Chaos in Classical and Quantum Mechanics (Interdisciplinary Applied Mathematics, 1) 1st Edition by Martin C. Gutzwiller (PDF)

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Describes the chaos apparent in simple mechanical systems with the goal of elucidating the connections between classical and quantum mechanics. It develops the relevant ideas of the last two decades via geometric intuition rather than algebraic manipulation. The historical and cultural background against which these scientific developments have occurred is depicted, and realistic examples are discussed in detail. This book enables entry-level graduate students to tackle fresh problems in this rich field.

User’s Reviews

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

⭐Martin Gutzwiller (12 October 1925 – 3 March 2014):”refused to follow any fashion; instead, he generated ideas that became the fashion.”(Physics Today, Volume 67, Number 6, page 60, 2014). That line encapsulates this text. As one reviewer has noted,the Bibliography of this text is a goldmine. Those sixteen pages of references conclude the book. Regards that bibliography, here is one instance of its efficacy: “Einstein turned the whole question around and asked which mechanical systems could be subject to the Bohr-Sommerfeld-Epstein rules. He came up with a much more direct view of the basic ingredients in terms of the invariant tori…there seems to be only one reference to this paper in the ensuing 40 years, such total neglect of an incisive comment is almost beyond comprehension.” (page 208).I found that remark interesting and worth perusing. I discover half page devoted to Einstein’s paper in Pais’ Biography, Subtle Is The Lord (page 412), there read: “Its relevance to modern physics…stems from the connection between the orbits of the old quantum theory and the semi-classical limit of quantum mechanics.” To learn more, peruse Chapter fourteen of Gutzwiller (the reference Gutzwiller writes of is Lanczos’ Variational Principles of Mechanics, pages 253-254, second edition !).What else does Gutzwiller offer ?(1) Bits and pieces of interesting, applicable, mathematics: geometry of continued fractions (page 393),hyperbolic geometry (page 345), Riemann-Zeta function (page 307), Wigner’s distribution function (page 241), rational and irrational numbers (page 129).(2) Bits and pieces of Gutzwiller’s modes of expression: “The limit of a small Planck’s quantum is a mathematical fiction when we come to the fields where quantum mechanics is important.”(page 41) and, “Since mechanics is closer to geometry than to algebra, one would like to have an example where the scrambling of phase space can be seen intuitively, the automorphisms of the torus fulfill this role.” (page 149) and, “Quantum Mechanics remains somewhat of a mystery, although it is well-confirmed by all the experimental evidence” (page 174) and, “The hard questions of classical mechanics can, therefore, not be dismissed as irrelevant because they are presumably superseded by modern physics.” (page 175). Finally: “The trace formula was originally derived in order to find an approximate spectrum for a classically chaotic system on the basis of the classical trajectories. Einstein was the first to recognize that a real problem lay ahead when the classical system was no longer integrable.” (page 315).(3) The book deals only with systems of finite degrees of freedom (Introduction). Examples abound: hydrogen atom in magnetic field, donor impurity in a semiconductor, motion of particle on a surface of negative curvature, and more.(4) As others have noted, this is not a textbook. There is unevenness in the exposition. Even so, there is plenty enough detail to send the reader on many a tangent. This is an interesting read full of much diversity, many fascinating connections, coupled with opinions and words of wisdom dotting the landscape. The manner in which details are left to the reader will certainly turn some away (but, pursue those references that Gutzwiller provides).I find Arnold’s Mathematical Methods of Classical Mechanics an excellent precursor to Gutzwiller.This book will provide instructive and entertaining reading. You will learn much in that pursuit !Recommended for enrichment.Addendum:Gutzwiller provides entree to Heller’s quantum scarring, Section #15.6 (page 249).Recent research shedding light on this phenomenon. From American Physical Society, we read: “Quantum scarring may seem like an abstract mathematical topic. But it is of practical interest for the field of quantum computing.” (Physical Review B, Volume 98, 2018).Gutzwiller is a source of inspiration:”… semiclassical methods based on the work of Martin Gutzwiller can provide stunning physical insights in regimes where quantum solutions are opaque.” (page 593, Kleppner and Delos, Foundations of Physics, Vol. 32, Nr 4, 2001).

