Course

MATH-495: Mathematical quantum mechanics

Lectures in this course (30)

Function Spaces: Review and Compact Operators

Covers the review of function spaces and explores the concept of compact operators.

Quantum Mechanics: Mathematical Framework

Introduces the need for a mathematical framework to describe linear operators on infinite-dimensional Hilbert spaces in quantum mechanics.

Quantum States and Operators

Covers the description of states in quantum mechanics and their relation to probability, expectation values, and the Schrödinger equation.

Orthonormal Sequences and Hilbert Spaces

Explores orthonormal sequences, Bessel's inequality, and unique decompositions in Hilbert spaces.

Fourier Transform and Schwartz Space

Explores the Fourier transform, Schwartz space, and Riemann-Lebesgue lemma in quantum physics.

Translation Groups on [0,1]

Covers the study of translation groups on the interval [0,1] with different phases and the Riesz representation theorem on Hilbert space.

Theory of Bounded Operators on Hilbert Space

Explores the theory of bounded operators on Hilbert space, including adjoint properties and self-adjointness.

Fourier Transform: Schwartz Space

Covers the Fourier transform on Schwartz space and its properties, including continuity and linearity, as well as the density of smooth compactly supported functions.

Plancherel's Identity and Schrödinger Equation

Covers Plancherel's identity and the solution of the free Schrödinger equation with initial data in Schwartz space.

Essential Operators: Spectrum and Resolvent Set

Covers the essential concepts of adjoint operators, spectrum, and resolvent sets in operator theory.

Self-Adjoint Operators

Covers the criteria for operators to be self-adjoint and the Friedrichs extension theorem.

Free Propagation and Heat Equation

Covers free propagation and the heat equation solution, discussing momentum preservation and Fourier multipliers.

Convolution and Fourier Transform

Explores convolution properties, heat equation application, and Fourier transform on tempered distributions.

Spectral Theory: Friedrichs Extension Theorem

Covers the proof of Friedrichs extension theorem and the properties of the spectrum and resolvent map.

Functional Calculus: Self-Adjoint Operators

Covers self-adjoint operators, Weyl criterion, and functional calculus in the context of symmetric operators and real spectrum.

Distributional Derivatives

Explores distributional derivatives, continuity, boundedness of linear operators, and weak-* continuity.

Functional Calculus: Simple Functions

Covers the extension of functional calculus to simple functions and the concept of *-homomorphism.

Extension of Linear Transformations

Covers the extension of bounded linear transformations and the free propagator in L^2 spaces.

Sobolev Spaces and Unitary Groups

Explores Sobolev spaces, unitary groups, Laplacian properties, and translation groups in L^2.

Functional Calculus and Borel Transform

Covers bounded and measurable functional calculus, Borel transform, and unbounded functions.