Table Of Content

Definition of some units Angstrom I A = 10-10 m (order of magnitude of the atomic dimensiQns) Fermi I F = 10-1 5 m (order of magnitude of the nuclear dimensions) Bam 1 b = 10-28 m2 = (10-4 A)2 = (10 F)2 Electron Volt l eV = 1.602189(5) X w-19 joule Useful orders of magnitude Electron rest energy : mec2 0.5 MeV [0.511 003(1) x 106 eV] :::::: Proton rest energy : MPc2 I 000 MeV [938.280(3) x 106 eV] :::::: { Neutron rest energy : Mnc2 I 000 MeV [939.573(3) x 106 eY] :::::: One electron volt corresponds to : = a frequency v :::::: 2.4 x 1014 Hz through the relationE hv [2.417 970(7) 1014 Hz) X a wavelength A :::::: 12 000 A through the relation A = cfv [12 398.52(4) A] a wave number 1l ~ 8 000 em-1 [8 065.48 (2) em -l] a temperature T :::::: 12 000 K through the relation E = kaT [11 604.5(4) K] In a l gauss magnetic field (I0-4 Testa): the electron cyclotron frequency vc = roc/21t = - qB/21tlne is vc ~ 2.8 MHz [2.799 225(8) x 106 Hz] the orbital Larmor frequency = u)£/21t = - !J B/h = Vc/2 "L 8 is vL:::::: l.4 MHz [1.399 612(4) x 106 Hz] (this corresponds by definition to a g = J Lande factor) Some general physical constants h - h 6.626 18(4) x 10-34 joule second Planck's constant { fl 2n = 1.054 589(6) 10-34 joule second := X Speed of light (in vacuum) c - 2.997 924 58(1) x 108 m/s Electron charge q = - 1.602 189(5) x 10-19 coulomb Electron mass m = 9.109 53(5) X 10-31 kg 11 Proton mass M, = 1.672 65(1) x to-27 kg Neutron mass Mn = 1.674 95(1) x 10-27 kg & = 1 836.1515(7) me Ac = h/mec = 2.426 309(4) w-2 A X Ele c t ron C ompt on waveI e ngt h { !c = 1i/mec = 3.861 591(7) X 10-3 A Fine structure constant 2 e2 1 (dimensionless) IX= q - = 47teJic - Tic 137.036 0(1) ! Bohr radius a = .....s. = 0.529 177 1(5) A 0 IX Hydrogen atom ionization energy (without proton recoil - Ei"' = IX2mc2/2 = 13.605 80(5) eV 11 effect} Rydberg's constant 2 = = "Classical" electron radius re q 2.817 938(7) fermi 41te m c2 0 11 Bohr magneton /Js = qflj2m =- 9.27408(4) X 10-24joulejtesla 11 Electron spin g factor (} = 2 X 1.001159 657(4) 11 Nuclear magneton /Jn = - q11/2M, = 5.05082(2) X w-27 joulejtesla Boltzmann's constant ks = 1.380 66(4) x 10-23 joule/K Avogadro's number N A = 6.022 05(3) X 1023 Useful Identities U : scalar field ; A, B, . . . : vector fields. v X (VU) = 0 V. (VU) = 11U v. ( v X A)= 0 Vx (Vx A)=V(V.A)-/1A 1i L=-rxV i a r i V=----rxL r or 1ir2 1 iP L2 11 =--r -- r or2 1i2r2 A X (B X C) = (A. C)B - (A. B)C c A X (B X C) + B X (C X A) + X (A X B) = 0 (A x B). (C x D) = (A. C)(B. D) - (A. D)(B. C) (A X B) X (C X D) = [(A X B). D]C - [(A X B). C]D = [(C X D). A]B - [(C X D). B]A V (UV) = uvv + v vu 11 ( UV) = U 11V + 2(VU). (VV) + V 11U V. (VA) =UV.A+A.VU V x ( U A) = U V x A + (VU ) x A V. (A x B) = B. (V x A)- A. (Vx B) v v v v v (A. B) =A X ( X B)+ B X ( X A)+ B. A+ A. B V x (A x B)= A (V. B)- B (V. A)+ B. V A- A. V B N.B.: B.V A vector field whose components are : a ox. (B .V A);= Bl\A; = ~ Bi A; J J (i = x, y, z) Coordinate systems Cartesian Cylindrical Spherical r :: ;: Le, ""e: ""e, M M~/~e., '-~/<.# .. e, ~j, •. ___ b.--~ -~ 3{~ ~~-~~--+ _/ :' v Jo-}' ~" x/ -"/~ . /"/ cp ,-~i y ///- p - /~/ ~ y£ "\, J' x¥' (z/) U = U(x. y, z) U = U(p, cp, z) U = U(r. 9, cp) 0 A = A:re:r + A1e, + A,e. A = A,e, + A.,e., + A.e. A = A,e, + A9e6 + A.,e., Ez Ax = A:r(x, y, z) A, = Ax cos cp + A, sin cp A, = A, sin 9 + A, cos 9 A, = A,(x, y, z) A., = - A" sin cp + A, cos cp A = A, cos (} - A, sin 9 w~ 6 0 A. = A=(x. y, z) A., = - A, sin (/) + A cos cp 1 1z- vu = (oUfox)e" (VU), = oUjop (VU), = au;ar w 0 + (outcr)ey (VU)., = [cU/ocp]!p (V U) = [ oUjo9]/r 9 <( a: + (oUfcz)e= (V U). = cUte: (VU)., = [cUiccp];(r sin 0) c:J ~<i~:z AUc2=U -c+2U- +c2-u ~V=-1 -c.: ( pa-u) +1 -l 2-U +c2-U u_,v,, =1- 2-2 ( r.'.'; ) +-1 --<, (s.m 0-cu) +-1- -<~2U <(U ilx2 cy2 CZ2 p cp cp Pz i':cpz cz2 r cr2 r2 sin 1:1 cO NJ, r2 sin2 8 i"cp2 ...J 'w a>Qw:cwuz:J V . A =c-eA+x, __oi'Ay. , _+ccAz- . v. A=--pI: :L;cP- (pAP ) + -pl -i'(:~ Acp-, . + -cc,Az- . V. A= -rl2 - cc(r r2A') +r- s-i1n - 0( cs<O~ in OAt, ) + _r siin_ 0 . :2:c..A:c..:p. .! l. ...J V x A = (cA=:<'r - ?Ar ·r~:)e_, 1v x A), = I<'A=;ilcpl!P - cA'I'Ic: ( v X A), = [o(sin 0 A,.)icO- oAefocp]/(r sin 0) :a::>: -i- (cA,;c: - cA=,cx)e). 1'v v x A)., = (~Ap_·c: - cA •. cp ( V x A}g = [cA,,<'cp -sin (J i'(rA,.)