P0 P P + (P 3 + P 3 ) arctan 3 P0 − (P 3 − P 3 ) arctan

P − P P0

P 2 + (P + P )2 3P0 P 2 + P03 + 3P0 P 2 log 02 − 4 P0 + (P − P )2

7.3.5.2 Zeroth approximation. C = 0; P 3 V P = P 3 V N ; P = constant, P = constant. k = P /P , t = Po /P :

.

359

NUCLEAR PHYSICS

TP (P, q) = TN (P , q) =

P2 , 2M k2 P 2 P 2 = . 2M 2M

2A 1+k k−1 VP (P, q) = − arctan + arctan π t t 2 2 (k + 1) + t t − log , 2 (k − 1)2 + t2 2A 1+k k−1 arctan − arctan VN (P , q) = − π t t 2 2 t (k + 1) + t − log . 2k (k − 1)2 + t2

3

P VP

2A 3 1+k = − P kt + (1 + k 3 ) arctan π t k−1 −(k 3 − 1) arctan t (k + 1)2 + t2 3(1 + k 2 ) + t2 log . −t 4 (k − 1)2 + t2

Particular case: k = 1, P = P . VP (P, q) = VN (P , q) = −

VP

2A =− π

2A π

arctan

t 4 + t2 2 − log t 2 t2

6 + t2 4 + t2 2 log t + 2 arctan − t t 4 t2

.

2A 2 6 + t2 4 + t2 3P 2 t + 2 arctan − t log + − π t 4 t2 5M 2 2 t 4+t P 2A 2 . = − arctan − log π t 2 t2 M

;

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E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

[16 ] 2P 2 5M P2 2M

= =

2A 2 + t2 4 + t2 t log −t , π 4 t2 5A 2 + t2 4 + t2 −t . t log 2π 4 t2

P02 P0 x P2 = Ay; = Ax; =t= ; 2M 2M P y 5 x + 4y x x + 2y y= log −1 . 2π y 4y x 5 y= 2π

2 + t2 4 + t2 t −t . log 4 t2

3 T = T (P ); 5 [17 ] t=

t=

16 @

P0 =

h . 2πε

x/y x = T (P0 )/A y = T (P )/A −V (P )/A 0.3 0.0213 0.237 0.540 0.4 0.0387 0.242 0.459 0.5 0.0587 0.235 0.394 0.6 0.0806 0.224 0.339 0.7 0.1039 0.212 0.293 0.8 0.1261 0.197 0.253

x/y 0.3 0.4 0.5 0.6 0.7 0.8

T /A −V /A [−V (P ) − T (P )]/A [−V /2 − T ]/A 0.142 0.892 0.303 0.304 0.145 0.724 0.217 0.217 0.141 0.600 0.159 0.159 0.134 0.498 0.115 0.115 0.127 0.417 0.081 0.0815 0.118 0.349 0.056 0.0565

In the original manuscript, the unidentiﬁed Ref. 1.03 appears here. The numerical values in the following tables have been obtained by using the appropriate equations above, with a given value of the parameter t. In particular, x has been calculated from x = t2 y. Note that, sometimes, the last digit in the numerical values appearing in the table is slightly erroneous. 17 @

361

NUCLEAR PHYSICS

For t = 0.6: A = 80 · 106 V; T (P0 ) = 6.5 · 106 V; 2πε = 11 · 10−13 ,

ε = 1.75 · 10−13 ;

−V (P ) = 27 · 106 V; T (P ) = 18 · 106 V; −V (P ) − T (P ) = 9 · 106 V; −V = 40 · 106 V; T = 11 · 106 V; −V /2 − T = 9 · 106 V. 2A V (0, q) = π t 0.3 0.4 0.5 0.6 0.7 0.8

1 2t 2 arctan − . t 1 + t2 −V (0, q)/A 1.280 1.076 0.900 0.750 0.624 0.519

——————– General case: k > 1, k = P /P , t = P0 /P . 2A 1+k VP =k VN =− kt + (1 + k 3 ) arctan π t 2 2 k−1 3(1 + k ) + t (k + 1)2 + t2 3 −(k − 1) arctan −t log . t 4 (k − 1)2 + t2 3

AP = −VP (P ) − TP (P ) 1+k k−1 t (k + 1)2 + t2 2A P2 arctan + arctan − log , − = π t t 2 (k − 1)2 + t2 2M

362

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

AN = −VN (P ) − TN (P ) 1+k k−1 t (k + 1)2 + t2 2A k2 P 2 arctan − arctan − log . = − 2 2 π t t 2k (k − 1) + t 2M 3P 2 ; 10M

TP =

TN =

3k 2 P 2 . 10M

−V P − T P − k 3 T N = AP + k 3 AN .

k5 P 2 3P 2 3k 5 P 2 P2 + − − 2M 2M 10M 10M 2A 1 + k 2 + t2 1 + k5 2 (k + 1)2 + t2 = − kt = log P , π 4 (k − 1)2 + t2 5M 5 A P2 = t 2M 1 + k5 π y= P0 = t= P

y=

5 t 1 + k5 π

1 P2 ; A 2M

x = y

1 + k 2 + t2 (k + 1)2 + t2 log −k . 4 (k − 1)2 + t2

T (P0 ) ; T (P )

x=

1 P02 ; A 2M T (P0 ) = t2 T (P ).

1 + k 2 + t2 (k + 1)2 + t2 log − k . 4 (k − 1)2 + t2

y = T (P )/A:

t = 0.5 0.6 0.7 0.8 0.9 1.0

k = 1 k = 21/19 k = 22/18 k = 23/17 0.235 0.204 0.157 0.109 0.225 0.196 0.154 0.111 0.211 0.187 0.149 0.109 0.195 0.174 0.142 0.106 0.179 0.162 0.133 0.101 0.165 0.194 0.124 0.096

363

NUCLEAR PHYSICS

k 2 y = T (P )/A: t = 0.5 0.6 0.7 0.8 0.9 1.0

7.3.6

k = 1 k = 21/19 k = 22/18 k = 23/17 0.236 0.249 0.234 0.199 0.225 0.240 0.231 0.202 0.211 0.228 0.223 0.200 0.195 0.213 0.212 0.194 0.179 0.198 0.199 0.185 0.165 0.182 0.186 0.175

Simple Nuclei I

In the following pages the author considered the nucleon interaction discussed in Sect. 7.3.4. h2 m = 2.9 · 10−12 = a0 , 2 2 4π M e M M 2π 2 M e4 = · 1 Rh = 25000 V, 2 h m

b0 = S = e2 b0

= 50000 V.

For deuterium 2 H: ψ0 = e−λx/2b0 ,

q = q1 − q2 ,

E0 = −

λ2 S. 2

For Z + Y = N > 2: ψ ∼ ψ1 (q1 )ψ2 (q2 ) . . . ψn (qn ), with q1 + q2 + . . . + qn = 0. Q=

1 (q1 + q2 + . . . + qn ). n

ψ = ψ(q1 − Q, q2 − Q, q3 − Q, . . . , qn − Q), q = q1 − Q,

q2 = q2 − Q,

...

q1 + q2 + . . . + qn = 0;

qn = qn − Q.

364

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

ψ = ψ(q1 , q2 , . . . , qn ).

pi =

h ∂ ; 2πi ∂qi

pi =

h ∂ . 2πi ∂qi

1 p1 = p1 − (p1 + p2 + . . . + pn ), n ... 1 pi = pi − (p1 + p2 + . . . + pn ); n pi = p1 −

p2i =

p2 i −

1 pi . n

2 2 1 2 2 1 2 pi + pi = pi . pi − n n n

T =

1 2M

p2 i −

1 . pi n

For an α particle: ψ ∼ e−s(r1 +r2 +r3 +r4 )/b0 , r1 = |q1 |,

p2 i

h2 ψ=− 2 4π

pi = (x1i , x2i , x3i ),

r2 = |q2 |,

r3 = |q3 |,

r4 = |q4 |.

2

2 s 2 s 2 s 2 s s − − − 4 2− ψ; r1 b0 r2 b0 r3 b0 r4 b0 b0

k

x1 s 4 xk2 xk3 xk4 ψ; = − + + + b0 2πi r1 r2 r3 r4 ! " # 2 q ·q h2 s2 i k − 2 2 4+2 pi ψ = 4π b0 ri rk i
2 2 2 2 2 s h + + + ψ. + b0 4π 2 r1 r2 r3 r4

pi ψ

365

NUCLEAR PHYSICS

Since n = 4:

18

! Hψ =

h2 − 2 8π M

−e

2

"

#

1 q i · q k s2 3 1 1 1 1 s 3− − + + + 2 ri rk b20 2 r1 r2 r3 r4 b0 i
λ λ λ 1 λ + + + + r13 r14 r23 r24 r12

,

where the indices 1,2 refer to the protons and 3,4 to the neutrons. E ∼ −4s2 S ∼ −s2 · 100000 V. Rough estimate:

5 6 h2 5 2 5 s∼e λ− ∼ e2 λ, b0 8π 2 M 2 8 2 2 5 8π M 5 ∼ λ, λ e2 b0 s∼ 2 12 h 6 25 2 E ∼ λ S ∼ −λ2 · 70000 V. 9 [19 ]

7.3.7

Simple Nuclei II

In the following notes the author considered the nucleon interaction discussed in Sect. 7.3.5. For deuterium 2 H (M = 1.65 · 10−24 , h2 /4π 2 M ): Hχ = Eχ,

M = M/2,

h2 /8π 2 M =

χ = ψr.

18 @ The following Hamiltonian was obtained by using the general expression for the kinetic energy T just reported above, specialized to the present case with 4 nucleons. 19 @ In the original manuscript there is also the following note:

2π 2 M e4 h2 1 = . 2 2 8π M b0 h2

366

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

∂ h2 −r/ε H=− +A e . 4π 2 M ∂r2

r −r/η . χ∼r 1+ e ξ

2 1 3 −2r/η r e r e dr + dr + 2 r4 e−2r/η dr ξ ξ

3 4 5 3 3η 3η η η η2 η + + 2 = 1+3 + 2 . 4 4ξ 4ξ 4 ξ ξ

2

χ dr = =

∂χ ∂r ∂2χ ∂r2

2 −2r/η

2 1 1 2 −r/η = 1+ − r− r e , ξ η ξη

2 2 4 1 1 2 −r/η − − − = . r + 2r e ξ η ξη η 2 ξη h2 4π 2 M

2 2 − ξ η

4 1 − − 2 −Hχ = ξη η

r −(1/ε+1/η)r . +A e ·r 1+ ξ

1 2 −r/η r + 2r e ξη

6 2 2 2 1 h2 − − r+ + r2 −χHχ = 4π 2 M ξ η ξ 2 ξη η 2

2 4 1 4 −2r/η 3 + − 2 + 2 r + 2 2r e ξ η ξη ξ η

2 1 + r2 + r3 + 2 r4 A e−(1/ξ+2/η)r . ξ ξ − χHχ dr =

2 η η3 h2 3η 2 η 3η 3 3η 2 3η 3 η − + 2− + − 2 + + 2 4π 2 M 2ξ 2 2ξ 2ξ 4 2ξ 4ξ 4ξ ⎫ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ 12 24 2 +A

3 +

4 +

5 . ⎪ 1 2 1 2 1 2 ⎪ ⎪ ⎪ ⎪ ⎪ + + + ξ ξ2 ⎭ ⎩ ε η ε η ε η

367

NUCLEAR PHYSICS

h2 1 B= 2 = 2 8π M ε 2M k=

η , ξ

h 2πε

2 =

η , ε

t=

P02 = T (P0 ); 2M

η = t ε,

ξ=

h2 = 2Bε2 . 4π 2 M

1 t η = ε. k k

t3 1 + 3k + 3k 2 , 4 kt k 2 t t 3 − χHχ dr = −Bε + + 2 2 2 3 12kt3 24k 2 t3 2t 3 . +Aε + + (2 + t)3 (2 + t)4 (2 + t)5 χ2 dr = ε3

−H = A

1 1+

+ t 3 2

3k

+ t 4

1+ 2 1 + 3k + 3k 2

3k 2 1+

t 5 2

−B·

2 1 + k + k2 · . t2 1 + 3k + 3k 2

−H k = 1 t = 0.6 0.3303A − 2.381B t = 0.7 0.2826A − 1.749B t = 0.8 0.2432A − 1.339B 20

7.3.7.1 Kinematics of two α particles (statistics). Mp ∼ = MN For one α particle: ψ(q1 , q2 ; Q1 , Q2 ) = ψ(B) ϕ(q1 , q2 ; Q1 , Q2 ), q1 = q1 − B,

q2 = q2 − B,

Q2 = Q1 − B,

Q2 = Q2 = B;

1 B = (q1 + q2 + Q1 + Q2 ). 4 For two α particles, without considering statistical eﬀects (ψ = ψ1 ): q1 + q2 + Q1 + Q2 = 0,

ψ(q1 , q2 ; Q1 , Q2 ) ψ1 (q3 , q4 ; Q3 , Q4 ); 20 @ From the original manuscript it is evident that the author intended to obtain a similar table for the value k = 0.8; however, no numerical value for H was reported.

368

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

including statistical eﬀects: ψ=

1 ± ψ(qi1 , qi2 ; Qk1 Qk2 ) ψ1 (qi3 qi4 ; Qk3 Qk4 ), 6

with ii < i2 , i3 < i4 , k1 < k2 , k3 < k4 . [21 ] i1 i2 1 2 1 3 1 4 2 3 2 4 3 4

7.4.

+ − + + − +

k1 k2 1 2 1 3 1 4 2 3 2 4 3 4

+ − + + − +

THOMSON FORMULA FOR β PARTICLES IN A MEDIUM

Majorana considered here the problem of the energy loss of β particles in passing through a medium, as discussed in the articles by E.J. Williams, 22 Proc. Roy. Soc. A130 (1930) 310, 328. By using the classical theorem of momentum

F dt =

dp, he ﬁrst obtained an expression for

the velocity v of β particles and then, from their kinetic energy T , the energy Q acquired by atomic electrons during the collision. Here, quantity a is the impact parameter and τ the Bohr’s time of collision. The classical number of collisions in which a certain β particle looses energy between Q and Q + dQ in traversing the medium (assumed to be a gas of free electrons, initially at rest) is denoted by ψ(Q) dQ, while J is the ionization potential.

21 @

In the original manuscript, three handwritten lines appear in the table below, connecting the 1st with the 6th row, the 2nd with the 5th row, the 3rd with the 4th row, respectively, pointing out the possible proton+neutron states in the two α particles. 22 @ In his notes the author quoted a paper by Williams and Terroux as present in the same issue of the above cited journal. However, no such a paper was published in that issue. Probably he referred to the important article of E.J. Williams and F.R. Terroux, Proc. Roy. Soc. A126 (1930) 289 which reported on some experimental observations.

