|Mathematics related to QM|
|Foundations and History of QM|
Last update 12 August 2015
(A) Classical Mechanics CM
A. Sommerfeld, Mechanics (Academic Press, New York, 1950). This is a translation of the first of 6 volumes of theoretical physics by a grand old master, who taught Schrödinger and many other physicists, many of whom later became illustrious. These books were not written in haste to help pay the rent, they were meticulously crafted after many years of teaching and lecturing in Munich. Lots of clear concise text to explain in physical terms what all those symbols really mean.
H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Massachusetts, 1965, 1980). This is the modern classic of classical mechanics for physicists. Reliable and clear, with interesting discussions of the relation of classical mechanics to wave mechanics, of the importance of symmetries, and some entertaining speculations as to what might have happened had Hamilton made that extra step in the dark towards QM.
L. Pars, A Treatise on Analytical Mechanics (John Wiley, New York, 1965). The title describes this book very well: it is a masterful presentation of the subject with copious fascinating problems worked out in leisurely detail. Strongly recommended!
C. Lanczos, The Variational Principles of Mechanics (University of Toronto Press, Toronto, 1970). This is mechanics set to music. It is a grand view of the mathematical and physical ideas.
R. Abraham and J. E. Marsden, Foundations of Mechanics (Benjamin-Cummings, London, 1978). Mathematicians have had a great time with mechanics. This book has the contemporary view: mechanics is an application of differential geometry. Calculus on a manifold is developed first and then follows a grand tour of mechanics leading to some famous solved and unsolved problems. This book is extravagantly complete but very abstract. It has some interesting historical remarks including pictures of leading contributers to the field.
W. Thirring, A Course in Mathematical Physics: (1) Mechanics (Springer-Verlag, New York, 1979). This book also provides an introduction to manifolds followed by a presentation of mechanics from the geometrical point of view. The exposition is extremely concise and the book forms part of a 4-volume series on theoretical physics. One very attractive feature is the inclusion of solutions to all the problems: a great help to the autodidact.
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(B) Quantum Mechanics QM
R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics III Quantum Mechanics (Addison-Wesley, Reading, Massachusetts, 1965). A broad introduction which stresses physical phenomena and their interpretation through QM.
D. A. McQuarrie, Quantum Chemistry (University Science Books, Mill Valley, California, 1983). A well-written elementary introduction presented in the context of applications to chemistry. Contains many carefully worked examples, and answers to all the exercises.
D. Bohm, Quantum Theory (1951, Dover republication 1989); K.Gottfried, Quantum Mechanics Volume I: Fundamentals (1966, Addison-Wesley republication 1989). These two books are rare examples of introductory books written by physicists who attempt to deal both with technical details and also with the difficult questions of interpretation. We still await Gottfried's Volume II.
J. M. Levy Leblond and F. Balibar (Translated from French by S. T. Ali), Quantics - Rudiments (Elsevier Science Publications, North Holland 1989). A very recent addition to the short list of elementary books that treat both technical and interpretational aspects of the theory. The approach is authoritative and original.
A. Messiah, Quantum Mechanics Vols. I and II, (North-Holland, Amsterdam, 1961). This is a standard textbook; it represents what most physicists need to know about QM, irrespective of their own special field. The treatments of angular momentum and symmetry are excellent. There is a chapter on the variational method in Volume II.
J. J. Sakurai, Modern Quantum Mechanics (Benjamin Cummins 1985). A very clear presentation of the theory in a thoroughly contemporary style. Hardly any essential basic topics are missing.
L. D. Landau and E. M. Lifshitz, Quantum Mechanics (Pergamon, Oxford, 3rd. Edition 1977). A Standard and very reliable textbook. Other volumes in this excellent series treat relativistic QM and field theory.
W. Greiner, Quantum Mechanics: An Introduction (Springer-Verlag, Berlin, 1989). This is Volume I of a series of books on QM by the same author. See also Quantum Mechanics: Symmetries, and Relativistic Quantum Mechanics. These books are excellent! They are up to date, extremely clear, and packed with solved examples and interesting problems (all with solutions). Perfect for self study!
