Group Theory and its Application to the Quantum Mechanics of Atomic Spectra by Eugene P. Wigner (PDF)

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⭐This is the classic, timeless text on group and representation theory as applied to atomic spectra with treatments of angular momentum in quantum mechanics, selection and intensity rules with electron spin, fine structure of spectral lines, and Racah coefficients. This Kindle eTextbook is a Print Replica, which means that the pages in their entirety are rendered as graphic images. This means that the text and equations resize together and it doesn’t happen that the equations are tiny and almost unreadable and can’t be resized as happens with so many Kindle eTextbooks that render the equations as graphic images but not the text. The Kindle readers do not yet support mathematical markup language, so equations are always graphic images. Although Print Replica eTextbooks result in larger file sizes, it is worth it to make the equations readable! By the way, I am using the latest Kindle for PC reader on a 15 inch Windows 10 laptop. Amazon should make all Kindle books with equations available in Print Replica format as well as the format where only the equations are graphic images!

⭐Wigner is a master of invariant groups, and his pedagogical style is very, very good. I’m just easing into it, but it certainly appears to be a classic text, likely followed by more modern presentations; however, reading one of the master’s, IMO, can’t be beat!

⭐Eugene Wigner: “If we understand something, its behavior, that is the events which it presents, should not produce any surprise for us. We should always have the impression that it could not be otherwise.” (Nobel Lecture, December 12, 1963).Abraham Pais: “Wigner’s is still an ideal introductory text.” (1986, page 267, Inward Bound).There was a time when three physics publications applying group theory commanded attention: Van Der Waerden’s Group Theory in Quantum Mechanics (1932), Weyl’s The Theory of Groups and Quantum Mechanics (1931) and Wigner’s Group Theory and its Applications to the Quantum Mechanics of Atomic Spectra (1931). One can hardly fail to note that all three publications originated with German- language printings, later translated to English. The field is now crowded with Group Theory books and it is difficult to decide which publications continue to command attention. My favorite of the earlier publications lands in the lap of Eugene Wigner. Now, before you decide to study his book, do yourself a favor and read the collection of essays entitled: Symmetries and Reflections (1967, Indiana University Press). That collection will whet your appetite for all that Eugene Wigner has to offer:(1) Of course, you get Group Theory. Also, gold nuggets of Quantum Mechanics. That is chapter four, five, six: synopsis, perturbation theory and transformation theory. Informative footnote: “The wave-function thus changes in two different ways. First, continuously in the course of of time according to the Schrodinger differential equation, and, secondly, during measurements which are applied discontinuously to the system according to the laws of probability.” (page 50). Regardless of orientation (whether it be physics or mathematics) these three chapters are worth the price of admission.(2) Returning to groups. Eugene Wigner begins simple enough (initial chapters) with vectors and matrices. Even if easy, do not skip these 13 pages. Chapter three is more demanding: Wigner mentions, in passing, Born and Jordan’s concept of super-matrices (page 18). Wigner reminds us that “we will use only unitary, Hermitian, and real orthogonal matrices.” (page 23). Abstract, group theory begins chapter seven. Basic definitions then representations (chapter eight). Reading: “The importance of irreducible representations rests on the fact that any representation can be decomposed into irreducible ones in a unique way.” (page 85).(3) Wigner progresses from the finite to infinite: that is, generalization to continuous groups (chapter 10, pages 88-101). An introduction to Hurwitz integral (page 95). Morton Hamermesh also describes this in his chapter eight (pages 279-320). I like to mention Hamermesh’s book Group Theory and its Physical Applications (1962), at this juncture, because his book distills much of Wigner’s book at more elementary, expository, level. In other words, study both books !(4) Eugene Wigner writes that the three foremost accomplishments brought forth in his monograph are the concept of parity, quantum theory of vector-addition model, also that “almost all the rules of spectroscopy follow from symmetry.” (preface). Read: “the concept of parity, which is very important for the understanding of spectra, has no analogy in classical theory comparable to the analogy between orbital quantum number and the angular momentum.” (page 182). Read: “the vector-addition-model is of very general validity and basic significance for all of spectroscopy.” (page 187).(5) A highlight: Chapter Nineteen: Partial determination of eigenfunctions from their transformation properties. Rigid rotators, quantum-mechanical top are used as physical models from which is extracted information by utilization of symmetry. Eugene Wigner also mentions that these solutions can be found in terms of hypergeometric functions (referring to Morse and Feshbach, Methods of Theoretical Physics, pages 388 and 542). Now read a remark made by Hamermesh: “all our results are obtainable without the formal methods of group theory.” (his preface).(6) Wigner presents the (by now) well-trod path: proceed from symmetric group, then rotation groups, then angular momentum (chapters 13-15). Read: “there exists a two-to-one homomorphism of the group of two-dimensional unitary matrices, with determinant one, onto the three-dimensional pure rotation group.” (page 161). See the textbook by Hamermesh which distills most of this material at introductory vantage (especially section #9.6, pages 348-357). Wigner’s footnote regards tensors and representations (page 168-170) is a most useful piece of wisdom !(7) In many respects, the capstone of the exposition is chapter 26: Time Inversion. Read: “Hence, reversal of the direction of motion, is perhaps a more felicitous, though longer, expression than time-inversion.” (page 325). Read: “Naturally, the considerations of this section do not prove that the quantum mechanical equations are invariant under the operation of time inversion. They do show, however, that if they are, the time inversion operator must be given….” (page 333). I let the prospective reader discover for him/herself what Wigner has to say on the matter, as no student should ignore what he has written !(8) Concluding my all-too-brief account of Eugene Wigner’s outstanding monograph, I say: Get a copy. Study it ! If you get lost at any juncture, turn to the pages of Morton Hamermesh. That which Wigner has expounded in 365 pages, Hamermesh will expound in 500 pages (and you get exercises to solve along the route travelled by Hamermesh). If you study both authors, rest assured, it will all fall into place.

⭐Wigner’s book (published in 1959) is an expanded and revised English translation of the book originally published in German in 1931. Starting from vectors and matrices the author moves on to the description of abstract group theory and its application to atomic spectra. This is a classical reference in the field of quantum mechanics and atomic physics which should be consulted by most physicists and chemists although the style is rather concise and the expected reader is someone that has already been exposed to the basic concepts of group theory and quantum mechanics. A nice and more pedagogical book on group theory is McWeeny’s “Symmetry: An Introduction to Group Theory and Its Applications” while for those interested in the application of group theory to molecules the book “Molecular Vibrations: The Theory of Infrared and Raman Vibrational Spectra” by Wilson, Decius, and Cross is highly recommended (both books are published by Dover).

⭐Eugene Wigner won the 1963 Nobel Prize in Physics, in part due to his contributions to symmetry principles in physics. In reading other books on group theory and quantum physics, you usually find a large number of references to Wigner’s book. In fact, other books often state a theorem and then refer to Wigner’s book for the proof. For me, I was trying to find a proof that I could understand for the Vector Addition Theorem for angular momentum. The proof involves breaking down the direct product of two angular momentum representations into a sum of irreducible representations–that is, D(i) x D(j) = D(i+j) + … + D(i-j). For example D(1) x D(1) = D(2) + D(1) + D(0). Wigner gives a simple proof for this theorem on pp.186-187 in which he just rearranges the components of the characters of the representations. I had already read the proofs for this theorem in Tinkham

⭐and Heine

⭐, but either I could not understand their proofs or I just could not find them convincing. Wigner’s proof, however, was clear and understandable. Please note that I have scanned pp.184-189 so that the reader can get a look at Wigner’s proof.

⭐The book arrived in very good condition

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