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Quantum theory for mathematicians / Brian C. Hall.

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Author/Creator:
Hall, Brian C., author.
Language:
English.
Publication date:
2013
Copyright date:
2013
Copyright notice:
New York : Springer, [2013]
Publication:
©2013
Format:
  • Book
  • xvi, 554 pages ; 24 cm.
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 runge-lenz 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
  • Time-evolution 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 time-independent schrödinger equation
  • Domain questions and the matching conditions
  • Finding square-integrable solutions
  • Tunneling and the classically forbidden region
  • Discrete and continuous spectrum
  • Exercises
  • Perspectives 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 operator
  • The 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
  • Exercises
  • The spectral theorem for bounded self-adjoint operators : proofs
  • Proof of the spectral theorem, first version
  • Proof of the spectral theorem, second version
  • Exercises
  • Unbounded 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
  • Exercises
  • The 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
  • 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 Stone-von 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
  • Finite-dimensional representations of lie groups and lie algebras
  • New representations from old
  • Infinite-dimensional 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
  • Runge-Lenz calculations
  • Exercises
  • Systems 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
  • Exercises
  • The 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
  • 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 half-forms
  • Quantization with half-forms : the real case
  • Quantization with half-forms : 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.
Series:
Graduate texts in mathematics, 0072-5285 ; 267
Graduate texts in mathematics ; 267.
Subjects:
ISBN:
9781461471158
146147115X

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