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 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.

Table of Contents

1 The Experimental Origins of Quantum Mechanics

2 A First Approach to Classical Mechanics

3 A First Approach to Quantum Mechanics

4 The Free Schrodinger Equation

5 A Particle in a Square Well

6 Perspectives on the Spectral Theorem

7 The Spectral Theorem for Bounded Self-AdjointOperators: Statements

8 The Spectral Theorem for Bounded Self-AdjointOperators: Proofs

9 Unbounded Self-Adjoint Operators

10 The Spectral Theorem for Unbounded Self-AdjointOperators

11 The Harmonic Oscillator

12 The Uncertainty Principle

13 Quantization Schemes for Euclidean Space

14 The Stone- von Neumann Theorem

15 The WKB Approximation

16 Lie Groups, Lie Algebras, and Representations

17 Angular Momentum and Spin

18 Radial Potentials and the Hydrogen Atom

19 Systems and Subsystems, Multiple Particles

20 The Path Integral Formulation of Quantum Mechanics

21 Hamiltonian Mechanics on Manifolds

22 Geometric Quantization on Euclidean Space

23 Geometric Quantization on Manifolds

Appendix A Review of Basic Material

About the Author

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