Quantum Theory for Mathematicians

Couverture
Springer Science & Business Media, 19 juin 2013 - 554 pages

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 Schrödinger equation in one space dimension; the Spectral Theorem for bounded and unbounded self-adjoint operators; the Stone–von Neumann Theorem; the Wentzel–Kramers–Brillouin approximation; 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.

 

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Table des matières

1 The Experimental Origins of Quantum Mechanics
1
2 A First Approach to Classical Mechanics
19
3 A First Approach to Quantum Mechanics
53
4 The Free Schrödinger Equation
91
5 A Particle in a Square Well
109
6 Perspectives on the Spectral Theorem
123
Statements
131
Proofs
153
15 The WKB Approximation
305
16 Lie Groups Lie Algebras and Representations
332
17 Angular Momentum and Spin
367
18 Radial Potentials and the Hydrogen Atom
393
19 Systems and Subsystems Multiple Particles
419
20 The Path Integral Formulation of Quantum Mechanics
441
21 Hamiltonian Mechanics on Manifolds
455
22 Geometric Quantization on Euclidean Space
467

9 Unbounded SelfAdjoint Operators
169
10 The Spectral Theorem for Unbounded SelfAdjointOperators
201
11 The Harmonic Oscillator
227
12 The Uncertainty Principle
239
13 Quantization Schemes for Euclidean Space
254
14 The Stonevon Neumann Theorem
279
23 Geometric Quantization on Manifolds
483
Appendix A Review of Basic Material
527
References
545
Index
549
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À propos de l'auteur (2013)

Brian C. Hall is a Professor of Mathematics at the University of Notre Dame.

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