369

NUCLEAR PHYSICS

e2 , r2

F =

r=

Fn dt =

e2 a dt = r3

+

3 dt =

x2 ) 2

a2 , cos2 ϕ

e2 a cos2 ϕ a dϕ = a3 v cos2 ϕ

Fn dt = =

(a2

2e2 av

= 2τ τ=

e2 a2

√

e2 a (a2 dx =

π/2 −π/2

+

v =

v=

a . τ

1 2e4 T = mv 2 = 2 2 , 2 a v m

2e4 Q=T = 2 2 a mv

e4 . a2 T a2 =

2e4 . Q mv 2

For n electrons per unit volume:23 23 @

In the original manuscript the typo “per centimeter” occurs.

dx . v

e2 cos ϕ dϕ av

2e2 , avm

T =

1

x2 ) 2

adϕ . cos2 ϕ

,

a , v

e2 a , r3

a2 + x2 .

e2 a

a2 + x2 =

x = a tan ϕ,

Fn =

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E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

2πe4 n dQ. Q2 mv 2 2πe4 n 1 . ψ(Q) = mv 2 Q2 ∞ 2πe4 n 1 ∼ , ψ(Q) dQ = 1= mv 2 J J

ψ(Q) dQ = −π n da2 =

that is, the Thomson formula.

7.5.

SYSTEMS WITH TWO FERMIONS AND ONE BOSON

In the following the author seems to consider a system formed by one boson and two fermions, with momentum γ 0 , γ , γ , respectively. It is not clear to what he precisely referred himself; the topic was only sketched. Let us consider three ﬁelds ψ(γ ),

ϕ(γ ),

χ(γ 0 ),

with: χ = (χ1 , χ2 ),

ψ = (ψ1 , ψ2 ),

ϕ = (ϕ1 , ϕ2 ).

χi (γ)χi (γ ) − χi (γ )χi (γ) = δ(γ − γ ), ψi (γ)ψ i (γ ) + ψ i (γ )ψi (γ) = δ(γ − γ ), ϕ(γ)ϕi (γ ) + ϕi (γ )ϕi (γ) = δ(γ − γ ).

R=

7.6.

0

0

χR ˜ χ dγ +

˜ ψ dγ + ψR

ϕR ˜ ψ dγ .

SCALAR FIELD THEORY FOR NUCLEI?

In the following pages the author apparently elaborated a relativistic ﬁeld theory for nuclei composed of scalar particles of two diﬀerent kinds (one

371

NUCLEAR PHYSICS

with positive charge and the other with negative charge), described by the complex scalar ﬁeld ψ and its conjugate P (this is the continuation of what reported in Sections 2.7 and 2.8). The total number of such constituents is denoted with N , while Z is the net charge; the number of “positive” particles is L, while that of the “negative” ones is M . Explicit expressions of some operators and their matrix elements were given. In particular, transitions between diﬀerent nuclei were described in the framework of the theory considered. For a more detailed discussion, see S. Esposito, Ann. Phys. (Leipzig) 16 (2007) 824. [ψ0 , P0 ] = 1,

[ψ0 , ψ1 ] = 0,

[P0 , P1 ] = 0,

[ψ1 , P1 ] = 1,

[ψ0 , P1 ] = 0,

[ψ1 , P0 ] = 0.

ψ=

ψ0 − iψ1 √ , 2

P =

P0 + iPi √ . 2

[24 ] N=

−

2πi ψP − ψ¯P¯ dV. h

2 2 ¯ = ψ 0 + ψ1 , ψψ 2

P 2 + P12 P¯ P = 0 . 2

ψP − ψ¯P¯ = i(ψ0 P1 − ψ1 P0 ). ψ0 = P0 =

N=

q0r ur ,

ψ1 =

pr0 ur ,

P1 =

q1r ur , pr1 ur .

2π r r (q p − q1r pr0 ). h r 0 1

¯ = 1 [ψ0 , ψ0 ] + [ψ, ψ] 2 1 [ψ, ψ] = [ψ0 , ψ0 ] − 2

1 [ψ1 , ψ1 ] + 2 1 [ψ1 , ψ1 ] − 2

i [ψ0 , ψ1 ] − 2 i [ψ0 , ψ1 ] − 2

i [ψ1 , ψ0 ], 2 i [ψ1 , ψ0 ]. 2

24 @ Note that, in subsequent pages, the author denotes with Z the following operator corresponding, eﬀectively, to the net charge rather than to the total number N of particles.

372

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

∇2 ur + kr2 ur = 0,

1 2 (p0r + p21r ), P¯ P dV = 2

ur2 dV = 1.

2 2 ¯ dV = 1 (q0r + q1r ), ψψ 2

1 2 2 2 kr (q0r + q1r ). ∇ ψ¯ · ∇ ψ dV = 2

The Hamiltonian H without external ﬁeld is (we write q0 , q1 , p0 , p1 , k instead of q0r , q1r , pr0 , pr1 , k r ): 4π 2 mc2

h2 1 2 2 2 2 2 2 (q + q ) + + q k mc q 0 1 0 1 h2 16π 2 m 4 r

4π 2 mc2 1 h2 k 2 2 + mc2 q02 p + = 0 2 2 h 16π m 4 r 2 2

4π 2 mc2 2 h k 1 2 2 + p1 + + mc q1 . h2 16π 2 m 4

H0 =

ν2 =

(p20 + p21 ) +

c2 k 2 m2 c4 + 2 , 4π 2 h

h2 ν 2 = m2 c4 +

c2 h2 k 2 , 4π 2

c2 h2 k 2 2 4 2 c2 = c m2 c2 + p2 , = m c + p 4π 2 Er = Nr hνr = Nr c m2 c2 + p2 . E= Er ,

hν =

m2 c4 +

W0r − hνr . hνr N= Nr , Z= Zr , Nr =

Nr = 0, 1, 2, . . . ;

Zr = Nr , Nr − 2, Nr − 4, . . . , −Nr .

|Zr | ≤ Nr ,

|Z| ≤ N.

——————–

373

NUCLEAR PHYSICS

With an external ﬁeld endowed with vector potential C = 0 and scalar potential ϕ = 0: ϕ= ϕr ur ,

ϕu2r dV,

ϕr =

H = H0 −

ϕrs =

ur us ϕdV,

2π e ϕrs (q0r ps1 − q1r ps0 ). h rs ——————– Zr

Nr 0

00

1

01 10

0 1, −1

⎫ 02 ⎬ 11 2, 0, −2 ⎭ 20

2

3

0 1 2 3

3 2 1 0

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

3, 1, −1, −3

By using units such that h = 2π, ν = 1/2π, hν = 1: 1 1 1 1 W = P02 + Q20 + P12 + Q21 , hν 2 2 2 2 1 1 1 1 N = P02 + Q20 + P12 + Q21 − 1, 2 2 2 2 Z = Q0 P1 − Q1 P0 . 1 1 P1 Q1 − Q1 P1 = , P0 Q0 − Q0 P0 = , i i P0 P1 − P1 P0 = 0, etc. N = 0, 1, 2, . . . ;

Z = N, N − 2, , . . . , −N.

374

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

N P0 − P0 N = iQ0 ,

−(ZP0 − P0 Z) = −iP1 ,

N Q0 − Q0 N = −iP0 ,

−(ZQ0 − Q0 Z) = −iQ1 ,

N P1 − P1 N = iQ1 ,

−(ZP1 − P1 Z) = iP0 ,

N Q1 − Q1 N = −iP1 ,

−(ZQ1 − Q1 Z) = iQ0 .

P0

Q0

P1

Q1

(N, Z); (N + 1, Z + 1)

f++ (N, Z)

if++ (N, Z)

+if++ (N, Z)

−f++ (N, Z)

(N, Z); (N + 1, Z − 1)

f+− (N, Z)

if+− (N, Z)

−if+− (N, Z)

+f+− (N, Z)

(N, Z); (N − 1, Z + 1)

f−+ (N, Z)

−if−+ (N, Z)

+if−+ (N, Z)

+f−+ (N, Z)

(N, Z); (N − 1, Z − 1)

f−− (N, Z)

−if−− (N, Z)

−if−− (N, Z)

−f−− (N, Z)

1 2 P + 12 Q20 2 0 + 12 P12 + 12 Q21

(N, Z); (N + 2, Z + 2)

(N, Z); (N + 2, Z)

−1

Q0 P 1 − Q 1 P 0

0

0

0

2f++ (N, Z) ·f+− (N + 1, Z + 1) −2f+− (N, Z) ·f++ (N + 1, Z + 1)

2|f++ (N, Z)|2 +2|f+− (N, Z)|2 +2|f−+ (N, Z)|2 +2|f−− (N, Z)|2 − 1

2 2|f++ (N, Z)| 2 (N, Z)| +2|f−− −2|f+− (N, Z)|2 −2|f−+ (N, Z)|2

(N, Z); (N + 2, Z − 2) (N, Z); (N, Z + 2)

(N, Z); (N, Z)

(N, Z); (N, Z − 2) (N, Z); (N − 2, Z + 2) (N, Z); (N − 2, Z)

0

(N, Z); (N − 2, Z − 2)

0

0

375

NUCLEAR PHYSICS

f++ (N, Z) = f¯−− (N + 1, Z + 1), f+− (N, Z) = f¯−+ (N + 1, Z − 1). |f++ (N, Z)|2 + |f−− (N, Z)|2 = |f+− (N, Z)|2 + |f−+ (N, Z)|2 =

N +Z +1 , 4 N −Z +1 . 4

f−− (N, Z) = f¯++ (N − 1, Z − 1), f−+ (N, Z) = f¯+− (N − 1, Z + 1). |f++ (N, Z)|2 + |f++ (N − 1, Z − 1)|2 = |f+− (N, Z)|2 + |f+− (N − 1, Z + 1)|2 =

N +Z +1 , 4 N −Z +1 . 4

(N + Z + 2)(N − Z + 2) − (N − Z + 2)(N + Z + 2) = 0. |f++ (N, Z)|2 = |f+− (N, Z)|2 =

N +Z +2 , 8 N −Z +2 . 8

N +Z +2 , 8 N −Z +2 , f+− = 8 N −Z , f−+ = 8 N +Z . f−− = 8 N +Z +2 δN +1,N δZ+1,Z P0 (N, Z; N , Z ) = 8 N −Z +2 δN +1,N δZ−1,Z + 8 N −Z δN −1,N δZ+1,Z + 8 N +Z δN −1,N δZ−1,Z + 8 = a + b + c + d, f++ =

376

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

Q0 (N, Z; N , Z ) = ia + ib − ic − id, P1 (N, Z; N , Z ) = ia − ib + ic − id, Q1 (N, Z; N , Z ) = −a + b + c − d.

——————– N =

Z=

0

0

0

0

0

0

...

0

1

0

0

0

0

...

0

0

1

0

0

0

...

0

0

0

2

0

0

...

0

0

0

0

2

0

...

0 ...

0 ...

0 ...

0 ...

0 ...

2 ...

... ...

,

... 0 1 0 0 0 0 ... 0 0 −1 0 0 0 ... . 0 0 0 2 0 0 ... 0 0 0 0 0 0 ... 0 0 0 0 0 −2 . . . ... ... ... ... ... ... ... 0

0

0

0

0

0

[25 ]

25 The columns and rows of the following matrix are ordered for N, Z equal to 0,0; 1,1; 1,-1; 2,2; 2,0; 2,-2; 3,3; 3,1; 3,-1; 3,-3; . . ., respectively.

377

NUCLEAR PHYSICS

0 1 2 1 2 0 0 P0 = 0 0 0 0 0 ...

1 2

1 2

0

0

0

0

0

0

0

2 2

1 2

0

0

0

0

0

0

1 2

0

0

0

0

0

0

√ 0 0

0 0

√

2 2

√

2 2 1 2

0

√ 0 1 2 √ 2 2

0

0

0

0

0

3 2

0 0

0

1 2 √ 2 2

√

2 2

0 √

0

0

0

0

1 2

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

...

...

...

...

3 2

√ 0

3 2

0

0

1 2

0

0

0

0

0

0

0

...

...

...

...

0

√

2 2 √ 2 2

1 2 √ 3 2 ...

——————–

... . . . . . . . . . . . . . . . . . . . . . . . . . . . ...

378

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

——————–

N +Z = L, 2

L = 0, 1, 2, . . . ;

N −Z = M, 2

M = 0, 1, 2, . . . .

L numbers the particles with positive charge, while M numbers the particles with negative charge. N 0 1 1 2 2 2 N = L + M, Z = L − M.

Z L M 0 0 0 1 1 0 −1 0 1 2 2 0 0 1 1 −2 0 2

379

NUCLEAR PHYSICS

N

Z

N +1 Z +1

L

M

L+1

N

Z

N +1 Z −1

L

M

L

N

Z

N −1 Z +1

L

M

L

N

Z

N −1 Z −1

L

M

L−1

M

M +1

M −1

M

√ L+1 L P0 (L, M ; L , M ) = δL+1,L δM M + δL−1,L δM M 2√ √2 M M +1 δLL δM +1,M + δLL δM −1,M . + 2 2 √

√ √

2 P0 =

2 Q0 = √ 2 P1 = √ 2 Q1 = L P0 √ = 2

0 1 − 2

P0L + P0M = QL 0 QL 0 L −P0

+ QM 0 − QM 0 M + P0

1 2

0 2 2

QL + QM ,

=

QL − QM ,

= −PL + PM . 0

2 2

0 √

√ 0

=

√ 0

0

PL + PM ,

3 2

√ 0

0

3 2

0

...

...

...

...

... 0 . . . , 0 . . . 1 . . . ... ... 0

380

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

L Q0 √ = 2

0 i − 2

i 2

0

0

√ 0

2 2

i

0 √

√

0

2 −i 2

0

i

3 2

√ 0

0

...

...

−i

3 2

...

0 ...

... 0 . . . . 0 . . . i . . . ... ... 0

[26 ] L L P0L QL 0 − Q0 P0 = −i.

√ 2 PL √ 2 QL √ 2 PM √ 2 QM

= = = =

P0 − Q1 , Q0 + P1 , P0 + Q1 , Q0 − P1 .

For h = 2π, ν = 1/2π: 1 1 1 2 1 W = PL2 + Q2L + PM + Q2M , hν 2 2 2 2 1 1 1 2 1 1 1 + Q2M − , N = L + M = PL2 + Q2L − + PM 2 2 2 2 2 2 1 1 1 L = PL2 + Q2L − , 2 2 2

1 1 1 2 + Q2M − , M = PM 2 2 2

1 1 1 2 1 − Q2M . Z = L − M = Q0 P1 − Q1 P0 = PL2 + Q2L − PM 2 2 2 2 ——————–

26 @ Notice that, by using the matrices given above, the following relation is not actually satisﬁed.

