This book provides a coherent introduction to Gutzwiller s trace formula accessible to well-prepared science, mathematics, and engineering students who have taken introductory courses in linear algebra, classical, and quantum mechanics. In addition to providing an enrichment of the undergraduate curriculum, this book may serve as the primary text for graduate courses on semiclassical methods. Since periodic-orbit expansions may be used to solve all types of wave systems that typically occur in mathematics, physics, and engineering, this book is attractive for professional scientists and engineers as well. Following a thorough review of elementary concepts in classical and quantum mechanics, the reader is introduced to the idea of classical periodic orbits, the foundation of Gutzwiller s approach to quantum spectra. The trace formula itself is derived following an introduction to Feynman s path integrals. Numerous applications, including the exact solutions of unsolvable one-dimensional quantum problems, illustrate the power of Gutzwiller s method. Worked examples throughout the text illustrate the material and provide immediate hands-on demonstrations of tools and concepts just learned. Problems at the end of each section invite the reader to consolidate the acquired knowledge.

1. Math Tools
2. Special Functions
3. Newton's Particle Mechanics
4. Schrodinger's Wave Mechanics
5. Semiclassics
6. Test Potentials
7. Periodic-Orbit Quantization
8. Quantum Graphs
9. Advanced Mathematical Techniques
10. Analysis Mechanics
11. Chaos
12. Advanced Concepts in Quantum Mechanics
13. Semiclassical Propagators
14. Applications