Quantum theory for mathematicians

• Verfasst von: Hall, Brian C. [VerfasserIn]
• Titel: Quantum theory for mathematicians
• Verf.angabe: Brian C. Hall
• Verlagsort: New York ; Heidelberg ; Dordrecht ; London
• Verlag: Springer
• E-Jahr: 2013
• Jahr: [2013]
• Umfang: xvi, 554 Seiten
• Illustrationen: Illustrationen, Diagramme
• Gesamttitel/Reihe: Graduate texts in mathematics ; 267
• Fussnoten: Literaturverzeichnis: Seiten 545-548
• ISBN: 978-1-4614-7115-8
• Schlagwörter: (s)Quantenmechanik / (s)Mathematische Methode
• Sprache: eng
• Bibliogr. Hinweis: Erscheint auch als : Online-Ausgabe: Hall, Brian C.: Quantum Theory for Mathematicians. - New York, NY : Springer, 2013. - Online-Ressource (XVI, 554 p. 30 illus., 2 illus. in color, digital)
• Bibliogr. Hinweis: Erscheint auch als : Online-Ausgabe: Hall, Brian C.: Quantum Theory for Mathematicians. - New York, NY : Springer, 2013. - Online-Ressource (XVI, 554 p. 30 illus., 2 illus. in color, digital)
• RVK-Notation: UK 1000
• RVK-Notation: SI 990
• RVK-Notation: UK 1200
• RVK-Notation: SK 950
• K10plus-PPN: 74633804X

Abstract

The experimental origins of quantum mechanics:Is light a wave or a particle? ;Is an electron a wave or a particle? ;Schrödinger and Heisenberg ;A matter of interpretation ;ExercisesA first approach to classical mechanics:Motion in R¹ ;Motion in R[superscript n] ;Systems of particles ;Angular momentum ;Poisson brackets and Hamiltonian mechanics ;The Kepler problem and the Runge-Lenz vector ;ExercisesFirst approach to quantum mechanics:Waves, particles, and probabilities ;A few words about operators and their adjoints ;Position and the position operator ;Momentum and the momentum operator ;The position and momentum operators ;Axioms of quantum mechanics : operators and measurements ;Time-evolution in quantum theory ;The Heisenberg picture ;Example : a particle in a box ;Quantum mechanics for a particle in R [superscript n] ;Systems of multiple particles ;Physics notation ;ExercisesThe free Schrödinger equation:Solution by means of the Fourier transform ;Solution as a convolution ;Propagation of the wave packet : first approach ;Propagation of the wave packet : second approach ;Spread of the wave packet ;ExercisesParticle in a square well:The time-independent Schrödinger equation ;Domain questions and the matching conditions ;Finding square-integrable solutions ;Tunneling and the classically forbidden region ;Discrete and continuous spectrum ;ExercisesPerspectives on the spectral theorem:The difficulties with the infinite-dimensional case ;The goals of spectral theory ;A guide to reading ;The position operator ;Multiplication operators ;The momentum operatorThe spectral theorem for bounded self-adjoint operators : statements:Elementary properties of bounded operators ;Spectral theorem for bounded self-adjoint operators, I ;Spectral theorem for bounded self-adjoint operators, II ;ExercisesThe spectral theorem for bounded self-adjoint operators : proofs:Proof of the spectral theorem, first version ;Proof of the spectral theorem, second version ;ExercisesUnbounded self-adjoint operators:Introduction ;Adjoint and closure of an unbounded operator ;Elementary properties of adjoints and closed operators ;The spectrum of an unbounded operator ;Conditions for self-adjointness and essential self-adjointness ;A counterexample ;An example ;The basic operators of quantum mechanics ;Sums of self-adjoint operators ;Another counterexample ;ExercisesThe spectral theorem for unbounded self-adjoint operators:Statements of the spectral theorem ;Stone's theorem and one-parameter unitary groups ;The spectral theorem for bounded normal operators ;Proof of the spectral theorem for unbounded self-adjoint operators ;ExercisesThe harmonic oscillator:The role of the harmonic oscillator ;The algebraic approach ;The analytic approach ;Domain conditions and completeness ;ExercisesThe uncertainty principle:Uncertainty principle, first version ;A counterexample ;Uncertainty principle, second version ;Minimum uncertainty states ;ExercisesQuantization schemes for Euclidean space:Ordering ambiguities ;Some common quantization schemes ;The Weyl quantization for R²[superscript n] ;The "No go" theorem of Groenewold ;ExercisesThe Stone-Von Neumann theorem:A heuristic argument ;The exponentiated commutation relations ;The theorem ;The Segal-Bargmann space ;ExercisesThe WKB approximation:Introduction ;The old quantum theory and the Bohr-Sommerfeld condition ;Classical and semiclassical approximations ;The WKB approximation away from the turning points ;The Airy function and the connection formulas ;A rigorous error estimate ;Other approaches ;ExercisesLie groups, Lie algebras, and representations:Summary ;Matrix Lie groups ;Lie algebras ;The matrix exponential ;The Lie algebra of a matrix Lie group ;Relationships between Lie groups and Lie algebras ;Finite-dimensional representations of Lie groups and Lie algebras ;New representations from old ;Infinite-dimensional unitary representations ;ExercisesAngular momentum and spin:The role of angular momentum in quantum mechanics ;The angular momentum operators in R³ ;Angular momentum from the Lie algebra point of view ;The irreducible representations of so(3) ;The irreducible representations of SO(3) ;Realizing the representations inside L²(S²)Realizing the representations inside L²(M³) ;Spin ;Tensor products of representations : "addition of angular momentum" ;Vectors and vector operators ;ExercisesRadial potentials and the hydrogen atom:Radial potentials ;The hydrogen atom : preliminaries ;The bound states of the hydrogen atom ;The Runge-Lenz vector in the quantum Kepler problem ;The role of spin ;Runge-Lenz calculations ;ExercisesSystems and subsystems, multiple particles:Introduction ;Trace-class and Hilbert-Schmidt operators ;Density matrices : the general notion of the state of a quantum system ;Modified axioms for quantum mechanics ;Composite systems and the tensor product ;Multiple particles : bosons and fermions ;"Statistics" and the Pauli exclusion principle ;ExercisesThe path integral formulation of quantum mechanics:Trotter product formula ;Formal derivation of the Feynman path integral ;The imaginary-time calculation ;The Wiener measure ;The Feynman-Kac formula ;Path integrals in quantum field theory ;ExercisesHamiltonian mechanics on manifolds:Calculus on manifolds ;Mechanics on symplectic manifolds ;ExercisesGeometric quantization on Euclidean space:Introduction ;Prequantization ;Problems with prequantization ;Quantization ;Quantization of observables ;ExercisesGeometric quantization on manifolds:Introduction ;Line bundles and connections ;Prequantization ;Polarizations ;Quantization without half-forms ;Quantization with half-forms : the real case ;Quantization with half-forms : the complex case ;Pairing maps ;ExercisesA review of basic material:Tensor products of vector spaces ;Measure theory ;Elementary functional analysis ;Hilbert spaces and operators on them.