381

NUCLEAR PHYSICS

ψP

=

1 {ψL PL + ψM PM + ψL PM + ψM PL 4 − PL ψL − PM ψM + PL ψM + PM ψL 2 2 + i ψL2 + PL2 − ψM − PM −ψL ψM + ψM ψL + PL PM − PM PL )} . ——————–

Versuchsweise:

27

PM = ψM = 0 (mc2 = 1, h = 2π). 1 ¯ = −i, [ψ, P ] = , [ψ, ψ] 2 We have, thus, the classical theory! 28

i [P, P¯ ] = . 4

2 2 ¯ = ψL + PL , ψψ 2 2 ψ + PL2 1¯ P¯ P = L = ψψ, 8 4 i ψP = (ψL2 + PL2 ). 4

27 @ This German word means “tentatively”, and refers to the successive assumptions. Note, however, that in the original paper the cited word is written as “versucherweiser”. 28 @ That is, a theory with only positively charged particle, without antiparticles.

PART IV

8 CLASSICAL PHYSICS

8.1.

SURFACE WAVES IN A LIQUID

The author studied the propagation of surface waves in liquids under the action of the gravitational potential U and the liquid pressure P . Some particular cases were considered in detail. μα = μ F − ∇ p. F = ∇ U: α = ∇U −

1 ∇ p. μ

μ = μ(p); P

dp , μ

∇P =

1 ∇ p. μ

α = ∇ (U − P ).

v = ∇ ϕ, ∂ϕ ∂ϕ ∂y ∂ϕ + vx ∇ + vy ∇ + vz ∇ α = ∇ ∂t ∂x ∂y ∂r ∂ϕ 1 + ∇ V 2. = ∇ ∂t 2 ∇

1 ∂ϕ + ∇ V 2 − ∇ U + ∇ P = 0, ∂t 2 ∂ϕ 1 2 + V − U + P = 0. ∂t 2

For a liquid:

385

386

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

U = g z,

P =

p . μ

p ∂ϕ 1 2 + V − g z + = 0, ∂t 2 μ ∇2 ϕ = 0. ϕ = Aeωi(t−x/v) ekiz . ∇ ϕ = −ϕ 2

ω2 2 +k . v2

Since ∇2 ϕ = 0, we have: ω k = ± i, v ϕ = eωi(t−x/v) Aeωzv + Be−ωzv . For small amplitudes: p ∂ϕ − g z + = 0. ∂t μ For z = 0,

dp = 0: dt ∂ϕ ∂2ϕ = 0. −g ∂t2 ∂z

For z = : ∂ϕ = 0. ∂z ω −ω 2 eωi(t−x/v) (A + B) = g (A − B)eωi(t−x/v) , v g (A − B) = −ω(A + B). v Aeω/v − Be−ω/v = 0, B = Ae2ωv .

387

CLASSICAL PHYSICS

g B+A = . B−A ωv eωv + e−ωv g . = ωv ωv e − e−ωv λ=v

ω 2πv = , 2π ω

v=ω

λ , 2π

v λ = , ω 2π 2π . λ

ω=v

e2π/λ + e−2π/λ λ g = , 2π v 2 e2π/λ − e−2π/λ e2π/λ − e−2π/λ 2π 2π v 2 = 2π/λ , = tanh −2π/λ λ g λ e +e

2π λ tanh , v =g 2π λ 2

For

v=

λ : 2π v=

For

λ : 2π

8.2.

2π λ tanh . 2π λ

g .

v=

g

g

λ . 2π

THOMSON’S METHOD FOR THE DETERMINATION OF e/m

The equations of motion for the electron moving in the Thomson apparatus, aimed at the determination of the charge to mass ratio, e/m, are studied by the author in these pages.

388

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

For photoelectric electrons: m¨ x = E e + H e y, ˙ m¨ y = −H e x. ...

m x= H e y¨ = − ...

x= H e y¨ = −

H 2 e2 x, ˙ m

H 2 e2 x. ˙ m2

He t. m By the substitution above, the constant c is determined as follows: x˙ = c sin

c

Ee He = , m m

c H = E,

c=

E . H

He E sin t. H m He Em 1 − cos t , H 2e m x˙ =

x0 =

8.3.

2E m . H 2e

WIEN’S METHOD FOR THE DETERMINATION OF e/m (POSITIVE CHARGES)

The equations of motion for positively charged particles moving in the Wien apparatus, aimed at the determination of the charge to mass ratio,

389

CLASSICAL PHYSICS

e/m, are solved and compared with the experimental results by Thomson. m¨ y = H e x, my˙ = H e dx, dy = H e dx, m dt m v dy = dx H e dx, mvy = dx H e dx = e A. y=A

e . mv

d2 z = Z e, dt2 d2 z m v 2 2 = Z e, dx m v 2 z = B e. m

z=B z=

e . m v2

y B . v A

y2 A2 e = . z B m Thomson has repeated the experiment by Wien, obtaining, as a result, the parabola: Z

y

390

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS 2 m vmax = 2V e, B . zmin = 2V

8.4.

DETERMINATION OF THE ELECTRON CHARGE

In the following, the author studied several electrical eﬀects in gases, with particular reference to the Townsend eﬀect, that is, the increase of the photoelectric saturation current from an electrode as a function of the distance d between plane parallel electrodes for high values of the electric ﬁeld (whose strength was denoted with X). The quantity n gives the number of electric charges (electrons) per unit volume, while the Townsend coeﬃcient α is the number of new ion pairs produced per centimeter of path in the gas by electron impacts. The gas is at the pressure p and temperature T , while D is a diﬀusion coeﬃcient. This study was aimed to obtain determinations of the electron charge e (with diﬀerent experimental methods).

8.4.1

Townsend Eﬀect

8.4.1.1

Ion recombination. dm dn = = q − α m n. dt dt

(1)

391

CLASSICAL PHYSICS

n = m:

dn = q − αn2 . dt q − αn2 = 0;

dn √ 2 q

(2)

n0 =

q . α

dn = dt, q − αn2 1 1 √ +√ √ √ q+n α q−n α

√ √ q+n α 1 √ √ log √ 2 qα q−n α √ √ q+n α √ √ q−n α α n q

(3)

= dt,

= t, √ 2t qα

= e

√

√

e2t qα , = 1

e2t qα − 1 , = 2t√qα e +1 √ q e2t qα − 1 √ n = , α e2t qα + 1 √

e 4αqt − 1 . n = n0 √4αqt e +1 e2n0 αt − 1 e2n0 αt + 1 (formula applying to a source active for a time t). n = n0

(4)

——————– dn dt dn n2 1 1 − n0 n 1 n n=

= −αn2 , = −α dt, = −αt, =

1 + αt, n0

n0 1 = 1 1 + n0 αt + αt n0

(5)

392

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

(formula applying to a source extinguished at time t). ——————–

For the determination of α we can use the following setup, where iA , iB are the saturation currents measured by setting alternately the electrical 1 tension in A and B, respectively, with iB = iA . 2 V = σv, T = d/v. nA nB = , 1 + nA αT 1 and since nB = nA , 2 nA αT = 1. iA = nA V e,

iA αT = V e. Ve α= . iA T For air we have α = 1.65 · 10−6 = 3480e (Townsend).

8.4.1.2

Ion diﬀusion. dn = q − αn2 + D∇2 n. dt d2 n dn = q − αn2 + D 2 . dt dx

For

dn = 0 and neglecting α, dt D

d2 n − q = 0, dx2 q d2 n = − , dx2 D q 2 − x2 . n = 2D

393

CLASSICAL PHYSICS

n dx =

q3 1 q3 2 q 3 − = . D 3 D 3D Q=

2 q 3 e. 3D 1 2 e = t. 3D

Q = 2q t,

D coeﬃcients (Townsend) + ions - ions dry air 0.028 0.043 wet air 0.032 0.026 dry CO2 0.023 0.026 dry H2 0.123 0.190

8.4.1.3

Velocity in the electric ﬁeld. dn , dx

N1 = D V n=D

dn , dx

V =D V =

p = n kT,

N1 = V n. 1 dn , n dx

V =D

1 dp , p dx

D n e X. p n 1 N = = , p kT π

[1 ] N D eX = eX, π kT The relation utilized by Townsend relation is for X = 1: V =D

V =D

8.4.1.4

D N e= e. π kT

Charge of an ion. N e, π πn Ne= , D

n=D

1N

is the total number of charged particles, while π is the atmospheric pressure (see below).

394

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

where π is the atmospheric pressure. Townsend has found: N e =

8.4.2

96540 · 3 109 = 1.3 · 1010 , 22400 e = 1.04. e

Method of the Electrolysis (Townsend)

The oxygen and hydrogen which are formed at the electrode are strongly electriﬁed, positively or negatively depending on the kind of electrolysis. From the Stokes law: v = k a2 .

4 3 πa . n=q 3 Q e= , n where q is evaluated thermodynamically.

8.4.3

Zaliny’s Method For The Ratio Of The Mobility Coeﬃcients

V − k u = 0, V − k1 v = 0, u k1 1 = = . v k 1.24

395

CLASSICAL PHYSICS

dry air wet air dry CO2 wet CO2 dry H2 wet H2

Mobility + ions 1.36 1.37 0.76 0.81 6.70 5.30

coeﬃcients - ions ratio 1.87 1.375 1.51 1.10 0.81 1.07 0.75 0.915 7.95 1.19 5.60 1.05

T (o C) 13.5 14 17.5 17 20 20

1 K u. y log b/a K K x 1 2 1 2 b − y = ut = u . 2 2 log b/a log b/a V 2K u x. V b2 − a2 = log b/a 2Kπ u x. Q= Q = π b2 − a2 V, log b/a Q log b/a u= . 2πKx −y˙ =

8.4.4

Thomson’s Method

396

8.4.5

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

Wilson’s Method

It is as the Thomson’s method, with the addition of an electric ﬁeld to the gravity. The charge e is obtained from the ratio between the fall velocities with and without the ﬁeld: 4 πρ g a3 + Xe v1 3 . = 4 v 3 πρ g a 3 By determining a from the Stokes formula (see below), we van obtain the value of e.

8.4.6

Millikan’s Method

The Stokes law: 2 ga2 (σ − ρ) 9 μ has been corrected by Cunningham for droplets with small radius: 2 ga2 (σ − ρ) 1 + A , v= 9 μ a where A is a numerical constant and is the mean free path. By setting B = A we have: B 2 ga2 (σ − ρ) 1 + . v= 9 μ a v=

397

CLASSICAL PHYSICS

8.5.

ELECTROMAGNETIC AND ELECTROSTATIC MASS OF THE ELECTRON

The expressions for the electromagnetic and the electrostatic mass of the electron are derived, by evaluating the magnetic energy W and the analogous electrostatic energy W/c2 . H = H2 = H2 8π 4πr2

=

H2 e2 u2 dr. dr = 8π 3r2

∞ a

W =

8.6.

e u sin θ , r2 e2 u2 sin2 θ , r4 e2 u2 sin2 θ . 8πr4

1 dr = . r2 a

1 e2 u2 = m u2 . 3a 2

m=

2 e2 3 a

(electromagnetic),

m=

2 e2 3 a c2

(electrostatic).

THERMIONIC EFFECT

In the following the author studied electron emission induced by thermionic eﬀect, obtaining the Richardson formula for the electron current. Moreover, he subsequently considered also the Langmuir eﬀect (for low voltage) induced by the cloud of (slowly moving) electrons (space charge) around the cathode, which limits the electron emission from the cathode.

398

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

Let V e be the extraction work; in order that an electron comes out of the metal, the following relation must hold: Ve≤

1 m u2 . 2

The Maxwell distribution gives: dn = C e−m u /kT du, h m −h m u2 du, e dn = n π 2

h = 1/2kT . 1 V e = m u20 , 2

u0 =

2V e . m

The number of electrons emitted is then given by: ∞ hm ∞ 2 2 dn = 2n e−h m u du √ √ π 2V e/m 2V e/m 1 hm e−b/T = n π h m 2 V E/m 1 = n e−b/T 2V ehπ kT −b/T e = n . πV e From this, the Richardson formula for the electron current i follows (Richardson eﬀect): i = a T 1/2 e−b/T . Instead, with the photoelectric theory, it has been found that: i = a T 2 e−b/T . Electron emission starts around 1000o C; for several elements (sodium) it starts around 200o C. If T is small, the value for the saturation current is reached very quickly. ——————–

399

CLASSICAL PHYSICS

V e=

1 m u2 . 2

V = u/300: 1 u e = m u2 , 300 2 √ 2e . u= u 300m e = 4.77 · 10−10 , m = 0.9 · 10−27 , 2e = 5.53 · 1015 , 300m u=

8.6.1

√

u · 594 km/s.

Langmuir Experiment on the Eﬀect of the Electron Cloud

At low values of the potential, the electron current does not change with varying T . d2 V = −4πρ. dx2 i = ρ v = const. √ v =k V,

c −4πρ = √ . V

c d2 V =√ , dx2 V d dV d2 V ; = 2 dx dx dx dV dx dV dV d dx dx d

dV dx

2

= =

c √ dx, V c √ . V

√ = c V + const.

400

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

V (0) = 0, √ v = v0 V ,

V () = V1 . i ρ= √ . v0 V

[2 ] V 3/2 . x2 ——————–

i=k

Eﬀects that are an obstacle to the reaching of the value of the saturation current are the following. 1) the cloud of slowly moving electrons around the cathode (Langmuir eﬀect): imax = k V 3/2 ; 2) the magnetic ﬁeld produced by the ﬁlament (a voltage of the order of 1 volt is required): mx ¨ = E e − H z, ˙ m z¨ = H x, ˙ E= mx ¨ = m z¨ =

A , x

B A e − z, ˙ x x B x; ˙ x

H=

B , x

mxx ¨ = A e − B z, ˙

3) a non-vanishing gradient of the voltage along the ﬁlament (of the order of 1 volt/cm).

2@

It is not clear how the author solved the diﬀerential equation for V , thus obtaining the expression for ρ and, ﬁnally, the following expression for the current i. Nevertheless, the expression for i is correct, choosing in a given way the integration constant in the diﬀerential equation above.

CLASSICAL PHYSICS

401

If the eﬀects 1), 2) and 3) are removed in some way, the saturation of the current is reached at a very lower voltage. This has been veriﬁed experimentally by Schottky.3 The eﬀect 3) is removed by switching oﬀ the voltage and measuring i at the same time instant.

3 In

the original manuscript, the author writes this name (between brackets) as “Sciochi”.

9 MATHEMATICAL PHYSICS

In the following six Sections, the author studied a number of topics dealing with tensor calculus, following closely the text T. Levi-Civita, Lezioni di calcolo diﬀerenziale assoluto (Stock, Rome, 1925), which was present in the Majorana personal library. For the notations used and further comments on the topics treated, we refer the reader to this book (we denote it as Levi-Civita I) or to its English translation (denoted as LeviCivita E) in T. Levi-Civita, The Absolute Diﬀerential Calculus – Calculus of Tensors (Blackie & Son, London, 1926). Some explicit references to chapters (III and IV) or pages (pp. 48, 60, 123, 137, 140, 141, 143, 160, 173, 174, 178, 197 of Levi-Civita I or pp. 36, 47, 107, 119, 121, 123, 131, 140, 152, 153, 156, 172 of Levi-Civita E) of this book are reported throughout the manuscript. A few results, on the contrary, do not appear in the mentioned book; they were obtained by Majorana, or he simply reported what was expounded in the university course taught by Levi-Civita at the University of Rome and followed by Majorana himself.

