Ebook: Mathematical concepts of quantum mechanics

The book gives a streamlined introduction to quantum mechanics while describing the basic mathematical structures underpinning this discipline.

Starting with an overview of key physical experiments illustrating the origin of the physical foundations, the book proceeds with a description of the basic notions of quantum mechanics and their mathematical content.

It then makes its way to topics of current interest, specifically those in which mathematics plays an important role. The more advanced topics presented include many-body systems, modern perturbation theory, path integrals, the theory of resonances, quantum statistics, mean-field theory, second quantization, the theory of radiation (non-relativistic quantum electrodynamics), and the renormalization group.

With different selections of chapters, the book can serve as a text for an introductory, intermediate, or advanced course in quantum mechanics. The last four chapters could also serve as an introductory course in quantum field theory.

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This book describes the main methods of one- and two-dimensional high-resolution nuclear magnetic resonance spectroscopy in liquids within the quantum-mechanical formalism of the density matrix. In view of the increasing importance of NMR in chemistry and biochemistry, the work is particularly addressed to those scientists who do not have a working knowledge of quantum calculations. This knowledge is provided in the first part of the book through a description of quantum mechanics as applied to spin systems. This section is self-contained and limited to essentials. A final chapter is devoted to the principles of relaxation theory. By providing readers with practice in the theoretical tools of NMR, this book opens the way for a critical analysis of comprehensive treatises or articles on new methods. The emphasis throughout is on the actual use of NMR, so the text is neither oversimplified nor overly theoretical. The aim is to describe the basic theoretical tools needed to understand and fully exploit the valuable potential of high-resolution NMR 1. Physical background -- 2. Dynamics -- 3. Observables -- 4. Quantization -- 5. Uncertainty principle and stability of atoms and molecules -- 6. Spectrum and dynamics -- 7. Special cases -- 8. Bound states and variational principle -- 9. Scattering states -- 10. Existence of atoms and molecules -- 11. Perturbation theory: Freshbach-Schur method -- 12. General theory of many-particle systems -- 13. Self-consistent approximations -- 14. The Feynman path integral -- 15. Quasi-classical analysis -- 16. Resonances -- 17. Quantum statistics -- 18. The second quantization -- 19. Quantum electro-magnetic field -- photons -- 20. Standard model of non-relativistic matter and radiation -- 21. Theory of radiation -- 22. Renormalization group -- 23. Mathematical supplement: spectral analysis -- 24. Mathematical supplement: the calculus of variations -- 25. Comments on literature, and further reading