Galindo and P. Pascual, Quantum Mechanics I and II (Springer-Verlag, Berlin, 1990). These books offer a modern, thorough, and complete introduction to QM. Many contemporary topics such as Bell's inequality and hidden variable theories are treated in appendicies. On a par with Greiner; a modern Messiah.
K. Hannabuss, An Introduction to Quantum Theory (Clarendon Press, Oxford, 1997). How can a student of mathematics learn something about quantum mechanics without being overwhelmed by operator theory or by the mysteries of physics? This delightful book might just be the answer.
A. Sudbery, Quantum Mechanics and the Particles of Nature: An Outline for Mathematicians (Cambridge University Press 1988). Similar goals to Hannabuss; from the mathematical point of view, it is rather elementary; some accessible details concerning symmetries and the particle zoo.
W. Heisenberg, The Physical Principles of Quantum Mechanics (University of Chicago Press, 1949; Reprinted by Dover, New York, 1967). A short book explaining general ideas and very little detail.
E. Schrödinger, Collected Papers on Wave Mechanics, (Chelsea Publishing Company. New York, 1982). Model papers for their rich content, their clarity, and their attention to detail. One should look at these before composing any new article on QM. Also contains his lectures in Dublin.
P. A. M. Dirac, Quantum Mechanics, (Oxford University Press, 1958). Masterly elegant, with many pages of prose and a seemingly effortless tour through the foundations of the subject and certain applications. Everything is there but work is required to follow the steps in detail and to keep up with the master.
W. Pauli, Pauli Lectures on Physics Vol 5, (MIT Press, Cambridge, Massachesetts, 1973). Beautiful logic and economy of effort; original and refreshing. Pauli seem always able to insert with ease any mathematical steps demanded by his train of thought in physics.
W. Pauli, General Principles of Quantum Mechanics, (Springer-Verlag, Berlin, 1958, 1980). A classic text by a master.
S. Flügge, Practical Quantum Mechanics, (Paperback edition in one volume, Springer-Verlag, New York, 1974). A rich collection of problems and solutions. If you are looking for an exactly solved problem in QM, it's probably in Fluegge. Soluble problems are invaluable for testing hypotheses and illustrating results. See also Lieb and Mattis in the mathematics section.
O. L. de Lange and R. E. Raab, Operator Methods in Quantum Mechanics (Clarendon Press, Oxford, 1991). A delightful compendium of exact solutions of, and operator factorization methods for solving, bound-state problems in QM. Interesting results for the Klein-Gordon equation and the Dirac equation are also included.
S. Sternberg, Group Theory and Physics. (Cambridge University Press, 1994). A clear modern presentation of an old and profound subject. The viewpoint is from mathematics, but by an author who obviously has a deep understanding of the connection with physics. Useful as a reference as well as for complete background coverage.
J. P. Blaizot and G. Ripka, Quantum Theory of Finite Systems, (MIT Press, Cambridge, Massachusetts, 1986). An up-to-date and comprehensive compendium of results and methods for finite many-body systems.
K. Blum, Density Matrix Theory and its Applications, (Plenum Press, New York, 1981). A clear treatment of a special topic which is often neglected or treated only cursorily in other texts.
J. Glimm and A. Jaffe, Quantum Physics : A Functional Integral Point of View, (Springer, New York, 1981). An alternative formulation of QM (due oginally to Feynman) which leads (often with extra effort) to the same results as the conventional theory. The advantage is that it provides a good starting point for the study of quantum field theory.
R. G. Newton, Scattering Theory of Waves and particles (Springer Verlag, New York, 1966, 1982). This is now a comprehensive classic which is not restricted to QM. There is much useful reference material on exact solutions to problems in QM, and a very good summary of inverse-scattering theory up to publication.