9.1.

LINEAR PARTIAL DIFFERENTIAL EQUATIONS. COMPLETE SYSTEMS

X1 , . . . , Xn :

Xi dxi = 0.

y(x1 , . . . , xn ) = C, dy = ∂y = pXi , ∂xi

∂y dxi . ∂xi p = p(x1 . . . xn ).

403

404

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

dy =

pXi dxi =

Ai dxi .

∂Ai ∂Aj − = 0. ∂xj ∂xi ∂p ∂p ∂Xi ∂Xj − − Xj + Xi p ∂xj ∂xi ∂xj ∂xi ∂Xj ∂Xk ∂p ∂p − − Xk + Xj p ∂xk ∂xj ∂xk ∂xj ∂Xi ∂p ∂p ∂Xk − − Xi + Xk p ∂xi ∂xk ∂xi ∂xk

Xk

9.1.1

∂Xi ∂Xj − ∂xj ∂xk

+ Xi

∂Xj ∂Xk − ∂xk ∂xj

+ Xj

= 0, = 0, = 0;

∂Xi ∂Xk − ∂xi ∂xk

Linear Operators Auv = vAu + uAv = (−Au)v + uAv.

A=

N r=1

AB =

∂ ar , ∂xr

N r,s=1

=

N r,s=1

BA =

N r,s=1

∂ ar ∂xr

B=

N

br

r=1

∂ . ∂xr

∂ bs ∂xs

N ∂2 ∂bs ∂ ar bs + ar , ∂xr ∂xs ∂xr ∂xs r,s=1

N ∂2 ∂as ∂ ar bs + br , ∂xr ∂xs ∂xr ∂xs

AB − BA = (A, B) =

r,s=1

N r,s=1

∂bs ∂as ∂ − br . ar ∂xr ∂xr ∂xs

= 0.

405

MATHEMATICAL PHYSICS

∂ ∂2 + (Abs ) , ∂x ∂x ∂x r s s rs s ∂ ∂2 ar bs + (Bas ) , BA = ∂xr ∂xs ∂xs rs s AB =

ar bs

AB − BA = (A, B) =

N

(Abs − Bas )

s=1

∂ . ∂xs

——————–

A1 , . . . , An : B=

n

λ i Ai ,

C=

1

BC =

n

CB =

i,k

μi Ai .

1

λi Ai μk Ak =

i,k=1

n

λi μk Ai Ak +

i,k

λi μk Ak Ai −

λi (Ai μk )Ak ,

i,k

μi (Ai λk )Ak ,

i,k

(B.C) = BC − CB λi μk (Ai , Ak ) + (λi Ai μk − μi Ai λk ) Ak . = i,k

9.1.2

k

i

Integrals Of An Ordinary Diﬀerential System And The Partial Diﬀerential Equation Which Determines Them

x1 , . . . , xn : dxi = Xi (x|t). dt f (x|t) = constant: ∂f ∂f dxi + = 0, ∂t ∂xi dt ∂f ∂f + Xi = 0. ∂t ∂xi

(1)

406

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

A=

∂ ∂ , + Xi ∂t ∂xi Af = 0.

f (x|t) constant for any value of of 1 implies Af = 0. Conversely, Af = 0 implies f (x|t) constant for any value of of 1.

9.1.3

Integrals Of A Total Diﬀerential System And The Associated System Of Partial Diﬀerential Equation That Determines Them duα =

n

Xαi dxi ,

α = 1, . . . , m.

1

f (x|u) = constant: df

∂f ∂f dxi + duα ∂xi ∂uα n m ∂f ∂f = + Xα dxi . ∂xi ∂uα i

=

α=1

i=1

∂f ∂f + Xαi = 0 ∂xi ∂uα m

(i = 1, 2, . . . , n).

α=1

Ωi =

m α=1

Bi =

∂ + Ωi , ∂xi

Bi f = 0,

Xαi

∂ . ∂uα

(i = 1, 2, . . . , n). (i = 1, 2, . . . n).

——————– Complete systems: Ak f = 0,

407

MATHEMATICAL PHYSICS

Ak =

N

∂ ∂xν

akν

1

(Ai , Ak ) =

n

(k = 1, 2, . . . , n);

pikl = −pkil .

pikl Al ,

1

Jacobian systems: (Ai , Ak ) = 0. Reduction of a complete system to a Jacobian one: Bi f =

n

cik = 0.

cik Ak f,

k=1

N − n = m; xn+1 = u1 , xn+2 = u2 , . . . xN = um : Ak f =

N

∂f ∂f = aki + Uk f = 0, ∂xi ∂xi m

aki

1

1

Uk =

m

ak,n+r

r=1 n

aki

i=1

∂f + Uk f = 0, ∂xi n

k = 1, 2, . . . , n,

aki

i=1 n

αkr aki

i=1

where αri =

∂ . ∂ur aki = 0;

∂f = −Uk f. ∂xi ∂f = −αkr Uk f, ∂xi

Ari is the reciprocal element of ari : A αri aki = δik , αki akr = δir . i

i n n i=1 r=1

=

∂f ∂f αkr aki = δir ∂xi ∂xi

∂f =− ∂xr

n

i=1

n r=1

αkr Uk f,

408

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

that is a Jacobian system. Conversely, let us start from a Jacobian system: ∂f + Ωi f = 0, ∂xi

i = 1, 2, . . . n,

where Ωi are linear operators depending only on u1 , . . . , um , Ωi =

m

Xiα

α=1

∂ . ∂uα

By setting: Bi =

∂ + Ωi , ∂xi

we have: Bi f = 0, Bi =

i = 1, 2, . . . , n,

αki Ak .

i

The Poisson brackets of the B operators are linear combinations of the Poisson brackets of the A operators and of the A themselves, and since the A operators deﬁne a complete system and, in turn, are combinations of the B operators, we have: (Bi , Bk ) =

qik B .

i

Bi =

∂ + Ωi , ∂xi

Bk =

∂ + Ωk . ∂xk

∂2 ∂ ∂ + Ωi + Ω k + Ωi Ω k , ∂xi ∂xk ∂xk ∂xi ∂2 ∂ ∂ + Ωk + Ω i + Ωk Ω i , Bk Bi = ∂xi ∂xk ∂xi ∂xk ∂ ∂ ∂ ∂ + Ωi Ω k − Ω k Ω i (Bi , Bk ) = Ωi − Ωi + Ωk − Ωk ∂xk ∂xk ∂xi ∂xi = Ωik = 0. Bi Bk =

409

MATHEMATICAL PHYSICS

9.2.

ALGEBRAIC FOUNDATIONS OF THE TENSOR CALCULUS

9.2.1

Covariant And Contravariant Vectors x −→ x, u −→ u .

S : : S −1∗

Covariant: ui = uk

∂xk , ∂xi

ui uk

∂xk ∂xr ∂xk ∂xr ∂xk = u = ur = uk . r ∂xi ∂xk ∂xi ∂xi ∂xi

Contravariant: ui = uk

∂xi , ∂xk

ui uk

i i ∂xi ∂xk ∂xi r ∂x k ∂x = u = u = u . r ∂xk ∂xr ∂xk ∂xr ∂xk

9.3.

GEOMETRICAL INTRODUCTION TO THE THEORY OF DIFFERENTIAL QUADRATIC FORMS I

9.3.1

The Symbolic Equation Of Parallelism dR · δP = 0

(δP taken on the surface); 3

dYν δyν = 0

ν=1

(δyν being the most general ones).

9.3.2

Intrinsic Equations Of Parallelism

Deduction of the intrinsic equations: δyν

=

2 ∂yν ∂xk , ∂xk k=1

Yν

=

2 ∂yν i=1

∂xi

Ri

410

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

(Ri = Rλi ; R is the length of the vector; λi = dxi /ds), aik Ri Rk . R2 = 2

2 3 2 ∂yν i ∂yν d R δxk ∂x ∂x i ν ν=1 i=1 k=1 = τk δrk = 0,

dyν δyν

=

ν=1

τk =

k 3 2 ν=1 i=1

∂yν d ∂xk

∂yν i R . ∂xi

τk = 0. τk =

3 3 2 2 2 ∂yν ∂yν ∂yν ∂ 2 yν dRi + Ri dxj ∂xk ∂xi ∂xk ∂xi ∂xj ν=1 i=1

2

ν=1 i=1 j=1

3 ∂yν ∂ 2 yν = aik dR + R dxj ∂xk ∂xi ∂xj ν=1 i=1 i,j=1 ⎡ ⎤ 2 2 i j ⎦. = aik dRi + Ri dxj ⎣ k i=1 i,j=1 i

2

i

3 ∂ ∂yν ∂yν ∂yν ∂ 2 yν ∂yν ∂ 2 yν = − . ∂xk ∂xi ∂xj ∂xj ∂xk ∂xi ∂xi ∂xk ∂xj ν ν ν=1 ⎡ ⎡ ⎤ ⎤ k j i j ∂ ⎣ ⎦, ⎦= aik − ⎣ ∂xj i k ⎡ ⎤ ⎡ ⎤ i j j k ⎣ ⎦+⎣ ⎦ = ∂ aik , ∂xj k i ⎡ ⎤ ⎡ ⎤ j k k i ⎣ ⎦+⎣ ⎦ = ∂ aji , ∂xk i j ⎤ ⎡ ⎤ ⎡ i j k i ⎦ = ∂ akj , ⎣ ⎦+⎣ ∂xi k j

411

MATHEMATICAL PHYSICS

⎡

i j

⎤

⎣ ⎦= 1 2 k

∂ ∂ ∂ aki + aki − aij . ∂xi ∂xj ∂xk

dR · δP =

τk =

2

2

aik dRi +

i=1

τk δxk , ⎡ ⎣

⎤

i j

⎦ Ri dxj = 0.

k

i,j=1

τk is a covariant vector; in fact,

τk = 0,

τk δxk = invariant. ak τk , τ = k

τ

is a contravariant vector. τ = dRi +

⎧ ⎫ 2 ⎨ i j ⎬ ⎭

Ri dxj = 0.

⎧ ⎫ 2 ⎨ i j ⎬

i,j=1

dR = −

⎩

i,j=1

⎩

⎭

Ri dxj

(which is the equation of the parallelism).

9.3.3

Christoﬀel’s Symbols ⎡

⎤

∂ ∂ ∂ ⎦= 1 ak + akj − aj , 2 ∂xj ∂x ∂xk k ⎧ ⎫ ⎡ ⎤ j ⎨ j ⎬ ⎦, aik ⎣ = ⎩ ⎭ i k k ⎧ ⎫ ⎧ ⎫ ⎡ ⎤ ⎡ ⎤ j j ⎨ j ⎬ ⎨ j ⎬ ⎣ ⎦=⎣ ⎦, = ; ⎩ ⎭ ⎩ ⎭ k k i i ⎤ ⎡ ⎤ ⎡ j i j k ∂aik ⎦, ⎦+⎣ =⎣ ∂xj k i ⎣

j

412

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

⎡ ⎣

j

⎤ ⎦=

⎧ ⎫ ⎨ j ⎬ aik

⎩

k

⎭

i

.

——————– .

a11 . . . a1n a = . . . an1 . . . ann

∂ars ars ∂a = , ∂xi ∂xi a r,s

⎛⎡

=

ars

⎤

i r

∂ars = ars ⎝⎣ ∂x i r,s r,s ⎧ ⎧ ⎫ ⎨ i r ⎬ ⎨ i s + = ⎩ ⎩ ⎭ r s r s ⎧ √ ⎨ i r ∂ log a = ⎩ ∂xi r r

∂ log a ∂xi

9.3.4

⎡

⎦+⎣

s ⎫ ⎬ ⎭

=2

i s

⎤⎞ ⎦⎠

r ⎧ ⎫ ⎨ i r ⎬ r

⎩

r

⎭

,

⎫ ⎬ ⎭

.

Equations Of Parallelism In Terms Of Covariant Components

dR = −

⎧ ⎫ ⎨ i j ⎬ ij

⎩

⎭

Ri dxj

Rs = dRs =

das =

t

∂xt

dxt =

as R ,

as dRl +

∂asl

(contravariant components),

t

⎛⎡ ⎝⎣

t s

⎤

R das , ⎡

⎦+⎣

t s

⎤⎞ ⎦⎠ Rl dxt ,

413

MATHEMATICAL PHYSICS

dRs =

as dR +

= −

i,j

=

R

⎡ ⎣

t s ⎡

⎡

t

⎦+⎣ ⎛⎡

R ⎝⎣

s t s

⎤⎞ ⎦⎠ dxt

⎤

⎡

⎦+⎣

t

,t

⎤

⎤⎞ ⎦⎠ dxt

s

⎦ dxt .

t s

⎤ ⎦=

,t,r

⎤

⎦ Ri dxj +

s

⎣

dRs =

t s

R ⎝⎣

⎤

i j

⎣

⎛⎡

,t

⎡

,t

ar R dxt

⎧ ⎫ ⎨ t s ⎬ ar

r

⎩

⎧ ⎫ ⎨ t s ⎬ ⎩

⎭

r

=

r

⎭

, ⎧ ⎫ ⎨ t s ⎬

Rr dxt

t,r

⎩

r

⎭

.

Equations of the parallelism contravariant components : dRi = −

⎧ ⎫ ⎨ k ⎬ ,k

covariant components :

dRi =

i

⎩

Some Analytical Veriﬁcations xi = xi (s),

R˙ i = −

⎧ ⎫ 2 ⎨ k ⎬ ⎭

i ⎧ ⎫ ⎨ k ⎬

ell=1

V˙ i = −

i = 1, 2,

⎩

,k

⎩

i

⎭

⎫ ⎧ ⎨ i k ⎬ ,k

9.3.5

⎩

⎭

R x˙ k

V x˙ k ;

⎭

R dxk

R dxk

414

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

⎧ ⎫ ⎨ k ⎬

R˙ i Vi = −

i

i ⎧ ⎫ ⎨ i k ⎬ i,,k

R˙ i =

⎩

,k

V˙ i =

R˙ i Vi = −

⎭

⎭

⎫ ⎧ ⎨ i k ⎬ ⎩

,k

⎩

⎭

R Vi x˙ k ;

R x˙ k ,

V x˙ k ;

⎧ ⎫ ⎨ k ⎬

R V x˙ k , ⎩ ⎭ i i,,k ⎧ ⎧ ⎫ ⎫ ⎨ i k ⎬ ⎨ k ⎬ Ri V˙ i = Ri V x˙ k = R Vi x˙ k , ⎩ ⎩ ⎭ ⎭ i i i,,k i,,k d d i R˙ i Vi + R Vi = Ri V˙ i = 0. (R · V ) = ds ds i

9.3.6

i

i

Permutability

dδxi = −

⎫ ⎧ ⎨ k ⎬ k,

⎩

i

⎭

δxi = −

δxk dx ,

⎫ ⎧ ⎨ k ⎬ k,

⎩

i

⎭

dδxi = δdxi . xi + dxi + δxi + dδxi = xi + δxi + dxi + δdxi .

