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Mathematical Concepts of Quantum Mechanics

E-BookPDF1 - PDF WatermarkE-Book
253 Seiten
Englisch
Springer Berlin Heidelbergerschienen am06.12.20122003

Produkt

Details
Weitere ISBN/GTIN9783642557293
ProduktartE-Book
EinbandartE-Book
FormatPDF
Format Hinweis1 - PDF Watermark
FormatE107
Erscheinungsjahr2012
Erscheinungsdatum06.12.2012
Auflage2003
ReiheUniversitext
Seiten253 Seiten
SpracheEnglisch
IllustrationenX, 253 p. 2 illus.
Artikel-Nr.8775499
Rubriken
Genre9200

Inhalt/Kritik

Inhaltsverzeichnis
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.- 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.- 9.4The 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.- 11.3 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.- 14.2 Klein-Gordon Theory as a Hamiltonian System.- 14.3 Maxwell's Equations as a Hamiltonian System.- 14.4 Quantization of the Klein-Gordon and Maxwell Equations.- 14.5 Fock Space.- 14.6 Generalized Free Theory.- 14.7 Interactions.- 14.8 Quadratic Approximation.- 15 Quantum Electrodynamics ofNon-relativistic Particles: the Theory of Radiation.- 15.1 The Hamiltonian.- 15.2 Perturbation Set-up.- 15.3 Results.- 15.4 Mathematical Supplements.- 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.mehr