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.
Author(s): Brian C. Hall (auth.)
Series: Graduate Texts in Mathematics 267
Edition: 1
Publisher: Springer-Verlag New York
Year: 2013
Language: English
Pages: 554
Tags: Mathematical Physics;Mathematical Applications in the Physical Sciences;Quantum Physics;Functional Analysis;Topological Groups, Lie Groups;Mathematical Methods in Physics
Front Matter....Pages i-xvi
The Experimental Origins of Quantum Mechanics....Pages 1-17
A First Approach to Classical Mechanics....Pages 19-52
A First Approach to Quantum Mechanics....Pages 53-90
The Free Schrödinger Equation....Pages 91-108
A Particle in a Square Well....Pages 109-122
Perspectives on the Spectral Theorem....Pages 123-130
The Spectral Theorem for Bounded Self-Adjoint Operators: Statements....Pages 131-152
The Spectral Theorem for Bounded Self-Adjoint Operators: Proofs....Pages 153-168
Unbounded Self-Adjoint Operators....Pages 169-200
The Spectral Theorem for Unbounded Self-Adjoint Operators....Pages 201-226
The Harmonic Oscillator....Pages 227-238
The Uncertainty Principle....Pages 239-253
Quantization Schemes for Euclidean Space....Pages 255-277
The Stone–von Neumann Theorem....Pages 279-304
The WKB Approximation....Pages 305-331
Lie Groups, Lie Algebras, and Representations....Pages 333-366
Angular Momentum and Spin....Pages 367-391
Radial Potentials and the Hydrogen Atom....Pages 393-418
Systems and Subsystems, Multiple Particles....Pages 419-440
The Path Integral Formulation of Quantum Mechanics....Pages 441-454
Hamiltonian Mechanics on Manifolds....Pages 455-466
Geometric Quantization on Euclidean Space....Pages 467-482
Geometric Quantization on Manifolds....Pages 483-526
Back Matter....Pages 527-554