K. Chadan and P.C. Sabatier, Inverse Problems in Quantum Scattering Theory (Springer Verlag, New York, 1977, 1989). A very complete and authoritative account of quantum mechanical inverse problems. Useful as a reference as for initial study of this rich and interesting field which links non-linear problems to inverse solutions of the linear Schrödinger equation.Back to the top
S. H. Friedberg, A. J. Insel, and L. E. Spence, Linear Algebra, (Prentice- Hall, Englewood Cliffs, New Jersey, 1979). This book is very elementary but correct and modern in style. On page 432 there is a proof of The Spectral Theorem.
T. F. Jordan, Linear Operators for Quantum Mechanics, (John Wiley, New York, 1969; Since 1986 this book has been out of print but it is available from the Author at The Physics Department, University of Minnesota, Duluth). In this book a selection of results from the theory of linear operators are presented from an elementary point of view. The presentation is motivated by QM.
S. Berberian, Introduction to Hilbert Space, (Chelsea Publishing, New York, 1976). A very concise and careful introduction. No QM!
E. Kreyszig, Introductory Functional Analysis with Applications, (John Wiley, New York, 1978). A remarkable book written by a fine teacher. Many worked examples, interesting discussions, and helpful exercises. Chapter 10 deals with unbounded operators and Chapter 11 is an introduction to QM.
M. Schechter, Operator Methods in Quantum Mechanics, (North Holland, New York, 1981). The author uses QM in one dimension as a motivation for the study of functional analysis. An excellent introduction to the mathematics necessary for QM.
E. H. Lieb and M. Loss, Analysis, (AMS, 1997 & 2001, Grad. Studies in Math. 14). This is a most unusual book. It connects analysis to physics in a most approachable, attractive, efficient, and fascinating way. In a volume of modest size there is fundamental material on measure theory, function spaces, Fourier transforms, inequalities, potential theory, distributions, and Schrödinger operators. The second edition has a new chapter on inequalities. Highly recommended!
E. Prugovecki, Quantum Mechanics in Hilbert Space, (Academic Press, New York, 1981). A thorough introduction to QM from a mathematically rigorous standpoint. Based on an undergraduate course given at The University of Toronto.
W. Thirring, A Course in Mathematical Physics: (3) Quantum Mechanics of Atoms and Molecules, (4) Quantum Mechanics of Large Systems, (Springer-Verlag, New York, 1979, 1980). It could be argued that much of non-relativistic QM has now become a branch of applied functional analysis. These books are rigorous and modern and move swiftly through the subject. They can be read after Kreyszig or Schechter. They are also a good reference source for mathematical results in QM. Volumes 1 and 2 treat mechanics and classical field theory within a framework of differential geometry.
R. Courant and D. Hilbert, Methods of Mathematical Physics, Vols. 1,2. (Interscience, New York, 1953). These classic volumes are now dated but they are very readable and the style makes the material (which is mathematically sound) more accessible to physicists than the more recent abstract works. If one is working on a particular problem then it's a good idea to have a look at Courant-Hilbert to see if a similar case is treated there.
M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vols. 1- 4, (Academic Press, New York, 1972-1978). Some people would call these volumes the modern Courant-Hilbert. The titles are respectively: (1) Functional Analysis, (2) Fourier Analysis and Self Adjointness, (3) Scattering Theory, (4) Analysis of Operators. These books provide a comprehensive account of the mathematics behind non-relativistic QM. The style is a very contemporary mixture of free general discussion and precise definitions and theorems. The notes at the ends of each chapter are a rich source of historical and background material. Volume (4) serves as a handbook of known mathematical results in QM : for example, the Rayleigh-Ritz (min-max) principle for eigenvalues is given on page 78; in the following sections there are large collection of general eigenvalue bounds, including also many such theorems for many-body systems.
H. L. Cycon, R. G. Froese, W. Kirsch, and B. Simon, Schrödinger Operators (Springer-Verlag 1986). One could describe this book as Reed-Simon IV brought up-to-date. It is based on some lectures given by Barry Simon.