⭐There are all kinds of physics books. There are books for didactics; nobody actually uses, say, Jackson’s book on Classical Electrodynamics for anything but didactics. You can tell the books that are useful for teaching in a classroom by the presence of extensive problem sets. This book has none. So, with this, let me dispose of the notion that you should be able to teach a ‘quantum chaos’ course based on this book. There are also books which explain in great technical detail all about a given small subject area; Fritz Haake’s book on ‘Quantum Signatures of Chaos’ is a good example of this (though it’s really about Random Matrix theory, which is but one quantum signature of chaos; a whole book on just one small area of this subject -covered in a few pages in Gutzwiller’s book). Then there are books which are reviews of a subject. Gutzwiller wrote a book which is a review of a subject. It’s a very good one.One could actually go through this, read it, know about the existence of many things, and read up on them as is appropriate for one’s research. That is how I used the book. Gutzwiller writes very well, and communicates his enthusiasm for the subject matter. The bibliography is extensive, and you could indeed learn all about anything you needed to by going through the bibliographic references for the details. An actual review which takes you from knowing about Hamilton Jacobi theory and the WKB approximation and covering all the material he presented in every detail needed to understand the whole mess …. I doubt as any such book exists. It would be an encyclopedia. If you simply want to understand the Gutzwiller trace formula starting at that level, there are books which do it reasonably well (Cvitanovic’s is a great effort in that direction). This is not such a book; this is a review. A review of a very important body of work. That is this book’s purpose, and it serves it admirably well.As for why Gutzwiller is important; if you think the correspondence principle is interesting, the Gutzwiller trace formula is the main useful way we know of thinking about it in detail. Most physicists think about the area where quantum mechanics becomes classical as a sort of intellectual blur. This is a mistake. This is arguably a much more interesting place to think about than, say, what happened in the first 10^-35 seconds of the big bang. For example, one can actually do experiments in the semiclassical regime. Experiments which only involve optics. There’s gold in them there hills, and nobody is looking there. Mesoscopic physics as applied to device physics may eventually force people to think about these issues in detail, but I somehow doubt the device guys are going to reinvent quantum mechanics. Gutzwiller thought about these issues in the 1960s while working on practical problems for IBM. These are the fundamental mysteries, folks: if you’re an ambitious young physicist who wants to make a real contribution to science; leave the noodle theory alone, and look into this stuff. Following the herd will get you a nice career, but immortality belongs to the one who plays Einstein or Poincare to Gutzwiller’s Minkowski.

⭐The first thing a non-specialist should do is to hire a reputable consultant or tutor to translate this book into approximately ordinary English, but it would be a bad mistake to miss this book because it is full of remarkable ideas that apply to almost every field. To show you how translation would work, it turns out that doughtnuts (torus, plural tori in math) are key to classical and quantum chaos. You might ask: why doughtnuts, instead of, say, balls (beachballs, tennis balls, microscopic balls, etc.)? Well, let’s consider a person’s head as a ball. You can comb a person’s hair flat on his head, but not without a part or irregularity (eddy) somewhere – see page 33 of the book, which says this is slightly different words. On a doughnut, however, you can comb hair flat without any irregularities. The hair with the directions in which they point are called vectors in mathematics, and all together they form a vector field. Chaos occurs when these doughtnuts get disturbed, say by raising the energy (perturbations). Gutzwiller of IBM New York pioneered the study of the relationship between classical and quantum chaos, but anybody who has read IBM books or manuals know that they usually require heavy translation. If you absolutely can’t find a math or physics consultant or tutor, go through Gutzwiller’s book page by page until you find something like this which you can understand, and keep reading the comprehensible paragraphs.

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