9.3.7

Line Elements ds2 =

n

aik dxi dxk .

i,k=1

dxi , λ = ds i

λi =

n k=1

aik λk ,

λi =

aik λk ,

dxk δx ,

415

MATHEMATICAL PHYSICS n

aik λi λk =

n

λi λi =

i=1

i,k=1

n

aik Ri Rk =

Ri = Rλi , n

n

Ri Ri =

i=1

i,k=1

cos θ =

aik λi λk = 1,

i,k=1

Ri = Rλi , R2 =

n

aik λi μk =

n

λi μi =

R·V =

aik Ri Rk .

i,k=1

i=1

i,k=1

n

n

λk μk =

k=1 n

n

aik λi μk ,

i,k=1

R i Vi .

i=1

9.3.8

Euclidean Manifolds. Any Vn Can Always Be Considered As Immersed In A Euclidean Space

Wp (immersed in Vn ): xi = fi (ui , . . . , up )

2

ds

=

=

n i,k=1 p

(i = 1, 2, . . . , n; p < n).

aik dxi dxk =

p n

aik

i,k=1 r,s=1

∂xi ∂xk dur dus ∂ur ∂us

brs dur dus ,

r,s=1

brs =

n i,k=1

aik

∂xi ∂xk . ∂ur ∂us

——————– An arbitrary Vn can always be considered as immersed in a Euclidean space. Vn immersed in SN , N > n.

416

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

y1 (x), y2 (x), . . . , yN (x). n

aik dxi dxk =

n ∂yν i=1

dyν2

dyν2 ,

ν=1

i,k=1

dyν =

N

∂xi

dxi =

n ∂yν dxk , ∂xk k=1

n ∂yν dyν = dxi dxk . ∂xi dxk i,k=1

n i,k=1

N n dyν ∂yν aik dxi dxk = dxi dxk , ∂xi ∂xk ν=1 i,k=1

aik =

N ∂yν ∂yν ∂xi ∂xk

(i, k = 1, 2, . . . , n).

ν=1

If N =

n(n + 1) , the problem has a solution. 2 n N = n(n + 1)/2 1 1 2 3 3 6 4 10

C = min(N − n), min N ≤ C≤

9.3.9

n(n + 1) , 2

n(n − 1) n(n + 1) −n= . 2 2

n max (Nmin ) Cmax n(n + 1)/2 n(n − 1)/2 1 1 0 2 3 1 3 6 3 4 10 6

Angular Metric R2 =

aik Ri Rk ,

V2 =

aik V i V k ,

417

MATHEMATICAL PHYSICS

|R + V |2 =

R·V =

aik (Ri + V i )(Rk + V k ) = R2 + V 2 + 2

aik Ri V k =

R i Vi =

i

i,k

Ri V i =

i

aik Ri V k ,

aik Ri Vk .

ik

For a deﬁnite form a, and taking xi and yi not proportional, it follows: 2 aik xi xk · aik yi yk . aik xi yk < zi = λxi + μyi .

λ2

aik xi xk + 2λμ

cos θ =

R·V =

9.3.10

n

aik zi zk > 0,

2 aik xi yk

<

aik λi μk =

n

i,k=1

i=1

aik Ri V k =

aik xi yk + μ2 aik xi xk ·

λi μi =

n i=1

R i Vi =

aik yi yk > 0,

aik yi yk .

λi μi =

n

aik λi μk ,

i,k=1

Ri V i =

aik Ri Vk .

Coordinate Lines

For the coordinate line i (aj = constant for j = i), the parameters λi are: ⎧ (j = i), ⎨ 0 dxj j = λ = ⎩ √ ds 1/ aii (j = i). The moments of the normal to the surface xi = constant are: 1 μi = √ . aii The angle between the coordinate lines i and k is given by: aik . cos θ = ars λr λs = √ aii akk μj = 0

for j = i,

418

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

The angle between the hypersurfaces xi = constant and xk = constant is given by: cos θ = √

aik

. aii akk Let si be a unitary vector along the line i (the parameters are then equal √ to the contravariant components: λj = 0 for j = i, λi = 1/ aii ). Let ni be a unitary vector normal to the hypersurface xi = constant (the moments √ are then equal to the covariant components: μj = 0 for j = i, μi = 1/ aii ). R · si = R · ni = Ri =

9.3.11

√

Ri Rj (si )j = √ , aii Ri Rj ni,j = √ ; aii

aii R · si ,

Ri =

√

aii R · ni .

Diﬀerential Equations Of Geodesics

xi = xi (t): aik dxi dxk = ds. s= I=

B

B

ds = A

aik dxi dxk ,

A

B

δds.

δI = A

ds2 = ds δds =

aik dxi dδxk + δaik =

δds =

aik dxi dxk , 1 δaik · dxi dxk , 2

δaik

aik x˙ i δdxk +

δxj

δxj ,

1 δaik δxj x˙ i x˙ k ds. 2 δxj i,k,j

419

MATHEMATICAL PHYSICS

δI =

1 aik x˙ i δdxk + 2

B

δI =

A

i,k,j

B aik x˙ i δxk −

aik x˙ i dδxk =

δaik δxj x˙ i x˙ k ds. δxj

A

⎛ δxk · ⎝

k

1 ∂aij 2

i,j

δxk

B

(a˙ ik x˙ i + aik x ¨i )δxk .

A

x˙ i x˙ j −

i,k

a˙ ik x˙ i −

i

⎞ aik x ¨i ⎠ ds.

i

——————–

a˙ ik =

∂aik ∂xj

j

B

dI = A

k

x˙ j ,

⎛

⎞ ∂aik ∂aij 1 δxk ⎝ x˙ i x˙ j − x˙ i x˙ j − aik x ¨i ⎠ ds. 2 ∂xk ∂xj i,j

i,j

dI = −

i

pk δxk ds,

B

δI +

pk δxk ds = 0, A

pk =

⎡ ⎣

i

aik x ¨i +

⎡ ⎣

i,j

⎤ ⎦ x˙ i x˙ j +

k

i,j

i j

i j

aik x ¨i .

i

⎤ ⎦ x˙ i x˙ j = 0

(k = 1, 2, . . . , n).

k pi =

aik pk ,

k

pk =

⎫ ⎧ ⎨ i j ⎬ i,j

⎩

k

⎭

x ¨i x ¨j + x ¨k .

420

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

Equations of the geodesic lines

dI = −

pk δxk ds,

pk =

AB

pi =

⎡ ⎣

aik xk ,

⎩

ij

pk = 0,

that is:

⎡ ⎣

i j

k

⎭

⎤ ⎦ x˙ i x˙ j +

aij x ¨i

¨k x˙ i x˙ j + a

n

k

i,j=1

i

⎧ ⎫ ⎨ i j ⎬

pk =

n

⎦ x˙ i x˙ j +

k

ij

⎤

i j

aik x ¨i = 0

i=1

(k = 1, 2, . . . , n), or pk = 0,

that is: x ¨k +

⎧ ⎫ n ⎨ i j ⎬ i,j=1

⎩

k

⎭

x˙ i x˙ j = 0

(k = 1, 2, . . . , n).

9.3.12

Application ds2 = dx21 + r2 dx22 . a11 = 1,

a22 = r2 ,

a12 = 0;

a11 = 1,

a22 =

1 , r2

a12 = 0.

∂a11 ∂a11 = = 0, ∂x1 ∂x2 ⎡

∂a22 = 2rr , ∂x1 ⎤

∂a22 = 0, ∂x2

∂a12 ∂a12 = = 0. ∂x1 ∂x2

∂a ∂a ∂a 1 11 11 11 ⎣ ⎦= + − = 0, 2 ∂x ∂x ∂x 1 1 1 1 ⎡ ⎤ 1 2 ∂a ∂a 1 ∂a 11 12 12 ⎣ ⎦= + − = 0, 2 ∂x2 ∂x1 ∂x1 1 1 1

421

MATHEMATICAL PHYSICS

⎡ ⎣ ⎡ ⎣

2 2 1 1 2

⎤

⎡

⎦ = −rr ,

⎣

⎤

⎡

⎦ = rr ,

⎣

⎤

1 1

⎦ = 0,

2 2 2

⎤ ⎦ = 0.

2

2

⎧ ⎫ ⎨ 1 1 ⎬

⎧ ⎫ ⎨ 1 2 ⎬

⎩ ⎭ 1 ⎧ ⎫ ⎨ 2 2 ⎬ ⎩ ⎭ 1 ⎧ ⎫ ⎨ 1 2 ⎬ ⎩ x ¨1 − r

2

⎭

= 0,

= −rr ,

=

r , r

dr 2 x˙ = 0, dx1 2

⎩ ⎭ 1 ⎧ ⎫ ⎨ 1 1 ⎬

= 0,

⎩ ⎭ 2 ⎧ ⎫ ⎨ 2 2 ⎬

= 0,

⎩

= 0.

r2 x ¨2 + 2r

2

⎭

dr x˙ 1 x˙ 2 = 0, dx1

or dr 2 x˙ = 0, dx1 2

x ¨2 +

sin α = rx˙ 2 ,

r sin α = r2 x˙ 2 ,

x ¨1 − r

2 dr x˙ 1 x˙ 2 = 0. r dx1

dr d (r sin α) = 2r x˙ 1 x˙ 2 + r2 x ¨2 = 0, ds dx1 r2 x˙ 2 = constant. ⎧ dr 2 ⎪ ⎪ ¨1 = r x˙ , ⎪ ⎨ x dx1 2

x ¨1 = r

⎪ ⎪ 2 dr ⎪ ⎩ x x˙ x˙ 2 , ¨2 = − r dx1

r2 x˙ 2 = c.

dr c2 , dx1 r4

422

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

9.4.

GEOMETRICAL INTRODUCTION TO THE THEORY OF DIFFERENTIAL QUADRATIC FORMS II

9.4.1

Geodesic Curvature

[1 ] xi = xi (s); x1 , x2 , . . . , xn are the coordinates in the space Vn . dI = − pk δxk ds

AB

k

= − AB

pk δxk ds;

k

(pk dxk − pk δxk )ds = 0, k

pk δxk =

k

pk δxk .

k

geodesic curvature

pk =

aki x ¨i +

i

⎡ ⎣

i j

⎫ ⎧ ⎨ i j ⎬ i,j

⎩

k

⎤ ⎦ x˙ i x˙ j ,

covariant components;

k

i,j

¨i + pk = x

9.4.2

⎭

x˙ i x˙ j ,

contravariant components.

Vector Displacement s

s + ds

parallel displacement x˙ i x˙ i −

⎫ ⎧ ⎨ l k ⎬ i,k

line displacement 1@

⎩

i

⎭

x˙ l x˙ k ds = ui

x˙ i x˙ i + x ¨i ds = vi

In the original manuscript, a reference appears (p. 154) of a unspeciﬁed text.

423

MATHEMATICAL PHYSICS

⎧ ⎫ ⎡ ⎤ ⎨ k ⎬ vi − u i = ⎣ x˙ k˙ ⎦ ds = pi ds. ⎩ ⎭ l i ,k ui + pi ds; x˙ i + pi ds.

ui , x˙ i ,

aik x˙ i x˙ k = 1.

aik (a˙ ik (a˙ i + pi ds)(x˙ k + pk ds) =1+2 aik x˙ i pk ds + aik pi pk ds2 , i,k

aik x˙ i pk =

i,k

aik x˙ i x ¨k +

i,k

=

⎧ ⎫ ⎨ m ⎬ aik

⎩

i,k,,m

aik x˙ i x ¨k +

i,k

=

⎡ ⎣

i,,m

⎡ 1 ⎣ d aik (x˙ i x˙ k ) 2 ds

m

k ⎤

⎭

x˙ i x˙ x˙ m

⎦ x˙ x˙ m x˙ i

i

i,k

+

i,k,

=

x˙ i x˙ k

∂ai ∂ak ∂aik + − ∂xk ∂xi ∂x

⎤ x˙ ⎦

⎡ d 1 ⎣ aik (x˙ i x˙ k ) 2 ds i,k

+

i,k

=

⎤ ∂aik ∂aik ⎦ ∂aik + + x˙ i x˙ k − ∂s ∂s ∂s

1 d aik x˙ i x˙ k = 0. 2 ds ——————– i,k

aik pi = ρ2 .

424

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

t · t = 1,

t · (t + ρ ds) = 1,

(t + ρ ds)(t + ρ ds) = 1 + ρ2 ds2 ; 1 1 = 1 − ds2 , 1 + (1/2)ρ2 ds 2

cos(t, t + ρ ds) =

sin(t, t + ρ ds) = ρ ds.

9.4.3

Autoparallelism Of Geodesics pk =

aik x ¨i +

⎡ ⎣

i j

i,j

¨k + pk = x

⎧ ⎫ ⎨ i j ⎬ ⎩

i,j

⎭

k

Vk =

⎦ x˙ i x˙ j = 0,

k x˙ i x˙ j = 0.

λi = x˙ i , ⎧ ⎫ ⎨ i j ⎬ ¨ ds = − dλk = x λi dxj ⎩ ⎭ k i,j

9.4.4

⎤

(antiparallelism)

Associated Vectors dRk ds

+

⎧ ⎫ ⎨ i j ⎬ ⎩

i,j

τ k = 0 : dRk +

for Rk = x˙ k : ¨k + V k = pk = x

⎩

⎩

⎧ ⎫ ⎨ i j ⎬ ⎩

k

⎭

k

⎭

k

τk ds

Ri dxj = 0 (parallelism);

⎧ ⎫ ⎨ i j ⎬ i,j

i,j

⎭

⎫ ⎧ ⎨ i j ⎬ i,j

¨k + pk = x

k

Ri x˙ j =

⎭

x˙ i x˙ j

x˙ i x˙ j = 0

(geodesic curvature);

(equation of the geodesic lines).

425

MATHEMATICAL PHYSICS

9.4.5

Remarks On The Case Of An Indeﬁnite ds 2 ds2 =

ds2 > 0 ds2 < 0 ds2 = 0

time directions: space directions: null interval directions:

9.5.

9.5.1

aik = 0.

aik dxi dxk ,

(∞n−1 ); (∞n−1 ); (∞n−1 ).

COVARIANT DIFFERENTIATION. INVARIANTS AND DIFFERENTIAL PARAMETERS. LOCALLY GEODESIC COORDINATES Geodesic Coordinates

xi = xi (x1 , x2 , . . . , xn ) (i = 1, 2, . . . , n). ∂aik =0 ∂xj

(i, k, j = 1, 2, . . . , n).

