Quantum theory for mathematicians 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 ;Stones 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.