/tr],(r sin (J) u + (cA, ex - rA, r'y)e, x A), = [clpA.,)/op - <~Ap,ccp].p ( v X A),. = [c(rAei cr - tA,,cO]r Introduction Structure and level of this text It is hardly necessary to emphasize the importance of quantum mechanics in modern physics and chemistry. Current university programs naturally reflect this importance. In French universities, for example, an essentially qualitative introduction to fundamental quantum mechanical ideas is given in the second year. In the final year of the undergraduate physics program, basic quantum mechanics and its most important applications are studied in detail. This book is the direct result of several years of teaching quantum mechanics in the final year of the undergraduate program, first in two parallel courses at the Faculte des Sciences in Paris and then at the Universites Paris VI and Paris VII. We felt it to be important to mark a clear separation, in the structure of this book, between the two different but complementary aspects (lectures and recitations) of the courses given during this time. This is why we have divided this text into two distinct parts (see the "Directions for Use" at the beginning of the book). On the one hand, the chapters are based on the lectures given in the two courses, which we compared, discussed and expanded before writing the final version. On the other hand, the "complements" grew out of the recitations, exercises and problems given to the students, and reports that some of them prepared. Ideas also came from other courses given under other circumstances or at other levels (particularly in the graduate programs). As we pointed out in the "Directions for Use", the chapters as a whole constitute, more or less, a course we would envisage teaching to fourth-year college students or those whose level is equivalent. However, the complements are not intended to be treated in a single year. The reader, teacher or student, must choose between them in accordance with his interests, tastes and goals. Throughout the writing of this book, our constant concern has been to address ourselves to students majoring in physics, like those we have taught over the past several years. Except in a few complements, we have not overstepped those limits. In addition, we have endeavored to take into account what we have seen of students' difficulties in understanding and assimilating quantum 3 INTRODUCTION mechanics, as well as their questions. We hope, of course, that this book will also be of use to other readers such as graduate students, beginning research workers and secondary school teachers. The reader is not required to be familiar with quantum physics : few of our students were. However, we do think that the quantum mechanics course we propose (see "General approach", below) should be supplemented by a more descriptive and more experimentally oriented course, in atomic physics for example. General approach We feel that familiarity with quantum mechanics can best be acquired by using it to solve specific problems. We therefore introduce the postulates of quantum mechanics very early (in chapter III), so as to be able to apply them in the rest of the book. Our teaching experience has shown it to be preferable to introduce all the postulates together in the beginning rather than presenting them in several stages. Similarly, we have chosen to use state spaces and Dirac notation from the very beginning. This avoids the useless repetition which results from presenting the more general bra-ket formalism only after having developed wave mechanics uniquely in terms of wave functions. In addition, a belated change in the notation runs the risk of confusing the student, and casting doubts on concepts which he has only just acquired and not yet completely assimilated. After a chapter of qualitative introduction to quantum mechanical ideas, which uses simple optical analogies to familiarize the reader with these new concepts, we present, in a systematic fashion, the mathematical tools (chapter fl) and postulates of quantum mechanics as well as a discussion of their physical content (chapter III). This enables the reader, from the beginning, to have an overall view of the physical consequences of the new postulates. Starting with the complements of chapter III we take up applications, beginning with the simplest ones (two-level systems, the harmonic oscillator, etc.) and becoming gradually more