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 |