T. Kato, Perturbation Theory for Linear Operators, (Springer-Verlag, Berlin, 1980). This is the mathematical classic for spectral theory related to QM. The early sections include a summary of the main results of elementary linear algebra and a delightful introduction to Hilbert spaces. Barry Simon once remarked that it was with some reluctance that he eventually realized that in order to understand these heady operator matters associated with QM, one had first to read Kato.
A. Weinstein and W. Stenger, Methods of Intermediate Problems for Eigenvalues, (Academic Press, New York, 1972). This is a special book dealing with variational and comparison results for estimating eigenvalues.
E. Lieb and D. Mattis, Mathematical Physics in One Dimension, (Academic Press, New York, 1965). A collection of exact results for systems in one spatial dimension.
S. Ciulli, F. Scheck, and W. Thirring (Eds.), Rigorous Methods in Particle Physics (Springer-Verlag, Berlin, 1990). An interesting collection of concrete results in QM, in honour of A. Martin, who had a taste for this sort of thing.
G. G. Emch, Mathematical and Conceptual Foundations of 20th-Century Physics, (Elsvier Science Publishers B.V. NY 1984). Here we have classical mechanics, relativity, and QM all between the covers of one book. The approach is via differential geometry and the book is not elementary. A considerable effort has been made to include the physical and historical aspects along with this elegant (perhaps grand) mathematical exposition.
V. S. Varadarajan, Geometry of Quantum Theory, (Springer-Verlag, New York, 1985). This book presents the contemporary mathematician's viewpoint of quantum mechanics. It is complementary to Reed-Simon and Kato, who stress functional analysis and operator theory. In Varadarajan we find a tour of quantum theory expressed in terms of group theory, differential geometry, projective geeometry, and logic. It's not for the faint at heart, nor for those who want to find the spectrum of the ammonia molecule.
V. Guillemin and S. Sternberg, Symplectic Techniques in Physics (Cambridge University Press, 1984). A magnificent approachable presentation of many branches of physics in terms of geometry and group theory.Back to the top
The books mentioned here are mostly to do with computing in C++. Alternative high-level environments such as Mathematica also allow one to write programs in a variety of different styles. Most people use such packages as sophisticated calculators, and ignore the rich support given to programming. Low-level programming in C++ is not for everyone, but it is often interesting and can be very useful. Experts in computer languages find much fault with C++. However, it has been adopted by a large number of people; some of them are very clever and have provided us with an impressive collection of ideas and methods. One inevitably learns some C, but the idea now is to jump right in to C++, with classes and all, and go with the flow.
W.H.Press, B.P.Flannery, S.A.Teukolsky, W.T.Vetterling Numerical Recipes, (Cambridge University Press, Cambridge, 1988). This book is well described by its title. It is a large compendium of numerical algorithms which are described reasonably well, but not developed in detail mathematically: for more theoretical results it is necessary to see, for example, some of the references given in the book. However, the algorithms themselves are expressed in Fortran or Pascal or C, and they are available from CUP on disks (see below). I have usually found it to suit me better to use the book to learn and understand the basic idea, and then to write the code for myself. The contents of the book, including the algorithms in C, can be downloaded in pdf or ps format from the site given below:
R.Sedgewick Algorithms in C++, (Addison-Wesley, New York, 1992). Here the main point is not the C++; there are a number of language flavours of this excellent book. No, the issue here is algorithms. Computer people are a little fixated on sorting and searching, simply because they have to do a great deal of it; and many fascinating methods are now available. The book is a fine example of the genre: it treats the elements of many different problems, and provides code fragments to get one started. If one’s background is in mathematics, physics, or engineering, one must have a look at a book like this. Although computing was originally ‘started’ by people in the above three fields, it now flourishes in a very interesting world of its own.
D.M.Capper, Introducing C++ for Scientists Engineers and Mathematicians, Springer-Verlag, New York, 1994). Here is a sensible practical book written for professionals outside computing. The index ‘works’, so that you can look things up when you forget them. Meanwhile the examples are suggestive of real scientific applications, rather than code for a kindergarten vending machine. Some things are a little dated already, but that’s not a real problem for the beginner. For more information on the the current standard C++ extensions, see the links to the Standard Template Library (STL) below.
