Wolfgang Nolting Theoretical Physics 6 Quantum Mechanics - Basics Theoretical Physics 6 www.com Wolfgang Nolting Theoretical Physics 6 Quantum Mechanics - Basics 123 www.com Wolfgang Nolting Inst. Physik Humboldt-Universität zu Berlin Berlin, Germany ISBN 978-3-319-54385-7 ISBN 978-3-319-54386-4 (eBook) DOI 10.1007/978-3-319-54386-4 Library of Congress Control Number: 2016943655 © Springer International Publishing AG 2017 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc.
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Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland www.com General Preface The nine volumes of the series Basic Course: Theoretical Physics are thought to be text book material for the study of university level physics. They are aimed to impart, in a compact form, the most important skills of theoretical physics which can be used as basis for handling more sophisticated topics and problems in the advanced study of physics as well as in the subsequent physics research. The conceptual design of the presentation is organized in such a way that Classical Mechanics (volume 1) Analytical Mechanics (volume 2) Electrodynamics (volume 3) Special Theory of Relativity (volume 4) Thermodynamics (volume 5) are considered as the theory part of an integrated course of experimental and theoretical physics as is being offered at many universities starting from the first semester. Therefore, the presentation is consciously chosen to be very elaborate and self-contained, sometimes surely at the cost of certain elegance, so that the course is suitable even for self-study, at first without any need of secondary literature.
At any stage, no material is used which has not been dealt with earlier in the text. This holds in particular for the mathematical tools, which have been comprehensively developed starting from the school level, of course more or less in the form of recipes, such that right from the beginning of the study, one can solve problems in theoretical physics. The mathematical insertions are always then plugged in when they become indispensable to proceed further in the program of theoretical physics. It goes without saying that in such a context, not all the mathematical statements can be proved and derived with absolute rigor.
Instead, sometimes a reference must be made to an appropriate course in mathematics or to an advanced textbook in mathematics. Nevertheless, I have tried for a reasonably balanced representation so that the mathematical tools are not only applicable but also appear at least “plausible”.com vi General Preface The mathematical interludes are of course necessary only in the first volumes of this series, which incorporate more or less the material of a bachelor program. In the second part of the series which comprises the modern aspects of theoretical physics, Quantum Mechanics: Basics (volume 6) Quantum Mechanics: Methods and Applications (volume 7) Statistical Physics (volume 8) Many-Body Theory (volume 9), mathematical insertions are no longer necessary. This is partly because, by the time one comes to this stage, the obligatory mathematics courses one has to take in order to study physics would have provided the required tools.
The fact that training in theory has already started in the first semester itself permits inclusion of parts of quantum mechanics and statistical physics in the bachelor program itself. It is clear that the content of the last three volumes cannot be part of an integrated course but rather the subject matter of pure theory lectures. This holds in particular for Many-Body Theory which is offered, sometimes under different names, e., Advanced Quantum Mechanics, in the eighth or so semester of study. In this part, new methods and concepts beyond basic studies are introduced and discussed which are developed in particular for correlated many particle systems which in the meantime have become indispensable for a student pursuing a master’s or a higher degree and for being able to read current research literature.
In all the volumes of the series Theoretical Physics, numerous exercises are included to deepen the understanding and to help correctly apply the abstractly acquired knowledge. It is obligatory for a student to attempt on his own to adapt and apply the abstract concepts of theoretical physics to solve realistic problems. Detailed solutions to the exercises are given at the end of each volume. The idea is to help a student to overcome any difficulty at a particular step of the solution or to check one’s own effort.
Importantly these solutions should not seduce the student to follow the easy way out as a substitute for his own effort. At the end of each bigger chapter, I have added self-examination questions which shall serve as a self-test and may be useful while preparing for examinations. I should not forget to thank all the people who have contributed one way or another to the success of the book series. The single volumes arose mainly from lectures which I gave at the universities of Muenster, Wuerzburg, Osnabrueck, and Berlin (Germany), Valladolid (Spain), and Warangal (India).
The interest and constructive criticism of the students provided me the decisive motivation for preparing the rather extensive manuscripts. After the publication of the German version, I received a lot of suggestions from numerous colleagues for improvement, and this helped to further develop and enhance the concept and the performance of the series. In particular, I appreciate very much the support by Prof. Ramakanth, a long-standing scientific partner and friend, who helped me in many respects, e., what concerns the checking of the translation of the German text into the present English version.com General Preface vii Special thanks are due to the Springer company, in particular to Dr.
