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Beautiful Models: 70 Years of Exactly Solved Quantum Many-Body Problems First Edition Edition

5 out of 5 stars 1 customer review
ISBN-13: 978-9812388971
ISBN-10: 9812388974
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Editorial Reviews

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?This book is self-contained and unified in presentation. It may be used as an advanced textbook by graduate students."
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Product Details

  • Paperback: 400 pages
  • Publisher: World Scientific Pub Co Inc; First Edition edition (September 2004)
  • Language: English
  • ISBN-10: 9812388974
  • ISBN-13: 978-9812388971
  • Product Dimensions: 0.8 x 6 x 8.8 inches
  • Shipping Weight: 1.2 pounds
  • Average Customer Review: 5.0 out of 5 stars  See all reviews (1 customer review)
  • Amazon Best Sellers Rank: #1,934,879 in Books (See Top 100 in Books)

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The main virtue of this book is that it clears up any confusion regarding the notion of integrability in a quantum system. After an historical overview of the theory of exactly solvable systems in chapter 1, the author recalls the notion of integrability in classical mechanics, restricting his discussion to systems that are governed by a Hamiltonian. Using the standard action-angle canonical transformation he shows that the integrability of a Hamiltonian system is, as is well known, indicated by the presence of a finite set of quantities that are in `involution', i.e. they are constants of motion.

This notion of integrability will not work for finite-dimensional quantum systems, as the author shows by using a system that hypothetically has a set L of mutually commuting operators, this set also including the Hamiltonian. He shows that no two commuting operators are algebraically independent, and at most D commuting operators are linearly independent, where D is the dimension of the eigenspace of one of the operators. The author then presents a notion of integrability that is less trivial, in that it will give information on the dynamics of the quantum system.

Since quantum systems are typically systems of particles that are interacting with each other, the dynamical events of interest are the scattering events. Indeed, the scattering theory of quantum systems is highly developed, and has inspired an enormous amount of research in both physics and mathematics.
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