Daoqi Yang, C++ and Object-Oriented Numeric Computing for Scientists and Engineers, Springer-Verlag, New York, 2001). A thorough and sophisticated book which, like the book by Capper, manages to wed computing to mathematics. The problem of functional objects is discussed on p266: some work-arounds are proposed for this significant hurdle for C++. The text also deals at length with exception handling, namespaces, templates, containers, and standard libraries. This book needs much more than a ‘careful leafing through’ (to use that nice phrase of Edmund Landau's).
STL Standard Template Library
M.A. Ellis and B. Stroustrup The Annotated C++ Reference Manual, (Addison-Wesley, N.Y., 1990). Here you get the authoritative account from the original creator (Stroustrup), along with examples, and some explanations as to why things are as they are. The creator's home page is also very interesting:
You can find out how to pronounce his name, some history of him and the language, and much much more.
B.Eckel Thinking in C++, (Prentice Hall, New Jersey, 1995).This is a deep and fascinating book about computing and the language C++. Whenever I think that I have finally 'got the point' about some aspect of computing in C++, reading this book (and the one by Koenig and Moo) remind me that I am still at the beginning. However, as time goes on, I find that I benefit more and more from these excellent volumes.
A.Koenig and B.Moo Ruminations on C++, (Addison-Wesley, New York, 1997). "A decade of programming insight and experience." This is sophisticated bed-time reading for the keen user of C++. Stroustrup once said that when he gets into trouble he calls on Koenig.Graphics The computing books mentioned below this point are special to the Windows environment. This is not everyone's choice. The Unix folk like to use X-Windows and the Macintosh people have the toolbox and Road Warrior. Others try to ride above all these OS and machine-specific limitations by using Java. The professional should be able to cope calmly with all this diversity, but life is short, and it is natural to make some choices and get on with one's work. I am very much attached to the graphical expression of mathematics, and to using graphics as an exploratory tool in my work. I have opted for Windows. However, even this is only the first of a sequence of choices that have to be made. One does not wish to re-invent the wheel; at the same time, using other people's conceptual framework and code is not always agreeable. I use `stick-shift' Win32 programming, with the aid of the next two titles. The Win32 example gtiny, which may be found on my web page under miscellaneous downloads, uses my small graphics package gwtiny and allows one to write programs quickly and easily. At the cost of even more entanglement with other people's code, one can also work generally at a higher (and arguably more effective) level by going the whole way with the Microsoft Foundation Class (MFC). Borland have their own scheme called Object Windows Libary (OWL). It is very difficult to find the time to explore all these different options so that one can make a reliable choice between them. Eventually time pressure and personal tast force a decision, which subsequently constrains one's path through the maze. The books below each come with a CD full of examples. Unless one has unlimited time, the obvious way to go for small-scale computing projects is to start with a suitable professionally-written example, and modify it to fit the problem at hand.
C.Petzold Programming Windows, (Microsoft Press, Redmond, 5th Edition 1998, or 4th Edition 1996).This book is regarded by many as the best book about Win32 programming. It's not as restricted as the title would suggest since most of the code works fine with Win98 and WinNT too. I have only used WinNT for my own work.
R. Simon Windows 95 WIN32 Programming Bible, (Waite Group Press, Corte Madera, 1996).Petzold does not give all the details of all the many Windows functions. For a more complete list and another set of examples, this book is an extremely handy reference. The issue of compatibility across the flavours of Windows is also dealt with.
A.R.Feuer MFC Programming, (Addison-Wesley, Reading, 1997).This seems to be an excellent account of MFC with a rich set of working examples. If you get tired of the low-level fuss required by Win32 programming, this might be the way to go. The use of MFC allows one to add another feature very quickly and easily. For example, if you want to read a file from the hard disk, MFC will allow you to use a pre-defined dialog box just like the ones you find in popular word processors.
