Mathematical Analysis of Physical Problems (Dover Books on Physics) by Philip R. Wallace (PDF)

Description

Intended for the advanced undergraduate or beginning graduate student, this lucid work links classical and modern physics through common techniques and concepts and acquaints the reader with a variety of mathematical tools physicists use to describe and comprehend the physical universe.For the physicist, mathematics is a language, or shorthand, for constructing workable models (necessarily approximate and incomplete) of aspects of physical reality. The present text, by a noted professor of physics at McGill University, Montreal, deals in an exceptionally well-organized way with some of the crucial mathematical tools used to construct such models.Contents include: I: The Vibrating String; II. Linear Vector Spaces; III. The Potential Equation; IV: Fourier and Laplace Transforms and Their Applications; V. Propagation and Scattering of Waves; VI. Problems of Diffusion and Attenuation; VII. Probability and Stochastic Processes; VIII. Fundamental Principles of Quantum Mechanics; IX. Some Soluble Problems of Quantum Mechanics; X. Quantum Mechanics of Many-body Problems.A special helpful feature of this volume is a Prelude to each chapter, which outlines the topics with which the chapter deals. In addition to providing a guide to the organization of its contents, it indicates the mathematical background assumed and calls attention to those methods and concepts which have an application in different physical problems. Relevant test problems are interspersed throughout the text to test the student’s grasp of the material, while brief bibliographies at the chapter ends suggest further reading.Ideal as a primary or supplementary text, Mathematical Analysis of Physical Problems will reward any reader seeking a firmer grasp of the mathematical procedures by which physicists unlock the secrets of the universe.

User’s Reviews

Editorial Reviews: About the Author Canadian theoretical physicist Philip Wallace (1916–2006) was a longtime professor at McGill University. He is best remembered for his pioneering 1947 paper on graphite and graphene, later the subject of research by recipients of the 2010 Nobel Prize in Physics.

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐I really enjoy this book. I like the author’s style , his approach and the topics he’s chosen. However, IMO, it’s intended for those with plenty of experience but who are still looking for another viewpoint to help clear up those loopholes we all have when trying to understand something that isn’t easy.

⭐Well this is a good Mathematical Reference Books for Theoretical Physisicst but has nothing to do with Mathematical Analysis of Physical Problems. It has all the tools you need that is fine, there are many similiar books as a reference book but if you think you will find ideas and methods “how to structure the Physical Problems in Mathematical terms”, this is not the book.

⭐I rather enjoyed this text. It differentiates itself from others (compare: Byron and Fuller). The author: “to produce a generally useful book, valuable to the personal library of students of Physics.” The text is not aimed at any specific-titled course but useful (as it is) for self study. Prerequisites (not explicitly stated in the text) should consist of the textbook by Mary Boas, Mathematical Methods in the Physical Sciences. With that in mind, we proceed.(1) First, note the pedagogic utility of the problems: these are set within the body of the text, instead of being placed at end-of-chapter (as is Byron and Fuller). Immediate comprehension of the material is thereby tested !(2) Chapter Three: Potential equation, will be most welcome as a crutch for Jackson: Classical Electrodynamics.Spherical Harmonics, Legendre polynomials and Green’s functions are detailed. The author provides such detail as to enable computational facility to proceed with a minimum of effort. Well done.(3) Preceding this third chapter is a chapter (two) of linear vector spaces and (one) of vibrations and waves.An excellent, easy-to-follow, section on Rayleigh-Ritz variational principle leads the reader to eigenvalues and a first exposure to Green’s functions (one-dimensional–as Chapter Five expounds multi-dimensional). Bras and kets, Dirac notation, introduced heuristically in the chapter on linear vector spaces (bras and kets are absent in Byron and Fuller). Instructive examples of summation by way of Fourier transforms (pages 170-171) makes for a nice detour.(4) Next, useful examples: Integrals evaluated by way of Three-Dimensional transforms in chapter four, concluding with construction and utilization of special functions: Gamma, Beta, Confluent Hypergeometric, Laguerre and Hermite.(5) Fifth chapter: Wave propagation and Scattering. Begin with sound waves. Following which: separation of variables, spherical coordinates, sources and radiation. This is a demanding chapter. Happily, few computations are glossed over. A highlight: the section entitled “Radiation from a uniformly charged vibrating liquid drop” (pages 316-322). It is rare indeed to find all of this material presented in such an elementary manner outside the research journals.(6) An appendix touches upon asymptotic expansions and Hankel functions. Sixth: Diffusion. Green’s functions are the primary focus. A section details connection between Schrodinger’s equation and the so-called heat equation. Then, a nice discussion of neutron diffusion. The chapter concludes with derivations of equations applied to superconducting electrodynamics. Detail is supplied for intermediate calculations. Thus, with paper and pencil in hand, easy to follow.(7) Probability and Stochastics, next. Generating functions introduced. Central limit theorem derived (page 431). A section delineates connection between random walk and diffusion. The chapter culminates in exposition of random distribution of energy among N particles. Following: a summary of quantum mechanics (eighth chapter): “our aim is to provide a practical handbook for the use of those who wish to comprehend and to undertake quantum calculations.” (page 447). This ‘handbook’ occupies fifty pages (nice comparison to Byron and Fuller, section #5.11, pages 277-294).(8) That summary (chapter eight) is put to work in chapter nine: Soluble problems of Quantum Mechanics. A highlight: elementary exposition of Schwinger approach to angular momentum operators. Scattering is detailed. This makes a nice complement to Francis Low’s textbook: Classical Field Theory. Final chapter is Quantum Mechanics: Hartree-Fock methods and density matrix calculations are introduced.(9) There you have a summary of the book. Now, you will meet neither group theory (see Byron and Fuller) nor tensor analysis. The author says: not theoretical, but mathematical physics, that is his objective. A bit off the beaten path if compared to other textbooks in this genre. Some overlap with material in the text of Mathews and Walker is evident.However, Philip Wallace is explicit computationally, while eschewing mathematical proofs (see Byron and Fuller).All in all, this is an exceptional offering for all advanced students of physics.Highly recommended for collateral reading and study.

⭐I have a graduate physics degree (as well as an undergrad math,physics dual major deal..) What I was looking for (and having a hard time finding) – was a book that explained HOW certain equations came about. For e.g. – we all know the equation of a vibrating string or of an electron in a potential well – but if you were the FIRST person to try and discover the equation – how would you go about formulating it? In other words – what would be your ‘mathematical analysis’ of the ‘physical problem’ of the vibrating string etc?While this book does not go the whole 9 yards in this regard – it is one of the few books (in my limited experience) that actually DOES attempt to ‘derive’ these equations from scratch! For that reason – I give it 5 stars.

⭐OK, the cover is tacky and the print looks like it has been done on a 1960s Remington, but I am yet to see a book which explains mathematical methods so well, and also shows you how you can use them to solve physics problems. I have loved every Dover book I’ve read, and this one too is a masterpiece just like the others. Beautiful stuff!

⭐This is a very good feeling, when you handle a book which provides all the mathematics (separation of variables, Fourier Laplace analysis, complex plane integration, Green’s functions…) necessary to solve almost all the physical problems; especially dealing with partial derivative equations (from vibrating string to Schroedinger equation). The price is also a good point. Thank you Dover.

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