This volume develops quantum mechanics and its mathematical structure. It emphasizes Hilbert space methods, operators, and physical interpretation.
Part I. Foundations
Chapter 1. Quantum States
1.1 Wave functions 1.2 Hilbert space formulation 1.3 Probability interpretation 1.4 Normalization 1.5 Examples
Chapter 2. Observables and Operators
2.1 Linear operators 2.2 Self-adjoint operators 2.3 Spectral decomposition 2.4 Measurement postulates 2.5 Examples
Chapter 3. Schrödinger Equation
3.1 Time-dependent equation 3.2 Time-independent equation 3.3 Boundary conditions 3.4 Solutions 3.5 Examples
Part II. Basic Systems
Chapter 4. One-Dimensional Systems
4.1 Particle in a box 4.2 Potential wells 4.3 Harmonic oscillator 4.4 Applications 4.5 Examples
Chapter 5. Angular Momentum
5.1 Operators and commutation 5.2 Eigenvalues and eigenvectors 5.3 Spin 5.4 Applications 5.5 Examples
Chapter 6. Central Potentials
6.1 Radial equation 6.2 Hydrogen atom 6.3 Energy levels 6.4 Applications 6.5 Examples
Part III. Approximation Methods
Chapter 7. Perturbation Theory
7.1 Time-independent perturbation 7.2 Degenerate perturbation 7.3 Applications 7.4 Examples 7.5 Extensions
Chapter 8. Variational Methods
8.1 Variational principle 8.2 Trial wave functions 8.3 Applications 8.4 Examples 8.5 Connections
Chapter 9. WKB Approximation
9.1 Semiclassical methods 9.2 Turning points 9.3 Applications 9.4 Examples 9.5 Connections
Part IV. Quantum Dynamics
Chapter 10. Time Evolution
10.1 Unitary evolution 10.2 Propagators 10.3 Heisenberg picture 10.4 Applications 10.5 Examples
Chapter 11. Scattering Theory
11.1 Scattering states 11.2 Cross sections 11.3 Born approximation 11.4 Applications 11.5 Examples
Chapter 12. Open Quantum Systems
12.1 Density matrices 12.2 Decoherence 12.3 Master equations 12.4 Applications 12.5 Examples
Part V. Mathematical Structure
Chapter 13. Hilbert Space Methods
13.1 Basis and expansions 13.2 Operators and domains 13.3 Spectral theory 13.4 Applications 13.5 Examples
Chapter 14. Operator Algebras
14.1 Observables as algebras 14.2 Commutation relations 14.3 Representations 14.4 Applications 14.5 Examples
Chapter 15. Functional Integration (Overview)
15.1 Path integrals 15.2 Action formulation 15.3 Applications 15.4 Examples 15.5 Connections
Part VI. Advanced Topics
Chapter 16. Relativistic Quantum Mechanics
16.1 Klein–Gordon equation 16.2 Dirac equation 16.3 Spinors 16.4 Applications 16.5 Examples
Chapter 17. Quantum Field Theory (Overview)
17.1 Fields as operators 17.2 Quantization 17.3 Interactions 17.4 Applications 17.5 Examples
Chapter 18. Quantum Information
18.1 Qubits 18.2 Entanglement 18.3 Quantum computation 18.4 Applications 18.5 Examples
Part VII. Applications
Chapter 19. Atomic and Molecular Physics
19.1 Spectra 19.2 Transitions 19.3 Applications 19.4 Examples 19.5 Connections
Chapter 20. Condensed Matter Physics
20.1 Band theory 20.2 Many-body systems 20.3 Applications 20.4 Examples 20.5 Connections
Chapter 21. Quantum Technologies
21.1 Quantum devices 21.2 Sensors 21.3 Communication 21.4 Applications 21.5 Examples
Part VIII. Research Directions
Chapter 22. Advanced Topics
22.1 Quantum entanglement theory 22.2 Topological phases 22.3 Quantum simulation 22.4 Modern developments 22.5 Emerging areas
Chapter 23. Open Problems
23.1 Measurement problem 23.2 Many-body complexity 23.3 Quantum gravity links 23.4 Computational challenges 23.5 Future directions
Chapter 24. Historical and Conceptual Notes
24.1 Development of quantum theory 24.2 Key contributors 24.3 Evolution of formalism 24.4 Interpretations 24.5 Summary
Appendix
A. Operator identities reference B. Common potentials C. Proof techniques checklist D. Numerical methods reference E. Cross-reference to other MSC branches