D.J.Kruglinski Inside Visual C++, (Microsoft Press, Redmond, 1996). This is the final step towards automatic computing under the Microsoft umbrella. Kruglinski explains how to use the VC environment to introduce standard MFC programming elements quickly and easily into one's code.Back to the top
(E) Computational QM
Computational QM is a vast field. The following books are elementary but very helpful. It is worthwhile to cultivate some facility with a computer because it allows one to work comfortably with examples and it also enables one to explore beyond what is immediately possible by the use of purely analytical methods. For serious work it is adviseable to make use of combined symbolic, numerical, and graphical computing such as is available, for example, with Derive, Maple, or Mathematica.
J.P.Killingbeck, Microcomputer Quantum Mechanics, (Adam Hilger, Bristol 1983). This is a very practical book which has a large collection of methods from numerical analysis which can easily be implemented on small computers. The author has a long experience with iterative methods suitable for small computers. This is an elementary and very practical little book.
H.Gould and J.Tobochnik, An Introduction to Computer Simulation Methods: Applications to Physical Systems (2 volumes, Addison-Wesley, Reading Mass. 1988).
S. Brandt and H. D. Dahmen, Quantum Mechanics on the Personal Computer (Springer-Verlag, Berlin, 1992). This book includes a disk for an IBM compatible computer with a collection of interactive solutions to QM problems, often with graphics. If you can tolerate following someone else's selection of things to do, it could be instructive fun.Back to the top
(E) Foundations and History of QM
J. A. Wheeler and W. H. Zurek, Editors, Quantum Theory and Measurement, (Princeton University Press, Princeton, 1983). QM is perhaps the most successful physical theory ever created. In spite of this, there remain many fundamental unresolved problems to do with the interpretation of the theory. These questions are usually not stressed and are often simply neglected in technical books dealing with the physical or mathematical theory of QM (noteable exceptions are the books by Bohm and by Gottfried). Wheeler and Zurek have put together a remarkable collection of reprints from a large and now rapidly growing literature devoted to the philosophical problems of QM.
B.d'Espagnat, Conceptual Foundation of Quantum Mechanics (Benjamin, 1976). One of the most reliable books on the philosophy of QM. Many technical details are also included.
J.M.Jauch, Foundations of Quantum Mechanics (Addison-Wesley 1968). This book presents elementary QM from an advanced standpoint and is a classic in the foundations of QM written by a master of theoretical physics. The mathematical theory is presented concisely and fully (but without proofs of basic results from functional analysis). Jauch has also written (with F.Rohrlich) a more advanced technical book entitled The Theory of Photons and Electrons.
J. S. Bell, Speakable and Unspeakable in Quantum Mechanics (Cambridge University Press 1987). A collection of papers written by Bell along with new commentary by the author. Here we have an exposition of Bell's ideas and thoughts on important foundational issues in QM. When reading this book one feels one is in the presence of a very original thinker who expresses himself with great clarity and never wishes to force a point or insist on a particular resolution of a question, a scientist who is searching for the truth with intelligence and sensitivity.
M. Redhead, Incompleteness, Nonlocality, and Realism: A Prolegomenon to the Philosophy of Quantum Mechanics (Clarendon Press Oxford 1987). In its approach to the philosphical problems of QM this book is a curious mixture of the elementary and the advanced, the philosophical and the technical, the physical and the mathematical. If one is interested to see what `is happening' in the philosophy of QM today, one should definitely take a look at this book.
J.Mehra and H.Rechenberg, The Historical Development of Quantum Theory (Springer-Verlag, New York, 1982 ...., 7 volumes and more still to come). If one is interested in the history of QM then these extravagantly detailed books are a must.
E. Prugovecki, Stochastic Quantum Mechanics and Quantum Spacetime,
(D. Reidel, Boston, 1984). This book is included to counter the impression suggested by the technical reading lists that all the basic mathematical questions to do with non-relativistic QM are completely settled. Prugovecki develops the idea that since precise measurement is not possible, even in classical mechanics, distributions are already necessary in that theory. When this is carried over into QM it leads to a theory that can be developed in harmony with relativity more easily than can conventional QM.Back to the top