Quantum theory for mathematicians
 Author/Creator
 Hall, Brian C., author.
 Language
 English.
 Copyright notice
 New York : Springer, [2013]
 Publication
 ©2013
 Physical description
 xvi, 554 pages ; 24 cm.
 Series
 Graduate texts in mathematics ; 267.
Access
Available online

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QC174.12 .H35 2013

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QC174.12 .H35 2013
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Contents/Summary
 Bibliography
 Includes bibliographical references and index.
 Contents

 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
 Exercises
 A first approach to classical mechanics
 Motion in R1
 Motion in Rn
 Systems of particles
 Angular momentum
 Poisson brackets and hamiltonian mechanics
 The kepler problem and the rungelenz vector
 Exercises
 First 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
 Timeevolution in quantum theory
 The heisenberg picture
 Example : a particle in a box
 Quantum mechanics for a particle in Rn
 Systems of multiple particles
 Physics notation
 Exercises
 The 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
 Exercises
 Particle in a Square Well
 The timeindependent schrödinger equation
 Domain questions and the matching conditions
 Finding squareintegrable solutions
 Tunneling and the classically forbidden region
 Discrete and continuous spectrum
 Exercises
 Perspectives on the spectral theorem
 The difficulties with the infinitedimensional case
 The goals of spectral theory
 A guide to reading
 The position operator
 Multiplication operators
 The momentum operator
 The spectral theorem for bounded selfadjoint operators : statements
 Elementary properties of bounded operators
 Spectral theorem for bounded selfadjoint operators, I
 Spectral theorem for bounded selfadjoint operators, II
 Exercises
 The spectral theorem for bounded selfadjoint operators : proofs
 Proof of the spectral theorem, first version
 Proof of the spectral theorem, second version
 Exercises
 Unbounded selfadjoint operators
 Introduction
 Adjoint and closure of an unbounded operator
 Elementary properties of adjoints and closed operators
 The spectrum of an unbounded operator
 Conditions for selfadjointness and essential selfadjointness
 A counterexample
 An example
 The basic operators of quantum mechanics
 Sums of selfadjoint operators
 Another counterexample
 Exercises
 The spectral theorem for unbounded selfadjoint operators
 Statements of the spectral theorem
 Stone's theorem and oneparameter unitary groups
 The spectral theorem for bounded normal operators
 Proof of the spectral theorem for unbounded selfadjoint operators
 Exercises
 The harmonie oscillator
 The role of the harmonie oscillator
 The algebraic appfoach
 The analytic approach
 Domain conditions and completeness
 Exercises
 The uncertainty principle
 Uncertainty principle, first version
 A counterexample
 Uncertainty principle, second version
 Minimum uncertainty states
 Exercises
 Quantization schemes for euclidean space
 Ordering ambiguities
 Some common quantization schemes
 The weyl quantization for R2n
 The "No Go" theorem of groenewold
 Exercises
 The Stonevon Neumann Theorem
 A Heuristic argument
 The exponentiated commutation relations
 The theorem
 The segal bargmann space
 Exercises
 The 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
 Exercises
 Lie 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
 Finitedimensional representations of lie groups and lie algebras
 New representations from old
 Infinitedimensional unitary representations
 Exercises
 Angular 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
 Exercises
 Radial 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
 RungeLenz calculations
 Exercises
 Systems and subsystems, multiple particles
 Introduction
 Traceclass and hilbertschmidt 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
 Exercises
 The path integral formulation of quantum mechanics
 Trotter product formula
 Formal derivation of the feynman path integral
 The imaginarytime calculation
 The wiener measure
 The feynmankac formula
 Path integrals in quantum field theory
 Exercises
 Hamiltonian mechanics on manifolds
 Calculus on manifolds
 Mechanics on symplectic manifolds
 Exercises
 Geometrie quantization on euclidean space
 Introduction
 Prequantization
 Problems with prequantization
 Quantization
 Quantization of observables
 Exercises
 Geometrie quantization on manifolds
 Introduction
 Line bundles and connections
 Prequantization
 Polarizations
 Quantization without halfforms
 Quantization with halfforms : the real case
 Quantization with halfforms : the complex case
 Pairing maps
 Exercises
 A review of basic material
 Tensor products of vector spaces
 Measure theory
 Elementary functional analysis
 Hilbert spaces and operators on them
 References
 Index.
 Publisher's Summary
 Although ideas from quantum physics play an important role in many parts of modern mathematics, there are few books about quantum mechanics aimed at mathematicians. This book introduces the main ideas of quantum mechanics in language familiar to mathematicians. Readers with little prior exposure to physics will enjoy the book's conversational tone as they delve into such topics as the Hilbert space approach to quantum theory; the Schrodinger equation in one space dimension; the Spectral Theorem for bounded and unbounded selfadjoint operators; the Stonevon Neumann Theorem; the WentzelKramersBrillouin approximation; the role of Lie groups and Lie algebras in quantum mechanics; and the pathintegral approach to quantum mechanics. The numerous exercises at the end of each chapter make the book suitable for both graduate courses and independent study. Most of the text is accessible to graduate students in mathematics who have had a first course in real analysis, covering the basics of L2 spaces and Hilbert spaces. The final chapters introduce readers who are familiar with the theory of manifolds to more advanced topics, including geometric quantization.
(source: Nielsen Book Data)
Subjects
 Subject
 Quantum theory > Mathematics.
Bibliographic information
 Publication date
 2013
 Copyright date
 2013
 Responsibility
 Brian C. Hall.
 Series
 Graduate texts in mathematics, 00725285 ; 267
 ISBN
 9781461471158
 146147115X