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Graduate Texts in Mathematics #267
Quantum Theory for Mathematicians
Although ideas from quantum physics play an important role in many parts of modern mathematics, there are few books about quantum mechanics aimed at mathematicians. This book introduces the main ideas of quantum mechanics in language familiar to mathematicians. Readers with little prior exposure to physics will enjoy the book's conversational tone as they delve into such topics as the Hilbert space approach to quantum theory; the Schrodinger equation in one space dimension; the Spectral Theorem; for bounded and unbounded self-adjoint operators; the Stone-von Neumann Theorem; the Wentzel-Kramers-Brillouin approximations; the role of Lie groups and Lie algebras in quantum mechanics; and the path-integral approach to quantum mechanics.
The numerous exercises at the end of each chapter make the book suitable for both graduate courses and independent study. Most of the text is accessible to graduate students in mathematics who have had a first course in real analysis, covering the basics of L2 spaces and Hilbert spaces. The final chapters introduce readers who are familiar with the theory of manifolds to more advanced topics, including geometric quantization.
The numerous exercises at the end of each chapter make the book suitable for both graduate courses and independent study. Most of the text is accessible to graduate students in mathematics who have had a first course in real analysis, covering the basics of L2 spaces and Hilbert spaces. The final chapters introduce readers who are familiar with the theory of manifolds to more advanced topics, including geometric quantization.
570 pages, Hardcover
First published December 18, 2012
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Displaying 1 - 2 of 2 reviews
January 8, 2023
A better, less drawn-out, more modern version of the 'Quantum Fields and Strings; A Course for Mathematicians' series edited by Pierre Deligne. It's great if you're familiar with the mathematics and would prefer the pages be used to explain quantum theory through mathematical means. In most physics textbooks, the terms in physics are used while the terms in mathematics are explained. Here, it's the opposite. It's very clearly physics from a mathematician's perspective, including critiques of some systems of mathematics used in physics, such as geometric quantization:
- "In the first place, geometric quantization has too many definitions (bundles, connections, curvature, polarizations, half-forms) and too few theorems..." -p. 484
The other alternatives to this book are highly insufficient in comparison, seeming more like an explanation of the mathematics behind quantum theory as opposed to a genuine explanation of quantum theory itself through mathematical means. Spectacular work.
- "In the first place, geometric quantization has too many definitions (bundles, connections, curvature, polarizations, half-forms) and too few theorems..." -p. 484
The other alternatives to this book are highly insufficient in comparison, seeming more like an explanation of the mathematics behind quantum theory as opposed to a genuine explanation of quantum theory itself through mathematical means. Spectacular work.
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March 11, 2024 chapter 7
spectral theorem for bonuded self-adjoint operators I
mu:sigma-algebra->bounded_operators(H->H) is projection-valued measure
9.6 Counterexample: The operator A:D(A) in L^2[0,1]->L^2[0,1] with D(A)={u continuosly differentiable on [0,1] and u(0)=u(1)=0} and Au=-i h partial_x u is symmetric but not self-adjoint.
(h is the Plank's constant).
Important! If Au=partial_x u, the operator A is not symmetric!
chapter 10
Example 10.12 H=L^2(R^n) and (T(t)u)(x)=u(x+ta) is strongly continuous one parameter unitary group (a is a vector)
spectral theorem for bonuded self-adjoint operators I
mu:sigma-algebra->bounded_operators(H->H) is projection-valued measure
9.6 Counterexample: The operator A:D(A) in L^2[0,1]->L^2[0,1] with D(A)={u continuosly differentiable on [0,1] and u(0)=u(1)=0} and Au=-i h partial_x u is symmetric but not self-adjoint.
(h is the Plank's constant).
Important! If Au=partial_x u, the operator A is not symmetric!
chapter 10
Example 10.12 H=L^2(R^n) and (T(t)u)(x)=u(x+ta) is strongly continuous one parameter unitary group (a is a vector)
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