The book gives a streamlined introduction to quantum mechanics and describes the basic mathematical structures underpinning it.

From the reviews:

"This book is an introduction to the mathematics of quantum mechanics....The strength of the book is where it shows how the mathematical treatment of quantum mechanics brings insights to physics....It will be useful to the experienced reader as a guide to the impressive recent advances in mathematical quantum mechanics." --SIAM REVIEW

1 Physical Background.- 1.1 The Double-Slit Experiment.- 1.2 Wave Functions.- 1.3 State Space.- 1.4 The Schrödinger Equation.- 1.5 Mathematical Supplement: Operators on Hilbert Spaces.- 2 Dynamics.- 2.1 Conservation of Probability.- 2.2 Existence of Dynamics.- 2.3 The Free Propagator.- 2.4 Mathematical Supplement: Operator Adjoints.- 2.5 Mathematical Supplement: the Fourier Transform.- 2.5.1 Definition of the Fourier Transform.- 2.5.2 Properties of the Fourier Transform.- 2.5.3 Functions of the Derivative.- 3 Observables.- 3.1 Mean Values and the Momentum Operator.- 3.2 Observables.- 3.3 The Heisenberg Representation.- 3.4 Quantization.- 3.5 Pseudodifferential Operators.- 4 The Uncertainty Principle.- 4.1 The Heisenberg Uncertainty Principle.- 4.2 A Refined Uncertainty Principle.- 4.3 Application: Stability of Hydrogen.- 5 Spectral Theory.- 5.1 The Spectrum of an Operator.- 5.2 Functions of Operators and the Spectral Mapping Theorem.- 5.3 Applications to Schrödinger Operators.- 5.4 Spectrum and Evolution.- 5.5 Variational Characterization of Eigenvalues.- 5.6 Number of Bound States.- 5.7 Mathematical Supplement: Integral Operators.- 6 Scattering States.- 6.1 Short-range Interactions: µ > 1.- 6.2 Long-range Interactions: µ ? 1.- 6.3 Existence of Wave Operators.- 7 Special Cases.- 7.1 The Infinite Well.- 7.2 The Torus.- 7.3 A Potential Step.- 7.4 The Square Well.- 7.5 The Harmonic Oscillator.- 7.6 A Particle on a Sphere.- 7.7 The Hydrogen Atom.- 7.8 A Particle in an External EM Field.- 8 Many-particle Systems.- 8.1 Quantization of a Many-particle System.- 8.2 Separation of the Centre-of-mass Motion.- 8.3 Break-ups.- 8.4 The HVZ Theorem.- 8.5 Intra- vs. Inter-cluster Motion.- 8.6 Existence of Bound States for Atoms and Molecules.- 8.7 Scattering States.- 8.8 Mathematical Supplement: Tensor Products.- 9 Density Matrices.- 9.1 Introduction.- 9.2 States and Dynamics.- 9.3 Open Systems.- 9.4 The Thermodynamic Limit.- 9.5 Equilibrium States.- 9.6 The T ? 0 Limit.- 9.7 Example: a System of Harmonic Oscillators.- 9.8 A Particle Coupled to a Reservoir.- 9.9 Quantum Systems.- 9.10 Problems.- 9.11 Hilbert Space Approach.- 9.12 Appendix: the Ideal Bose Gas.- 9.13 Appendix: Bose-Einst ein Condensation.- 9.14 Mathematical Supplement: the Trace, and Trace Class Operators.- 10 The Feynman Path Integral.- 10.1 The Feynman Path Integral.- 10.2 Generalizations of the Path Integral.- 10.3 Mathematical Supplement: the Trotter Product Formula.- 11 Quasi-classical Analysis.- 11.1 Quasi-classical Asymptotics of the Propagator.- 11.2 Quasi-classical Asymptotics of Green's Function.- 11.2.1 Appendix.- 11.3 Bohr-Sommerfeld Semi-classical Quantization.- 11.4 Quasi-classical Asymptotics for the Ground State Energy.- 11.5 Mathematical Supplement: Operator Determinants.- 12 Mathematical Supplement: the Calculus of Variations.- 12.1 Functionals.- 12.2 The First Variation and Critical Points.- 12.3 Constrain ed Variational Problems.- 12.4 The Second Variation.- 12.5 Conjugate Points and Jacobi Fields.- 12.6 The Action of the Critical Path.- 12.7 Appendix: Connection to Geodesics.- 13 Resonances.- 13.1 Tunneling and Resonances.- 13.2 The Free Resonance Energy.- 13.3 Instantons.- 13.4 Positive Temperatures.- 13.5 Pre-exponential Factor for the Bounce.- 13.6 Contribution of the Zero-mode.- 13.7 Bohr-Sommerfeld Quantization for Resonances.- 14 Introduction to Quantum Field Theory.- 14.1 The Place of QFT.- 14.1.1 Physical Theories.- 14.1.2 The Principle of Minimal Action.- 14.2 Klein-Gordon Theory as a Hamiltonian System.- 14.2.1 The Legendre Transform.- 14.2.2 Hamiltonians.- 14.2.3 Poisson Brackets.- 14.2.4 Hamilton's Equations.- 14.3 Maxwell's Equations as a Hamiltonian System.- 14.4 Quantization of the Klein-Gordon and Maxwell Equations.- 14.4.1 The Quantization Procedure.- 14.4.2 Creation and Annihilation Opertors.- 14.4.3 Wick Ordering.- 14.4.4 Quantizing Maxwell's Equations.- 14.5 Fock Space.- 14.6 Generalized Free Theory.- 14.7 Interactions.- 14.8 Quadratic Approximation.- 14.8.1 Further Discussion.- 15 Quantum Electrodynamics of Non-relativistic Particles: the Theory of Radiation.- 15.1 The Hamiltonian.- 15.2 Perturbation Set-up.- 15.3 Results.- 15.4 Mathematical Supplements.- 15.4.1 Spectral Projections.- 15.4.2 Projecting-out Procedure.- 16 Supplement: Renormalization Group.- 16.1 The Decimation Map.- 16.2 Relative Bounds.- 16.3 Elimination of Particle and High Photon Energy Degrees of Freedom.- 16.4 Generalized Normal Form of Operators on Fock Space.- 16.5 The Hamiltonian H0(?, z).- 16.6 A Banach Space of Operators.- 16.7 Rescaling.- 16.8 The Renormalization Map.- 16.9 Linearized Flow.- 16.10 Central-stable Manifold for RG and Spectra of Hamiltonians.- 16.11 Appendix.- 17 Comments on Missing Topics, Literature, and Further Reading.- References.