complicated (the hydrogen atom, approximation methods, etc.). Our intention is to provide illustrations of quantum mechanics by taking many examples from different fields such as atomic physics, molecular physics and solid state physics. In these examples we concentrate on the quantum mechanical aspect of the phenomena, often neglecting specific details which are treated in more specialized texts. Whenever possible, the quantum mechanical results are compared with the classical ones in order to help the reader develop his intuition concerning quantum mechanical effects. This essentially deductive viewpoint has led us to avoid stressing the historical introduction of quantum mechanical ideas, that is, the presentation and discussion of experimental facts which force us to reject the classical ideas. We have thus had to forego the inductive approach, which is nevertheless needed if physics is to be faithfully portrayed as a science in continual evolution, provoked by constant confrontation with experimental facts. Such an approach seems to us to be better suited to an atomic physics text or an introductory quantum physics course on a more elementary level. Similarly, we have deliberately avoided any discussion of the philisophical 4 INTRODUCTION implications of quantum mechanics and of alternative interpretations that have been proposed. Such discussions, while very interesting (see section 5 of the biblio graphy), seem to us to belong on another level. We feel that these questions can be fruitfully considered only after one has mastered the "orthodox" quantum theory whose impressive successes in all fields of physics and chemistry compelled its acceptance. Acknowledgements The teaching experiences out of which this text grew were group efforts, pursued over several years. We want to thank all the members of the various groups and particularly, Jacques Dupont-Roc and Serge Haroche, for their friendly collaboration, for the fruitful discussions we have had in our weekly meetings and for the ideas for problems and exercises which they have suggested. Without their enthusiasm and their valuable help, we would never have been able to undertake and carry out the writing of this book. Nor can we forget what we owe to the physicists who introduced us to research, Alfred Kastler and Jean Brossel for two of us and Maurice Levy for the third. It was in the context of their laboratories that we discovered the beauty and power of quantum mechanics. Neither have we forgotten the importance to us of the modern physics taught at the C.E.A. by Albert Messiah, Claude Bloch and Anatole Abragam, at a time when graduate studies were not yet incorporated into French university programs. We wish to express our gratitude to Ms. Aucher, Baudrit, Boy, Brodschi, Emo, Heyvaerts, Lemirre, Touzeau for preparation of the manuscript. 5 Foreword This book is essentially a translation of the French edition which appeared at the end of 1973. The text has undergone a certain number of modifications. The most important one is an addition of a detailed bibliography, with suggestions concerning its use appearing at the end of each chapter or complements. This book was originally conceived for French students finishing their undergraduate studies or beginning their research work. It seems to us however that the structure of this book (separation into chapters and complements - see the "Directions for use") should make it suitable for other groups of readers. For example, for an undergraduate elementary Quantum Mechanics course, we would recommend using the most important chapters with their simplest complements. For a more advanced course, one could add the remaining chapters and use more difficult complements. Finally, it is hoped that some of the more advanced complements will help students in the transition from a regular Quantum Mechanics course to current research topics in various fields of Physics. We wish to thank Nicole and Dan Ostrowsky, as well as Susan Hemley, for the care and enthusiasm which they brought to this translation. Their remarks often led to an improvement of the original text. In addition, we are grateful to Mrs. Audoin and Mrs. Mathieu for their aid in organizing the bibliography. C. Cohen-Tannoudji B. Diu F. Laloe 6 Waves and particles. Introduction to the fundamental ideas of quantum mechanics