P = P0 (x01 , x02 , . . . , x0n ) = P 0 (x01 , x02 , . . . , x0n ) aik =

r,s

∂aik ∂aj

=

ars

∂xr ∂xs , ∂ri ∂xk

∂ars ∂xr ∂xs ∂xt ∂ 2 xr ∂xs + ars ∂xt ∂xi ∂xk ∂xj ∂xi ∂xj ∂xk r,s r,s,t +

r,s

ars

∂xr ∂ 2 xs . ∂xi ∂xk ∂xj

∂xi = aik , ∂xk

dxi =

aik dxk ,

dx = Sdx. x = U x ,

dx = U dx ,

426

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

dx = S U dx.

P

P0 1 r qik xi xk , 2

xr = xr +

r = q r , (x ) = 0, qik r 0 ki

i,k

∂xr = 1, ∂xj 0 2 ∂ xr r = qj . ∂xj ∂x 0

∂xr r = δrj + qjk xk , ∂xj k ∂ 2 xr r = qj , ∂xj ∂x

∂aik ∂xj

0

2 ∂ 2 xr ∂ xs = + ark + ais ∂xi ∂xj 0 ∂xk ∂xj 0 0 r s ∂aik r r = + akr qij + air qkj . ∂xj 0 r s

∂aik ∂xj

r air qkj

+

r

r akr qji +

r air qkj

r

si i a air

1 = 2

= −

r ajr qki = −

r

r ajr qik

+

r

r akr qij

r

r

= −

r air qjk

r

∂aik ∂xj

+ 0

∂aji ∂xk

− 0

∂aik ∂xj ∂akj ∂xi ∂aji ∂xk

∂akj ∂xi

, 0

,

. 0

0

= δrs .

s qkj

⎧ ⎫ ⎨ k j ⎬ =

⎩

s

=

⎭ 0

∂ r xs ∂xk ∂xj

0

. 0

⎡ =⎣

k j i

⎤ ⎦ . 0

427

MATHEMATICAL PHYSICS

geodesic coordinates xi for the point xi = xi = 0 ⎧ ⎫ k j ⎬ 1 ⎨ dxi = dxi + dx dxj , ⎩ ⎭ k 2 i k,j ⎧ ⎫ k j ⎬ 1 ⎨ dxi = dxi − dx dxj + . . . , ⎩ ⎭ k 2 i k,j ⎧ ⎫ ⎨ k j ⎬ ∂ 2 xi =− , ⎩ ⎭ ∂xk ∂xj i

⎧ ⎫ ⎨ k j ⎬ ∂ 2 xi = . ⎩ ⎭ ∂xk ∂xj i 0

geodesic coordinates xi ⎧ ⎫ ⎨ k j ⎬ ∂xi = δik − dxj , ⎩ ⎭ ∂xk i j 0 ⎧ ⎫ ⎨ k j ⎬ ∂xi = δik + dxj . ⎩ ⎭ ∂xk i j

i

1◦ parallelism: (dR = 0), (R0 ) = (Ri )0 ,

9.5.1.1 Applications. (dxi )0 = (dxi )0 . Ri =

dRi = −

⎧ ⎫ ⎨ k j ⎬ k,j

⎩

Ri =

dRi =

i

⎭ Rk

⎧ ⎫ ⎨ i j ⎬ ⎩

k

⎭

R

k

Rk dxj , ∂xk , ∂xi

Rk dxj ,

k

∂xi , ∂xk covariant components.

(Ri )0 = (Ri )0 ,

covariant components.

428

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

2◦ geodesic lines:

dxi ds

dxi ds

=

0

dxi ds

. 0

∂xi dxk , ∂xk ds

=

k

∂xi d2 xk ∂ 2 xi dxk dxj . = + ∂xk ds2 ∂xk ∂xj ds ds

d2 xi ds2

k

k,j

d2 xk = 0. ds ⎧ ⎫ ⎨ k j ⎬ x˙ x˙ = 0. x ¨i + ⎩ ⎭ k j i k,j ⎡ ⎤ k j ⎣ ⎦ x˙ k x˙ j = 0. ¨r + air x r k,j 3◦ geodesic curvature: ⎧ ⎫ ⎨ k j ⎬ 2k d k =x ¨k + pk = x˙ x˙ j . ⎩ ⎭ k ds2 i k,j 4◦ Associated vectors: i

Vi =

dR , ds

i

R =

k

Rk

i

dxi , dxk

dRi

dR = + ds ds

⎧ ⎫ ⎨ k j ⎬ k,j

⎩

i

5◦ Covariant diﬀerentiation: i ...i

Ak11 ...kμm | r = i ...iμ

Ak11 ...km =

p,q

p ...p

μ Aq11...qm

∂ i1 ...iμ A . ∂xr k1 ...km

∂xiμ ∂xq1 ∂xqμ ∂xi1 ... ... . ∂xp1 ∂xpm ∂xk1 ∂xkm

⎭

Rk x˙ j .

429

MATHEMATICAL PHYSICS

i ...i

Ak11 ...kμm |r =

∂ i1 ...iμ A + ∂xr k1 ...km −

i ...i

p

k ...k

μ Aii1...im|l

=

i ...i

Ak21 ...kμm

p

μ 1 Ap,k 2 ...km

⎧ ⎫ ⎨ p r ⎬

⎧ ⎨ k1 ⎩

p0

⎩ ⎭ i1 ⎫ r ⎬ + .... ⎭

+ ...

∂ k1 ...kμ A ∂x ii ...im

⎧ ⎫ k ...k ∂Ai11...imμ k1 ...kr−1 jkr+1 ...kμ ⎨ j ⎬ = + Ai1 ...im ⎩ ⎭ ∂x kr j ⎫ ⎧ ⎨ iρ ⎬ k ...kμ Ai11...iρ−1 jiρ+1 ...iμ − ⎭ ⎩ j j

9.5.2

Particular Cases

1) Ai|k =

∂Ai − ∂xk

⎧ ⎫ n ⎨ i k ⎬ p=1

Ai|k − Ak|i =

⎩

Ai|k =

∂xk

+

Ap ,

∂Ai ∂Ak − . ∂xk ∂xi

2) ∂Ai

⎭

p

n

Ap

p=1

⎧ ⎫ ⎨ p k ⎬ ⎩

i

⎭

.

3) f|i =

fi|k = fik = f|i|k =

∂f = fi . ∂xi ∂2f

∂xi ∂xk

−

fik = fki .

p

⎧ ⎫ ⎨ i k ⎬ fp

⎩

p

⎭

.

430

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

4) ∂Aik − ∂xj

Aik|j =

n

⎧ ⎫ ⎨ i j ⎬ Apk

p=1

5) Aik |j =

∂Aik ∂xj

+

n

Apk

p=1

6) aik|j

=

=

9.5.3

∂aik − ∂xj

n

⎩

⎭

p

−

⎧ ⎫ ⎨ p j ⎬ ⎩

i

+

⎭

n

⎧ ⎫ ⎨ k j ⎬ Aip

p=1

⎩

n

⎧ ⎫ ⎨ p j ⎬

Aip

p=1

⎧ ⎫ ⎨ i j ⎬

⎩

p

k

⎭

⎭

.

.

⎧ ⎫ ⎨ k j ⎬

n

aip − ⎩ ⎭ ⎩ ⎭ p p p=1 ⎡ ⎤ ⎡ ⎤ i j k j ∂aik ⎣ ⎦−⎣ ⎦=0 − (Ricci lemma). ∂xj k i apk

p=1

Applications Vi = V|ji =

aik Vk ,

Vi =

aik Vk|j ,

aik V k ;

Vi|j =

aik V|jk .

Covariant derivative of the scalar product: χ=U ·V =

U i Vi = χj =

i

U|ji Vi =

Ui V i =

i n

aik U i V k =

(U|ji Vi + U i V|j ),

aik Uk|j Vi =

n i=1

k=1

Ui|j V i ,

(Ui|j V i + U i Vi|j ). χj = i

U =V: χj = 2

Ui|j U i .

aik Ui Vk .

431

MATHEMATICAL PHYSICS

9.5.4

Divergence Of A Vector n

Θ=

aij Xi|j =

i,j=1

Xi|j =

X|ii ,

i

aik X|jk ,

k

Θ=

n

ij

a

aik Xjk

=

i,j,k=1

j,k=1

X|ii =

Θ=

X|ii

δjk X|jk

=

n ∂X i

∂xi

+

=

n

k X|k .

k=1

⎧ ⎫ ⎨ p i ⎬

∂X i p + X ⎩ ∂xi p

i=1

n

i Xp

i,p=1

, ⎭ ⎧ ⎫ ⎨ p i ⎬ ⎩

i

⎭

.

1 da = aki daik . a

dxr −→ da: 1 ∂a a ∂xr

n

⎡

i r

⎤

⎡

∂aik ⎦+ = aki ⎣ aki ⎣ ∂xr k k,i i,k i,k ⎡ ⎤ i r ⎦. aik ⎣ = 2 k i,k ⎧ ⎫ ⎡ ⎤ √ n n ⎨ i r ⎬ i r ∂ log a ⎦= = aik ⎣ . ⎩ ⎭ ∂xr k i i=1 i,k=1 ⎧ ⎫ √ n ⎨ p i ⎬ n ∂ log a p p X . X = ⎩ ⎭ ∂xp i =

aki

p=1

i,p=1

Θ=

n ∂X i i=1

√ 1 ∂ a i +√ X , ∂xi a ∂xi

√ n 1 ∂ aX i . Θ= √ ∂xi a i=1

k r i

⎤ ⎦

432

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

Special case: X = ∇ u,

Xi =

∂u . ∂xi

∇ · X = ∇2 u. n n n 1 ∂ √ i 2 ik i ∇ u= a uik = u|i = √ au , ∂xi a i=1

i,k=1

where ui =

n

i=1

n

aik uk =

k=1

9.5.5

aik

k=1

∂u . ∂xk

Divergence Of A Double (Contravariant) Tensor

Given X ik : Yi =

n

X ik|k

k=1

(which, in general, is diﬀerent from Yi =

n

n

ki k=1 X |k ),

ak Xik|l .

k,=1

Yi =

r

air Y =

r

=

n

air X rk|k

r,k=1

=

n k=1

Xik |k =

ak Xi|k

,k

k

a Xik| .

,k

Coming back to Yi =

n

X ik|k ,

k=1

let us suppose X to be antisymmetric:

X ik|j =

∂X ik ∂xj

X ik + X ki = 0. ⎧ ⎧ ⎫ ⎫ ⎨ p j ⎬ ⎨ p j ⎬ + X pk + X ip , ⎩ ⎩ ⎭ ⎭ p p i k

433

MATHEMATICAL PHYSICS

X ik|k

⎧ p k ∂X ik ⎨ = + ⎩ ∂xk p i ⎧ ⎨ p k ⎩

p,k

(if X is antisymmetric). Yi =

k

∂xk

⎭ ⎫ ⎬ ⎭

i

∂X ik

⎫ ⎬

+

X pk +

⎧ ⎫ ⎨ p k ⎬ p

⎩

X ip .

X pk = 0

⎧ ⎫ ⎨ p k ⎬ p,k

k

⎭

⎩

k

⎭

X ip .

⎧ ⎫ ⎨ p k ⎬ k

Y

i

= = =

⎩

√ 1 ∂ a . =√ ⎭ a ∂xp

k

∂X ik

√ 1 ∂ a ik +√ X ∂xk a ∂xk k √ 1 √ ∂X ik ik ∂ a √ a +X ∂xk ∂xk a k √ n 1 ∂( aX ik ) √ . ∂xk a k=1

9.5.6

Some Laws Of Transformation

For n covariant systems λα|i (i is the covariance index; α is the ordering number of the system): ∇ = |λα|i |,

∇ = |λα|i |, x1 . . . xn . D= x1 . . . xn

∇ = ∇D, dx = Sdx.

−1

λα = S ∗ λα .

P = piα , piα = λα|i , −1

P = S ∗ P,

x = S −1 x.

——————–

434

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

aik dxi dxk =

aik dxi dxk ,

dx∗ a dx = dx∗ a dx, dx∗ a dx = (S dx)∗ a S dx = dx∗ S ∗ aS dx. a = S ∗ aS,

−1

a = S ∗ aS −1 .

——————– ∇ = ∇D,

a = aD2 ,

9.5.7

∇ ∇ √ = √ . ± a ± a

ε Systems

Contravariant ε system: 1 √ a

n

S

λ1|i1 λ2|i2 · · · λn|in

i1 ...in =1

=

n

εi1 ,i2 ,...,in λ1|i1 λ2|i2 . . . λn|in = invariant,

1

εi1 ...in

is an antisymmetric contravariant tensor:

εi1 ...in =

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

if in are not all diﬀerent each other,

0 1 √ a

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎩ −√ a

if in form an even permutation of 1, 2, . . . , n, if in form an odd permutation of 1, 2, . . . , n.

Covariant ε system (it is the reciprocal of the previous one): ⎧ ⎨ √0 . . . a ... εi1 ...in = ⎩ √ − a ... ai1 k1 ai2 k2 . . . ain kn εk1 ...kn = a εi1 ...in . εi1 ...in = k1 ...kn

435

MATHEMATICAL PHYSICS

9.5.8

Vector Product

Vector product of v 1 . . . v n−1 : wi =

n

εi,i1 ...in−1 v1|i1 . . . vn−1|in−1 ,

i1 ...in1 =1

wi =

n

i

n−1 εi,i1 ...in−1 v1i1 v2i2 . . . vn−1 .

i1 ...in1 =1

0 1 v pik = 1 . .1. vn 0 v qik = 1|1 ... vn|1 1 W i = √ Q|i a √ Wi = aP|i

(Q|i is the algebraic complement of q|i ), (P|i is the algebraic complement of p|i ).

W i vr|i = 0,

(r = 1, 2, . . . , n − 1).

Wi vri = 0

1

i

9.5.9

. . . . . . 0 . . . . . . v1n , . . . . . . . . . . . . . . . vnn 0 ...... 0 v1|2 . . . . . . v1|n . . . . . . . . . . . . . vn|2 . . . . . . vn|n 0 v12 ... vn2

Extension Of A Field dV =

√

√

a dx1 dx2 . . . . . . xn .

C

√ C

a D dx1 . . . dxn .

C

a = D2 a,

√

a dx1 . . . dxn =

a dx1 . . . dxn =

√

√ a = D a.

√ C

a dx1 . . . dxn .

436

9.5.10

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

Curl Of A Vector In Three Dimensions

In general, in n dimensions the curl of a vector is the two indices antisymmetric system pi = Xi| − X|i .

