Mediaspace scheduled maintenance: Aug 25, 2026 07:00 - 12:00 AM. During this time, videos will be temporarily unavailable. Check status updates.

Lecture

Sobolev Spaces and Unitary Groups

In course

MATH-495: Mathematical quantum mechanics

Quantum mechanics is one of the most successful physical theories. This course presents the mathematical formalism (functional analysis and spectral theory) that underlies quantum mechanics. It is simultaneously an introduction to mathematical physics and ...

Description

This lecture covers the proof that initial data in H^2 solve the free Schrödinger equation in the L^2-sense, the definition of the generator of a one-parameter unitary group, and the properties of the Laplacian as the generator of free time evolution. It also introduces the momentum operator as the generator of the translation group on the real line, defines Sobolev spaces on an interval, presents the baby Sobolev embedding theorem, and discusses the translation group on L^2 over an interval.

This video is available exclusively on Mediaspace for a restricted audience. Please log in to MediaSpace to access it if you have the necessary permissions.

Watch on Mediaspace

Official source

Related concepts (32)

Translation

Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between translating (a written text) and interpreting (oral or signed communication between users of different languages); under this distinction, translation can begin only after the appearance of writing within a language community.

Translation studies

Translation studies is an academic interdiscipline dealing with the systematic study of the theory, description and application of translation, interpreting, and localization. As an interdiscipline, translation studies borrows much from the various fields of study that support translation. These include comparative literature, computer science, history, linguistics, philology, philosophy, semiotics, and terminology. The term "translation studies" was coined by the Amsterdam-based American scholar James S.

Number line

In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a point. The integers are often shown as specially-marked points evenly spaced on the line. Although the image only shows the integers from –3 to 3, the line includes all real numbers, continuing forever in each direction, and also numbers that are between the integers.

Schrödinger equation

The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. Its discovery was a significant landmark in the development of quantum mechanics. The equation is named after Erwin Schrödinger, who postulated the equation in 1925 and published it in 1926, forming the basis for the work that resulted in his Nobel Prize in Physics in 1933. Conceptually, the Schrödinger equation is the quantum counterpart of Newton's second law in classical mechanics.

Unitary group

In mathematics, the unitary group of degree n, denoted U(n), is the group of n × n unitary matrices, with the group operation of matrix multiplication. The unitary group is a subgroup of the general linear group GL(n, C). Hyperorthogonal group is an archaic name for the unitary group, especially over finite fields. For the group of unitary matrices with determinant 1, see Special unitary group. In the simple case n = 1, the group U(1) corresponds to the circle group, consisting of all complex numbers with absolute value 1, under multiplication.

Related lectures (40)

Position and Momentum Operators

Explores position and momentum operators, eigenstates, and probability density in three dimensions.

Real Numbers: Sets and Operations

Explores the fundamental concepts of real numbers, including sets, operations, and properties like supremum and infimum.

Cohomology Real Projective Space

Covers cohomology in real projective spaces, focusing on associative properties and algebraic structures.

Quantum Harmonic Oscillator: Basics and Operators

Introduces the quantum harmonic oscillator basics, including operators and energy states.

Mathematics: Analysis and Algebra Overview