Quantum theory for mathematicians

Content: 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 R[superscript n]
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 R [superscript n]
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 harmonic oscillator: The role of the harmonic oscillator
The algebraic approach
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 R²[superscript n]
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 --
Geometric quantization on Euclidean space: Introduction
Prequantization
Problems with prequantization
Quantization
Quantization of observables
Exercises --
Geometric 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.