Xi| =

X|i =

∂Xi − ∂x

⎧ ⎫ ⎨ i ⎬ Xp

p

⎩ ⎭ p ⎧ ⎫ ⎨ i ⎬

,

∂X − Xp . ⎩ ⎭ ∂xi p p

pi = In 3 dimensions: h

R =

∂Xi ∂X − . ∂x ∂xi 3

εhi X|i ,

i,=1

that is:

1 1 R1 = √ (X3|2 − X2|3 ) = √ p32 , a a

and analogous relations for R2 and R3 . Summing up: ⎧ ∂X3 ∂X2 1 1 ⎪ 1 ⎪ , − R = √ p32 = √ ⎪ ⎪ ⎪ ∂x3 a a ∂x2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ 1 1 ∂X1 ∂X3 2 R = √ p13 = √ − , ⎪ ∂x1 a a ∂x3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 1 ∂X2 ∂X1 ⎪ 3 ⎪ − . ⎩ R = √ p21 = √ ∂x2 a a ∂x1

9.5.11

Sections Of A Manifold. Geodesic Manifolds

Let us consider m directions λα (α = 1, 2, . . . , m). The directions ξ with parameters i

ξ =

m α=1

ρα λiα

437

MATHEMATICAL PHYSICS

and the moments ξi =

m

ρα λα|i

α=1

are deﬁned for arbitrary ρ provided that: m

ξ i ξi = 1

i=1

that is: m m

m

ρα ρβ λiα λβ|i =

α,β=1 i=1

=

α,β=1 m

ρα ρβ

m

λiα λβ|i

i=1

ρα ρβ cos(αβ) = 1.

α,β=1

The section2 G is deﬁned by means of m directions (it is a set of ∞m−1 directions). The geodesic surface of pole P is made of the geodesic curves outgoing from P along the section λ, μ. The geodesic manifold V m with m dimensions and with pole P is made of the ∞m−1 geodesic lines outgoing from P along a section Gm ; it contains ∞m points. Geodesic surfaces correspond to m = 2, while geodesic hypersurfaces to m = n − 1.

9.5.12

Geodesic Coordinates Along A Given Line xi = xi (x1 , x2 , . . . , xn ),

dy i = dxi +

xi = f1 (s). ⎫ ⎧ m ⎨ k j ⎬ k,j=i

dxi =

n =1

Si dy =

n =1

⎩

Si dxl +

i

⎭

n k,i,=1

dxk dxj . ⎧ ⎫ ⎨ k j ⎬

Si

⎩

l

⎭

dxk dxj .

2 @ The symbol G is introduced by the author in reference to the initial of the Italian word “giacitura”, which means “section”.

438

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

Si = Si (s). ∂ 2 xi

∂xi = Si , ∂x

∂xk ∂xj

=

n

⎧ ⎫ ⎨ k j ⎬ Si

=1

⎩

⎭

,

∂ 2 xi ∂Sik = , ∂xk ∂xj ∂xj ⎧ ⎫ m k j ⎬ ∂Si ⎨ = Sil , ⎩ ⎭ ∂xj =1 ⎧ ⎧ ⎫ ⎫ n n ⎨ k j ⎬ ⎨ j k ⎬ Si Si = . ⎩ ⎩ ⎭ ⎭ i=1 i=1 ∂Si ∂Sim = , ∂xm ∂x

∂Sik = ds

n ∂Sik j=i

∂xj

x˙ j =

n

⎧ ⎫ ⎨ k j ⎬ Si

i,j=1

⎩

⎭

x˙ j

(k = 1, 2, . . . , n; i = 1, 2, . . . , n). dxi ∂xi = x˙ k = Sik x˙ k . ds ∂xk

xi =

n

n

n

k=1

k=1

Sik x˙ k ds +

n

⎧ ⎫ ⎨ k j ⎬

δx δxj ⎩ ⎭ k ⎧ ⎫ ⎛ ⎞ n n n ⎨ k j ⎬ = Sik x˙ k ds + Si ⎝δx + δx δxj ⎠ ⎩ ⎭ k k=1 =1 k,j=1 k=1

Si δx +

n

=1

Si

k,j,=1

Second proof: xi

⎧ ⎫ ⎞ n ⎨ k j ⎬ 1 = pi (s) + Si (s) ⎝δx + δx δxj ⎠ ⎩ ⎭ k 2 l =1 k,j=1 + ﬁrst-order inﬁnitesimals. m

⎛

439

MATHEMATICAL PHYSICS

δxi = dxi + δ xi , dxi = x˙ i ds + 12 x ¨i ds2 , ¨i ds2 + δ xi . δxi = x˙ i ds + 12 x

(a)

xi

1 = pi (s) + Si (s) x˙ l ds + x ¨ ds2 2 =i ⎫ ⎧ ⎞ n k j ⎬ ⎨ 1 x˙ x˙ ds2 ⎠ + ⎭ k j ⎩ 2 l j=1 ⎫ ⎧ ⎛ ⎞ n n ⎨ k j ⎬ 1 δx δx ⎠ + Si (s) ⎝δ x + ⎭ k ⎩ 2 =1 k,j=1 ⎧ ⎫ n ⎨ k j ⎬ + Si (s) x˙ δx ds. ⎩ ⎭ k ,k,j=1 n

1 pi (s + ds) = pi (s) + p˙ i (s)ds + p˙i (s)ds2 , 2 Si (s + ds) = Si (s) + S˙ i (s)ds + . . . , ⎫ ⎧ ⎨ k j ⎬ ⎩

(b)

⎫ ⎧ ⎨ k j ⎬ =

⎭ s+ds

xi = pi (s + ds) +

n

⎩

+ ....

⎭ s

! Si (s + ds) δ x

⎧ =1 ⎫ ⎞ n k j ⎬ 1 ⎨ + δx δx ⎠ ⎩ ⎭ l j 2 l k,j=1 ⎫ ⎧ ⎛ ⎞ n n ⎨ k j ⎬ 1 δx δx ⎠ = pi (s) + Si (s) ⎝δ x + ⎭ k j ⎩ 2 i=1 k,j=1 n 1 + p˙i (s) ds + p¨i (s) ds2 + S˙ i (s) ds δ x . 2 =1

440 (a-b)

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS n 1 2 p˙i (s) ds + p¨i (s) ds + S˙ i (s) ds δ xl 2 =1 ⎧ ⎫ ⎛ ⎞ n n ⎨ k j ⎬ 1 1 = Si (s) ⎝x˙ ds + x ¨l ds2 + x˙ x˙ j ds2 ⎠ ⎩ ⎭ k 2 2 =1 k,j=1 ⎧ ⎫ n ⎨ k j ⎬ Si (s) x˙ δ x ds. + ⎩ ⎭ k j ,k,j=1

⎧ n ⎪ ⎪ st ⎪ 1 order: p˙i (s) ds = Si (s)x˙ ds, ⎪ ⎪ ⎪ ⎪ =1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ n ⎪ ⎪ 1 ⎪ 2 nd order: ⎪ S˙ i (s)ds δ xl p ¨ (s), ds + 2 ⎪ i ⎨ 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

⎧ ⎫ n ⎨ k j ⎬ 1 Si x ¨ (s)ds2 + Si (s) x˙ x˙ ds2 ⎩ ⎭ k j 2 =1 ,k,j=1 ⎧ ⎫ n ⎨ k j ⎬ x˙ δ xj ds. + Si (s) ⎩ ⎭ k ,k,j=1 =1

1 = 2

n

For arbitrary δ xj : ⎧ n n ⎪ ⎪ ⎪ p ˙ (s) = S (s) x ˙ = Sij (s)x˙ j i ⎪ i ⎪ ⎪ ⎪ j=1 =1 ⎪ ⎪ ⎪ ⎪ (i = 1, 2, . . . , n), ⎪ ⎪ ⎪ ⎪ ⎪ ⎧ ⎫ ⎪ ⎪ n n ⎪ k j ⎨ ⎬ ⎪ ⎪ ⎨ p¨i (s) = Si x ¨ (s) + Si (s) x˙ x˙ j ⎩ ⎭ k =1 ,k,j=1 ⎪ ⎪ ⎪ (i = 1, 2, . . . , n), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎧ ⎫ ⎪ ⎪ n ⎪ ⎨ k j ⎬ ⎪ ⎪ ⎪ Si (s) S˙ ij (s) = x˙ ⎪ ⎪ ⎪ ⎩ ⎭ k ⎪ ⎪ ,k=1 ⎪ ⎩ (i, j = 1, 2, . . . , n).

(1)

(2)

(3)

From (3), by summing over every value of i, we obtain the quantities Sij (with n2 arbitrary constants; for example they are given by the initial

441

MATHEMATICAL PHYSICS

values). By taking the derivative of (1) with respect to s and replacing Sij (s) with their expression in (3), we get (2) identically. Then, it is enough to satisfy only (1). We ﬁnd: n pi (s) = Si (s)x˙ (s)ds. =1

Since the integrals are deﬁned up to a constant, we thus have 2n2 arbitrary constants, n2 of which are trivial (additive constants). The ﬁnal formula coincides with the one already obtained above: n

xi =

Si x˙ ds +

=1

⎫ ⎧ n ⎨ k j ⎬

⎛

n

Si ⎝δx +

=1

k,j=1

⎩

⎭

⎞ δxk δxj ⎠

Si being the solutions of the n diﬀerential systems (3).

9.6.

RIEMANN’S SYMBOLS AND PROPERTIES RELATING TO CURVATURE

9.6.1

Cyclic Displacement Round An Elementary Parallelogram

xi → xi + δxi → xi + δxi + δ xi → xi + δ xi → xi , ui →

dui = −

uii

→

⎫ ⎧ n ⎨ k j ⎬ k,j=1

⎩

i

⎭

uk dxj =

Xji = −

is not a tensor). Up to

2nd

Xji dxj ,

⎫ ⎧ n ⎨ k j ⎬ k=1

(Xji

→

ui2

⎩

i

⎭

uk

order inﬁnitesimals:

ui3

→ ui4 .

442

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

δui = −

⎫ ⎧ n ⎨ k j ⎬ k,j=1

Xji = −

i 0 ⎫ ⎧ n ⎨ k j ⎬ ⎩

i ⎧ n ⎨

k=1

+

k,,m=1

Δui =

" n j=1 P1

=

(uk )0 δxj ,

⎭

⎩

⎩

⎫⎤ ⎧ ⎨ k j ⎬ ∂ ⎦ (ux )0 δx ⎣ (uk )0 − ⎭ ∂x ⎩ i ⎭ k,=1 0⎫ ⎧ 0 ⎫ k j ⎬ ⎨ m ⎬ (um )0 δx . ⎭ ⎩ ⎭ i k 0 0

Xji dxj =

P1

P P3

n

n j=1

⎡

Xji dxj +

P2

P1 P2

P4 =P

... + P2

P4 =P

P3

... +

... P3

... + ... + ... + ... P1 P3 ⎧ P2 ⎫ k j ⎬ ∂ ⎨ uk δx dxj = ∂x ⎩ i ⎭ k,j,=1 ⎫ ⎫⎧ ⎧ n ⎨ k j ⎬⎨ m ⎬ um δx dxj − ⎭ ⎭⎩ ⎩ k i k,,m,j=1 ⎫ ⎧ n k j ⎬ ∂ ⎨ uk dx δxj − ∂x ⎩ i ⎭ k,j,=1 ⎧ ⎫⎧ ⎫ n ⎨ k j ⎬⎨ m ⎬ + um dx δxj . ⎩ ⎭⎩ ⎭ i k k,,m,j=1 ⎫ ⎫⎧ ⎡ ⎛⎧ n n ⎨ i k ⎬⎨ p h ⎬ ⎝ ui dxk δxk ⎣ Δur = − ⎭ ⎭⎩ ⎩ r p p=1 i,h,k=1 ⎫ ⎫⎧ ⎧ ⎨ i h ⎬⎨ p k ⎬ − ⎭ ⎭⎩ ⎩ r p ⎧ ⎫ ⎧ ⎫⎞ ⎤ i k ⎬ i h ⎬ ∂ ⎨ ∂ ⎨ ⎠⎦ . + − ⎭ ∂xk ⎩ r ⎭ ∂xk ⎩ r P n

443

MATHEMATICAL PHYSICS

r

Δu = +

n

{ir, hk}ui dxh δxk

i,h,k=1

which is the Riemann curvature.

9.6.2

Fundamental Properties Of Riemann’S Symbols Of The Second Kind

⎧ ⎧ ⎫ ⎫ i k ⎬ i h ⎬ ⎨ ⎨ ∂ ∂ {ir, hk} = − + ∂xh ⎩ r ⎭ ∂xk ⎩ r ⎭ ⎫⎧ ⎫ ⎧ ⎫⎧ ⎫⎤ ⎡⎧ n ⎨ p h ⎬⎨ i k ⎬ ⎨ p k ⎬⎨ i h ⎬ ⎣ ⎦ − − ⎩ ⎭⎩ ⎭ ⎩ ⎭⎩ ⎭ r p r p p=1 Properties of Riemann’s symbols of the second kind: (covariance with respect to the indices i, h, k, contravariance with respect to the index r)

1)

{ir, hk} = arihk ,

2)

{ir, hk} = −{ir, kh},

3)

{ir, hk} + {hr, ki} + {kr, ih} = 0 .

Up to 2nd order inﬁnitesimals:

{1 1, 1 2} =

{2 1, 1 2} =

{1 2, 1 2} =

{2 2, 1 2} =

⎧ 1 2 ∂ ⎨ ∂x1 ⎩ 1 ⎧ 2 2 ∂ ⎨ ∂x1 ⎩ 1 ⎧ 1 2 ∂ ⎨ ∂x1 ⎩ 2 ⎧ 2 2 ∂ ⎨ ∂x1 ⎩ 2

⎫ ⎬ ⎭ ⎫ ⎬ ⎭ ⎫ ⎬ ⎭ ⎫ ⎬ ⎭

−

−

−

−

⎧ 1 1 ∂ ⎨ ∂x2 ⎩ 1 ⎧ 2 1 ∂ ⎨ ∂x2 ⎩ 1 ⎧ 1 1 ∂ ⎨ ∂x2 ⎩ 2 ⎧ 2 1 ∂ ⎨ ∂x2 ⎩ 2

⎫ ⎬ ⎭ ⎫ ⎬

= 0,

⎭ ⎫ ⎬

=

2 1 + = 1, 3 3

1 2 = − − = −1, ⎭ 3 3 ⎫ ⎬ = 0. ⎭

444

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

Δu = − r

n

{ir, hk}ui δxh δ xk .

i,h,k=1

[3 ] Δu1 = −u2 (δx1 δ x2 − δx2 δ x1 ), Δu2 = u1 (δx1 δ x2 − δx2 δ x1 ).

r=1: r=2:

9.6.3

Fundamental Properties And Number Of Riemann’s Symbols Of The First Kind

(ij, hk) =

n

ajr {ir, hk}

r=1

⎫ ⎫ ⎧ ⎧ n i k ⎬ i h ⎬ ⎨ ⎨ ∂ ∂ + ajr ajr = − ⎭ ∂xh ⎩ r ∂xk ⎩ r ⎭ r=1 r=1 ⎫ ⎫⎧ ⎡⎧ n ⎨ p h ⎬⎨ i k ⎬ − ajr ⎣ ⎭ ⎭⎩ ⎩ p r p,r=1 ⎧ ⎫⎧ ⎫⎤ ⎨ p k ⎬⎨ i h ⎬ ⎦ − ⎩ ⎭⎩ ⎭ r p ⎫ ⎧ ⎡ ⎤ n i k ⎨ i k ⎬ ∂a ∂ ⎣ jr ⎦+ = − ∂xh ∂x h ⎩ r ⎭ j r=1 ⎫ ⎧ ⎤ ⎡ n i h ⎬ i h ⎨ ∂ajr ∂ ⎣ ⎦− + ∂xk ∂xk ⎩ r ⎭ j r=1 ⎫⎞ ⎫ ⎡ ⎤⎧ ⎤⎧ ⎛⎡ n p k ⎨ i h ⎬ p h ⎨ i k ⎬ ⎠. ⎦ ⎦ ⎝⎣ −⎣ − ⎭ ⎩ ⎭ ⎩ p j p j rp=1 n

3@

In the original manuscript, the following note appears: Change the sign of Riemann’s symbols. Also, the following is pointed out, referring to equations reported in Levi-Civita I: Notes on the Tallis formulae: Eq. (3), p.201 is correct; Eq. (4), p.201, change the sign; Eq. (26), p.219 is correct.

