Quantum Mechanics www.com The Ecole Polytechnique, one of France’s top academic institutions, has a longstand- ing tradition of producing exceptional scientific textbooks for its students. The origi- nal lecture notes, the Cours de l’Ecole Polytechnique, which were written by Cauchy and Jordan in the nineteenth century, are considered to be landmarks in the develop- ment of mathematics. The present series of textbooks is remarkable in that the texts incorporate the most recent scientific advances in courses designed to provide undergraduate students with the foundations of a scientific discipline. An outstanding level of quality is achieved in each of the seven scientific fields taught at the Ecole: pure and applied mathe- matics, mechanics, physics, chemistry, biology, and economics.
The uniform level of excellence is the result of the unique selection of academic staff there which in- cludes, in addition to the best researchers in its own renowned laboratories, a large number of world-famous scientists, appointed as part-time professors or associate professors, who work in the most advanced research centers France has in each field. Another distinctive characteristic of these courses is their overall consistency; each course makes appropriate use of relevant concepts introduced in the other textbooks. This is because each student at the Ecole Polytechnique has to acquire basic knowl- edge in the seven scientific fields taught there, so a substantial link between depart- ments is necessary. The distribution of these courses used to be restricted to the 900 students at the Ecole.
Some years ago we were very successful in making these courses available to a larger French-reading audience. We now build on this success by making these textbooks also available in English.com Jean-Louis Basdevant Jean Dalibard Quantum Mechanics Including a CD-ROM by Manuel Joffre With 84 Figures and 92 Exercises with Solutions ABC www.com Professor Jean-Louis Basdevant Professor Jean Dalibard École Polytechnique École Normale Supérieure Département de Physique Département de Physique Laboratoire Leprince-Ringuet Laboratoire Kastler Brossel 91128 Palaiseau, France 24, rue Lhomond E-mail: 75231 Paris Cedex 05, France jean-louis.edu Email: jean. Manuel Joffre École Polytechnique Laboratoire d’Optique et Biosciences 91128 Palaiseau, France E-mail:manuel.edu Cover Figure: Shows a schematic drawing of a Young double slit interference experiment performed with ultracold atoms(drawing by the authors); see also Chap.5 Corrected Second Printing 2005 First Edition 2002 Library of Congress Control Number: 2005929876 ISBN-10 3-540-27706-4 Springer Berlin Heidelberg New York ISBN-13 978-3-540-27706-4 Springer Berlin Heidelberg New York ISBN 3-540-42739-2 c 2002 published in the former Springer series Advanced Texts in Physics This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks.
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Typesetting: by the authors and F. Herweg, EDV Beratung, using a Springer LATEX macro package Cover design: Erich Kirchner, Heidelberg Printed on acid-free paper SPIN: 11525127 56/3141/jl 543210 www.com Preface Felix qui potuit rerum cognoscere causas (Lucky are those who have been able to understand the causes of things. Uderzo, Asterix in Corsica, 1973, page 22; see also: Virgil, Georgics II Quantum mechanics has the unexpected feature that there is as yet no em- pirical evidence that it has limited applicability. The only hypothetical in- dication that some “new physics” might exist comes from cosmology, and concerns the first 10−43 s of the universe.
This is quite unlike the situation for other physical theories. Quantum physics was born at the beginning of the 20th century from the questioning of physicists faced with an incredi- ble variety of experimental facts which were steadily accumulating without any global explanation. This questioning was amazingly ambitious and fruit- ful. In fact, quantum theory is undoubtedly one of the greatest intellectual endeavors of mankind, perhaps the greatest of the 20th century.
It was born in an unexpected way. At the beginning of the 19th century, the sagacious French philosopher Auguste Comte claimed that one could never know the chemical composition of stars since it was impossible for us to visit them. Had he thought that the same remark could apply just as well to a hot oven, he would have described unintentionally, and by pure reasoning, the cradle of quantum physics. Quantum physics appeared fortuitously in an idea of Planck about the black-body radiation spectrum, which was acknowledged to be a fundamen- tal problem.
Quantum physics first developed by disentangling spectroscopic data. In that sense it owes much to astrophysics, which was developing at the same time, and revealed the complex spectra of elements. The phenom- enological analysis of the regularities of spectra (by Balmer, Rydberg, Ritz and Rayleigh) had led to a set of efficient recipes. But there was no indica- tion that this scrupulous classification would lead to such an upheaval of the foundations of physics.
In fact, the fate of quantum physics was unexpected. It started by ex- plaining the laws of radiation, and no one could have imagined that it would end up giving a complete explanation of the structure of matter, of atoms and molecules. Atomic theory ceased to be a qualitative controversy. It became a www.com VI Preface fact, and this struck the minds of people.
