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AuthorGustafson, Stephen J. author
TitleMathematical Concepts of Quantum Mechanics [electronic resource] / by Stephen J. Gustafson, Israel Michael Sigal
ImprintBerlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 2003
Connect tohttp://dx.doi.org/10.1007/978-3-642-55729-3
Descript X, 253 p. 2 illus. online resource

SUMMARY

The first fifteen chapters of these lectures (omitting four to six chapters each year) cover a one term course taken by a mixed group of senior undergraduate and junior graduate students specializing either in mathematics or physics. Typically, the mathematics students have some background in advanced analยญ ysis, while the physics students have had introductory quantum mechanics. To satisfy such a disparate audience, we decided to select material which is interesting from the viewpoint of modern theoretical physics, and which illustrates an interplay of ideas from various fields of mathematics such as operator theory, probability, differential equations, and differential geometry. Given our time constraint, we have often pursued mathematical content at the expense of rigor. However, wherever we have sacrificed the latter, we have tried to explain whether the result is an established fact, or, mathematically speaking, a conjecture, and in the former case, how a given argument can be made rigorous. The present book retains these features


CONTENT

1 Physical Background -- 1.1 The Double-Slit Experiment -- 1.2 Wave Functions -- 1.3 State Space -- 1.4 The Schrรถdinger 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 Schrรถdinger 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: ยต > 1 -- 6.2 Long-range Interactions: ยต ? 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 Greenโ{128}{153}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 Maxwellโ{128}{153}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 of Non-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


Mathematics Functional analysis Physics Quantum physics Mathematics Functional Analysis Theoretical Mathematical and Computational Physics Quantum Physics



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