445

MATHEMATICAL PHYSICS

Since:

⎧ ⎫ n i k ⎬ ⎨ ∂ajr r=1

∂xh ⎩ r

⎭

=

n

⎡ ⎣

p h j

p=1

⎫ ⎫ ⎡ ⎤⎧ ⎤⎧ n j h ⎨ i k ⎬ ⎨ i k ⎬ ⎣ ⎦ ⎦ + ⎩ ⎩ ⎭ ⎭ p p p p=1

etc., we ﬁnally have: ⎡ ⎤ ⎡ ⎤ i k i h ∂ ⎣ ∂ ⎦+ ⎣ ⎦ (ij, hk) = − ∂xh ∂x k j j

+

n

⎛⎡ ⎝⎣

j h p

p=1

⎫ ⎡ ⎫⎞ ⎤⎧ ⎤ ⎧ j k ⎨ i k ⎬ ⎨ i h ⎬ ⎦ ⎦+ ⎠. −⎣ ⎩ ⎭ ⎩ ⎭ p p p

Properties of Riemann’s symbols of the ﬁrst kind: 1)

covariance with respect to every index,

2)

(ij, hk) = −(ij, kh),

3)

(ij, hk) = −(ji, hk).

In fact:

⎡

i k

∂ ⎣ ∂xh

j

⎤

⎡

⎦− ∂ ⎣ ∂xk

∂ 2 ajk

⎤

i h

⎦

j

prt2 aik ∂ 2 aih − − + ∂xi ∂xh ∂xi ∂xk ∂xj ∂xh ∂xj ∂xk ⎛ ⎡ ⎡ ⎤ ⎤⎞ j k j h ∂ ∂ ⎣ ⎣ ⎦− ⎦⎠ , = −⎝ ∂xh ∂xk i i 1 = 2

etc.; n p=1

⎡ ⎣

j h p

∂ 2 aih

⎫ ⎤⎧ ⎨ i k ⎬ ⎦ = ⎩ ⎭ p

n

⎡ apq ⎣

j h

⎤⎡ ⎦⎣

p ⎤⎧ n i k ⎨ j h ⎣ ⎦ = ⎩ p p p=1 p,q=1

⎡

i k q ⎫ ⎬ ⎭

,

⎤ ⎦

446

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

etc. 4)

(ij, hk) + (hj, ki) + (kj, ih) = 0,

5)

(ij, hk) + (ih, kj) + (ik, jh) = 0,

6)

(ij, hk) = (hk, ij) .

In fact:

⎡ ∂ ⎣ ∂xh

i k j

⎤

⎡

⎦− ∂ ⎣ ∂xk

⎤ ⎦

j

∂ 2 ajh ∂ 2 aih − + + ∂xi ∂xh ∂xj ∂xh ∂xi ∂xk ∂xj ∂xk ⎡ ⎡ ⎤ ⎤ h j h i ∂ ⎣ ⎦− ∂ ⎣ ⎦ = ∂xi ∂xj k k 1 = 2

∂ 2 ajk

i h

∂ 2 aik

etc.; for the remaining proof, see property 3). 7) Number of the independent symbols of ﬁrst kind. Given the indices i, j, h, k, irrespectively of their ordering, we have two independent symbols if all the indices are diﬀerent from each other; one independent symbol if three indices are diﬀerent and the fourth is equal to one of them; one independent symbol if we have two pairs of diﬀerent symbols; no non-vanishing symbol in the other cases. Thus the total number of independent symbols results to be:

2

n(n − 1)(n − 2) n(n − 1) n(n − 1)(n − 2)(n − 3) +3 + 24 6 2 n(n − 1) n2 (n2 − 1) = [(n − 2)(n − 3) + 6(n − 2) + 6] = . 12 12 n 1 2 3 4 5

n2 (n2 − 1) 12 0 1 6 20 50

447

MATHEMATICAL PHYSICS

9.6.4

Bianchi Identity And Ricci Lemma

The Bianchi identity for the covariant derivatives of the Riemann’s symbols is: {ir, hk} + {ir, k}h + {ir, k}k = Arihk = 0. It can be easily veriﬁed by performing the covariant derivatives in locally cartesian coordinates. The same holds for the Ricci lemma: (ij, hk) + (ij, k)h + (ij, h)k = 0.

9.6.5

Tangent Geodesic Coordinates Around The Point P0 xi = (λi )0 s,

s

λi ds

xi = 0

(λi are evaluated in the point P0 ; in order to have geodesic coordinates in P it is enough that the formula holds up to s2 terms, as we certainly assume). ⎫ ⎧ n ⎨ k j ⎬ x˙ x˙ = 0. x ¨i = ⎭ k j ⎩ i k,j=1 ⎧ ⎫ n ⎨ k j ⎬ x ¨i = − x˙ x˙ , ⎩ ⎭ k j i k,j=1 ⎫ ⎫ ⎧ ⎧ n n ⎨ k j ⎬ k j ⎬ ⎨ d ... xi = − x˙ k x˙ j x˙ r − 2 x˙ x ¨ ⎭ k j ⎩ dxr ⎩ i ⎭ i k,j=1 k,j=1 ⎫ ⎧ n k j ⎬ d ⎨ x˙ k x˙ j x˙ r = − ∂xr ⎩ i ⎭ k,j,r=1 ⎧ ⎫⎧ ⎫ n ⎨ k j ⎬⎨ r s ⎬ +2 x˙ x˙ x˙ . ⎩ ⎭⎩ ⎭ k r s i j k,j,r,s=1

448

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

At a point P0 : x˙ r = λr ,

x ¨r = −

⎧ ⎫ n ⎨ h k ⎬ h,k=1

⎩

r

⎭

λh λk ,

⎧ ⎧ ⎫ ⎫⎧ ⎫⎤ n ⎨ i p ⎬⎨ h k ⎬ h k ⎬ ⎨ ∂ ... ⎦. xr = x˙ i x˙ h x˙ k ⎣− +2 ⎩ ⎭⎩ ⎭ ∂xi ⎩ r ⎭ r p p=1 i,h,k=1 n

⎡

INDEX Acetylene vibration modes, 278 Action for the electromagnetic ﬁeld, 57 Alkali s terms, 190 polarization forces, 205 α particle, 364, 367 Angular metric, 416 Angular momentum for the electromagnetic ﬁeld, 78 Associated vectors, 424, 428 Atomic spectra complex atoms, 219, 223 hyperﬁne structure, 239 hyperﬁne structures, 211 Atomic wavefunction, 136, 197, 201 Atom one-electron magnetic moment, 229 Atom three-electron ground state, 183 two-electron, 125, 133, 136 1s1s term, 170 1s2s term, 174 2p2s term, 169 2s2p term, 155, 158 2s2s term, 169 2s terms, 144 X term, 153, 159, 179 Y term, 153, 179 energy levels, 144 self-consistent ﬁeld, 141 β particles traversing a medium, 368 Bianchi identity, 447 Bose-Einstein commutation relations, 94 Center-of-mass, 347 Christoﬀel’s symbols, 410–411 Complete systems of diﬀerential operators, 406 Compton eﬀect, 331 Coordinates locally cartesian, 447 Coulomb ﬁeld, 318, 324 screening factor, 198 Covariance index, 433 Covariant diﬀerentiation, 428 Cross section two-electron scattering, 330 Cunningham corrections to the Stokes’ law, 396 Curie point, 299

Curl of a vector, 436 Delta-function, 317 Derivative covariant, 428 Determination of e, 390 Determination of e/m, 387–388 Deuterium, 363, 365 Diﬀerential forms, 403 Diﬀerential operators complete systems, 406 Jacobian systems, 407 linear, 404 Dirac coordinates, 104, 339, Dirac equation, 25 16-component spinors, 48 4-component spinors, 47 5-component spinors, 55 6-component spinors, 48 non-relativistic approximation, 242 Dirac ﬁeld angular momentum, 40 electromagnetic interaction, 25 Hamiltonian, 46 interacting with the electromagnetic ﬁeld, 45 normal mode decomposition, 31 plane wave expansion, 44 quantization, 22 real, 35, 45 Dirac operators particular representations, 32 Divergence of a tensor, 432 Divergence of a vector, 431 Electric charge (determination) Millikan’s method, 396 Thomson’s method, 395 Townsend’s method, 394 Wilson’s method, 396 Zaliny’s method, 394 Electromagnetic and electrostatic mass, 397 Electromagnetic ﬁeld analogy with the Dirac ﬁeld, 59, 66 Dirac formalism, 68 Hamiltonian, 58 interacting with bound electrons, 112 interacting with electrons, 84 Lagrangian, 57 plane wave operators, 64 quantization, 71, 78, 82, 84, 95, 100 retarded, 116 total energy, 58 Electron, 397 bound, 112

449

450

E. MAJORANA: RESEARCH NOTES ON THEORETICAL PHYSICS

interaction with the electromagnetic ﬁeld, 84, 112 semiclassical theory, 4 Electron wavefunction, 112 Elliptic coordinates, 261 Exchange energy, 223, 234, 290 tables, 204 Exchange forces nuclear, 340 Extraction work, 398 Fall velocity, 396 Fermi-Dirac commutation relations, 94 Ferromagnetism, 289, 300, 307 Field extension, 435 Gas degenerate, 287, 352 Gauge invariance, 30 Geodesic coordinates, 425, 437 tangent, 447 Geodesic curvature, 428 Geodesic lines, 418, 428 autoparallelism, 424 Geodesic manifolds, 436 Geodesic surface, 437 Goudsmith method, 213 Ground state three-electron atom, 183 two-electron atom, 125 Hamiltonian formalism, 37, 43 Helium atomic wavefunction, 136 composed of two deuterium nuclei, 340 ionization energy, 128–129 molecule, 261 nuclear potential, 340 Houston formula, 213 Hydrogen, 329 Hydrogen atom, 327 Hyperﬁne structures, 246, 251 Ionization energy for a two-electron atom, 129 for a two electron atom, 128 Jacobian systems of diﬀerential operators, 407 j-j coupling, 214 Klein-Gordon equation, 7, 84, 370 Land´ e formula, 211 Langmuir experiment, 399 Lithium, 201 electrostatic potential, 184 ground state, 185 Lorentz transformations for the photon wavefunction, 70 Magnetic charges, 119 Magnetic moment, 298 atomic, 247, 251 for a one-electron atom, 229

nuclear, 247 Magnetic moments diagonal, 114 Maxwell-Dirac theory, 29 Maxwell distribution, 398 Maxwell equations, 27 variational approach, 28 Method of electrolysis (determination of e), 394 Metric indeﬁnite, 425 Millikan’s method (determination of e), 396 Minimum approach distance, 324, 329 Mobility coeﬃcients, 394 Molecules vibration modes, 275 Neutron-proton interaction, 340 Neutron susceptivity, 339 wave equation, 339 Nuclear magnetic moment, 247 Nuclear potential, 340 Nuclei scalar ﬁeld theory, 370 Nucleon density, 345 Nucleon interaction, 347, 352 interaction potential, 340, 345 kinetic energy, 345 Parallel displacement, 422 Parallelism, 427 symbolic equations, 409 Paramagnetism, 288 Partial wave method, 319 Pauli matrices, 3, 7 Perturbation method for a two-electron atom, 125, 157 scattering, 316–317 Phase advancement, 323 Photon wave equation, 100 Plane waves, 82 Poisson brackets, 408 Polarization forces, 205 Potential between nucleons, 340, 345 nuclear, 340 Potential well, 311 P triplets, 233 Quasi-stationary states, 332 Radioactivity tables, 339 Radions, 293 Reﬂecting power, 315 Relativistic kinematics, 330–331 Resonance between = 1 and electrons, 223

INDEX in the two-electron scattering, 330 Retarded ﬁelds, 116 Ricci lemma, 430, 447 Richardson formula, 398 Riemann curvature, 443 Riemann’s symbols ﬁrst kind, 444 second kind, 443 Russell-Saunders coupling, 214 Rutherford formula, 324, 329 Rydberg corrections, 212 relativistic, 239 Saturation current, 392, 398 Scattering between two nuclei, 340 Born method, 319 bound electron, 112 coherent, 112 Compton, 331 Coulomb, 321, 328 Dirac method, 318 Dirac method, 317 free electron, 104 from a potential well, 311 intensity, 324, 329 method of the particular solutions, 327 quasi coulombian, 324 resonant, 113 screened Coulomb, 197 simple perturbation method, 316 transition probability, 318 two-electron, 330 Schr¨ odinger equation, 325, 329 for a Coulomb ﬁeld, 321 Screening factor, 198 Slater determinants, 307 Space charge, 399 Spin-orbit coupling, 233 Spin function, 108 Stokes law, 396 Surface waves, 385

451 Susceptibility for a one-electron atom, 229 magnetic, 288 Susceptivity atomic, 209 for the neutron, 339 Symmetrization for fermion ﬁelds, 35 Tallis formulae, 444 Thermionic eﬀect, 397 Thomson formula β particles, 368 Thomson’s method (determination of e), 395 Thomson’s method (determination of e/m), 387 Three-fermion system, 282 Time delay constant, 118 Townsend coeﬃcient in air, 392 Townsend eﬀect, 390 Townsend relation, 393 Transformation laws for covariant systems, 433 Transition probability, 318 Triplets P , 233 Two-particle system Dirac equation, 242 ε systems, 434 Variational method, 126 for a two-electron atom, 128 Vector product, 435 Vector cyclic displacement, 441 Vibrating string, 3 Vibration modes in molecules, 275 Wave equation for the photon, 100 Wavefunction alkali atoms, 190 two-electron atom, 133 Wien’s method (determination of e/m), 388 Wilson’s method (determination of e), 396 Zaliny’s method (determination of e), 394