In an article published in 1948 en- titled “2400 years of quantum mechanics”, Schrödinger said that Democritus and the inventors of atomism were the “first quantum physicists”.1 He paid tribute to all those who had tried to understand the fundamental structure of matter. This had been difficult for many reasons. The Catholic Church, for instance, remained strongly opposed to the idea for a long time since atoms do not have souls. Even Leibniz thought he could disprove the existence of atoms.2 Our first quantitative ideas about atoms came on one hand from the chemists of the 19th century, who discovered that they could reduce chemical reactions to an interplay of integers, and on the other hand from the initiators of statistical physics, Maxwell and Boltzmann, who showed that the thermo- dynamic properties of gases found natural explanations within the molecular hypothesis.
Because it succeeded in describing quantitatively the structure of atoms, quantum mechanics consecrated their existence. The range of its applications was also unexpected. Quite rapidly, all physics and all chemistry became quantum theories. The theory accounts not only for atoms and molecules, but also for the structure of nuclei, for particle physics and cosmology, for the electrical and mechanical properties of solid-state materials, etc.
Astrophysics was well paid back and underwent spectacular developments because of quantum theory. These developments led to new observational means to probe the cosmos, and also to the explana- tion of truly macroscopic quantum objects such as white dwarfs and neutron stars. Since its beginning, quantum theory has also generated considerable intel- lectual and philosophical turmoil. For the first time, not only pure reasoning but also what we think to be common sense appeared to be falsified by exper- imental facts.
We needed a new way of thinking about reality, a new logic. It was necessary to develop a quantum intuition, which often seemed contrary to common intuition. As one can guess, an epistemological revolution took place. Philosophers such as Kirkegaard, Höffding, Husserl, Wittgenstein and many others had already discovered how treacherous common language may be.
It is full of a priori conclusions on the nature of things, and any new ex- perimental field can be analyzed only with new concepts and a new language. Quantum mechanics seems to have been invented to prove the philosophers were right. In some respects it goes against some aspects of rationalism. It is quite remarkable that, although at present everyone accepts the mathemat- ical and operational framework of the theory, there are still bitter disputes about its interpretation and its philosophical content.
Schrödinger, “2400 Jahre Quantenmechanik”, Ann. Leibniz, New Essays on Human Understanding, Leibnitii Opera Omnia, L. 3 See for instance Quantum Theory and Measurement, edited by J. Zurek, Princeton University Press, Princeton (1983).com Preface VII What was really unexpected in quantum theory was that it would tackle so directly and so successfully the fundamental structure of matter.
There is no experimental evidence at present that a more elaborate conceptual frame- work is necessary in order to understand the fundamental constituents of matter and their interactions. By its predictive power, quantum physics has been able to radically transform numerous technological sectors in the last 50 years. It has changed the orders of magnitude of what was conceivable. It is now possible to manufacture a material with a virtually unlimited range of thermal, optical, mechanical and electrical properties.
It is more and more feasible to detect a deficiency in a biological function and to cure it in a planned and reasoned manner. The results of the development of semicon- ductor physics and of microelectronics fill our daily life. In the history of mankind, it is a true revolution which multiplies the power of man’s mind, just as the industrial revolution multiplied our strength. This gigantic tech- nological progress is modifying deeply the structure of social, economic and political life, and the mere question of how to adapt our societies to these developments has become a major problem.
Obviously, the number of problems to be solved increases faster than those which have been solved. For instance, in order to go from elementary processes to macroscopic phenomena, one needs the concepts of statistical physics. It is one of the great discoveries of the past decades that it is impossible to reduce everything to microscopic processes. However, one cannot deny that the dimensions and perspectives of physics have changed radically since it has entered the quantum era.
Let us recall that the construction of quantum mechanics benefited consid- erably from the collaboration of mathematicians. The mathematical frame- work of the theory was discovered very soon by Hilbert and Von Neumann. The mathematical structure of quantum mechanics and of quantum field the- ory has always been a fruitful field of research for mathematicians. Conversely, one must admit that one of the difficulties one meets in ap- prehending the theory lies in the fact that the experimental reality of the quantum world is quite far from what is directly accessible.
Many interme- diate steps are necessary in order to build one’s own representation of a phenomenon. This is of course reflected by the mathematical structure of the theory, which certainly deserves the criticism of being abstract. In the epi- graph of his book An Introduction to the Meaning and Structure of Physics (Leon N. Cooper, Harper & Row, New York (1968)), Leon Cooper writes, in beautiful French, S’il est vrai qu’on construit des cathédrales aujourd’hui dans la Science, il est bien dommage que les gens n’y puissent entrer, ne puissent pas toucher les pierres elles-mêmes.4 4 “While it is true that we build cathedrals nowadays in Science, it is a great pity that people cannot enter them, and touch the stones they are made of.com VIII Preface How to teach quantum mechanics has been a source of discussion per- haps as rich as that of its foundations.
Many of the first textbooks were oriented along one of the two following lines. The first consisted in explaining at length the failure of classical conceptions and in using similarities which were often as long as they were obscure.