Schneider and his team. I remember many useful motivations and stimulations. I have the feeling that my books are well taken care of. Berlin, Germany Wolfgang Nolting August 2016 www.com Preface to Volume 6 The main goal of the present volume 6 (Quantum Mechanics: Basics) corresponds exactly to that of the total basic course in Theoretical Physics.
It is thought to be accompanying textbook material for the study of university-level physics. It is aimed to impart, in a compact form, the most important skills of theoretical physics which can be used as basis for handling more sophisticated topics and problems in the advanced study of physics as well as in the subsequent physics research. It is presented in such a way that it enables self-study without the need for a demanding and laborious reference to secondary literature. For the understanding of the text it is only presumed that the reader has a good grasp of what has been elaborated in the preceding volumes.
Mathematical interludes are always presented in a compact and functional form and practiced when they appear indispensable for the further development of the theory. For the whole text it holds that I had to focus on the essentials, presenting them in a detailed and elaborate form, sometimes consciously sacrificing certain elegance. It goes without saying, that after the basic course, secondary literature is needed to deepen the understanding of physics and mathematics. For the treatment of Quantum Mechanics also, we have to introduce certain new mathematical concepts.
However now, the special demands may be of rather conceptual nature. The Quantum Mechanics utilizes novel ‘models of thinking’, which are alien to Classical Physics, and whose understanding and applying may raise difficulties to the ‘beginner’. Therefore, in this case, it is especially mandatory to use the exercises, which play an indispensable role for an effective learning and therefore are offered after all important subsections, in order to become familiar with the at first unaccustomed principles and concepts of the Quantum Mechanics. The elaborate solutions to exercises at the end of the book should not keep the learner from attempting an independent treatment of the problems, but should only serve as a checkup of one’s own efforts.
This volume on Quantum Mechanics arose from lectures I gave at the German universities in Würzburg, Münster, and Berlin. The animating interest of the students in my lecture notes has induced me to prepare the text with special care. The present ix www.com x Preface to Volume 6 one as well as the other volumes is thought to be the textbook material for the study of basic physics, primarily intended for the students rather than for the teachers. The wealth of subject matter has made it necessary to divide the presentation of Quantum Mechanics into two volumes, where the first part deals predominantly with the basics.
In a rather extended first chapter, an inductive reasoning for Quantum Mechanics is presented, starting with a critical inspection of the ‘pre- quantum-mechanical time’, i., with an analysis of the problems encountered by the physicists at the beginning of the twentieth century. Surely, opinions on the value of such a historical introduction may differ. However, I think it leads to a profound understanding of Quantum Mechanics. The presentation and interpretation of the Schrödinger equation, the fundamental equation of motion of Quantum Mechanics, which replaces the classical equations of motion (Newton, Lagrange, Hamilton), will be the central topic of the second chapter.
The Schrödinger equation cannot be derived in a mathematically strict sense, but has rather to be introduced, more or less, by analogy considerations. For this purpose one can, for instance, use the Hamilton-Jacobi theory (section 3, Vol. 2), according to which the Quantum Mechanics should be considered as something like a super-ordinate theory, where the Classical Mechanics plays a similar role in the framework of Quantum Mechanics as the geometrical optics plays in the general theory of light waves. The particle-wave dualism of matter, one of the most decisive scientific findings of physics in the twentieth century, will already be indicated via such an ‘extrapolation’ of Classical Mechanics.
The second chapter will reveal why the state of a system can be described by a ‘wave function’, the statistical character of which is closely related to typical quantum-mechanical phenomena as the Heisenberg uncertainty principle. This statistical character of Quantum Mechanics, in contrast to Classical Physics, allows for only probability statements. Typical determinants are therefore probability distributions, average values, and fluctuations. The Schrödinger wave mechanics is only one of the several possibilities to represent Quantum Mechanics.
The complete abstract basics will be worked out in the third chapter. While in the first chapter the Quantum Mechanics is reasoned inductively, which eventually leads to the Schrödinger version in the second chapter, now, opposite, namely, the deductive way will be followed. Fundamental terms such as state and observable are introduced axiomatically as elements and operators of an abstract Hilbert space. ‘Measuring’ means ‘operation’ on the ‘state’ of the system, as a result of which, in general, the state is changed.
This explains why the describing mathematics represents an operator theory, which at this stage of the course has to be introduced and exercised. The third chapter concludes with some considerations on the correspondence principle by which once more ties are established to Classical Physics.