**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.

Lecture# Linear Algebra: Quantum Mechanics

Description

This lecture covers the application of linear algebra concepts to change of coordinates, the postulates of Quantum Mechanics, and important linear algebra concepts such as Hilbert spaces, self-adjoint operators, and unitary operators. It also delves into the spectral theorem for Hermitian operators and the diagonalization of commutative Hermitian operators. The lecture emphasizes the importance of linear algebra in understanding fundamental concepts in quantum treatment of materials properties.

Official source

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related concepts (166)

Spectral theory

In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces. It is a result of studies of linear algebra and the solutions of systems of linear equations and their generalizations. The theory is connected to that of analytic functions because the spectral properties of an operator are related to analytic functions of the spectral parameter.

Self-adjoint operator

In mathematics, a self-adjoint operator on an infinite-dimensional complex vector space V with inner product (equivalently, a Hermitian operator in the finite-dimensional case) is a linear map A (from V to itself) that is its own adjoint. If V is finite-dimensional with a given orthonormal basis, this is equivalent to the condition that the matrix of A is a Hermitian matrix, i.e., equal to its conjugate transpose A^∗. By the finite-dimensional spectral theorem, V has an orthonormal basis such that the matrix of A relative to this basis is a diagonal matrix with entries in the real numbers.

Vector space

In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called vectors, may be added together and multiplied ("scaled") by numbers called scalars. Scalars are often real numbers, but can be complex numbers or, more generally, elements of any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called vector axioms. The terms real vector space and complex vector space are often used to specify the nature of the scalars: real coordinate space or complex coordinate space.

Spectral radius

In mathematics, the spectral radius of a square matrix is the maximum of the absolute values of its eigenvalues. More generally, the spectral radius of a bounded linear operator is the supremum of the absolute values of the elements of its spectrum. The spectral radius is often denoted by ρ(·). Let λ1, ..., λn be the eigenvalues of a matrix A ∈ Cn×n. The spectral radius of A is defined as The spectral radius can be thought of as an infimum of all norms of a matrix.

Linear algebra

Linear algebra is the branch of mathematics concerning linear equations such as: linear maps such as: and their representations in vector spaces and through matrices. Linear algebra is central to almost all areas of mathematics. For instance, linear algebra is fundamental in modern presentations of geometry, including for defining basic objects such as lines, planes and rotations. Also, functional analysis, a branch of mathematical analysis, may be viewed as the application of linear algebra to spaces of functions.

Related lectures (554)

Linear Algebra: Bases and SpacesMATH-111(e): Linear Algebra

Covers linear independence, bases, and spaces in linear algebra, emphasizing kernel and image spaces.

Vector Spaces: Subspaces and Bases

Covers subspaces, bases, linear independence, and dimensionality in vector spaces.

Polynomials: Operations and PropertiesMATH-111(e): Linear Algebra

Explores polynomial operations, properties, and subspaces in vector spaces.

Vector Spaces: Axioms and ExamplesMATH-111(e): Linear Algebra

Covers the axioms and examples of vector spaces, including matrices and polynomials.

Characteristic Polynomials and Similar MatricesMATH-111(e): Linear Algebra

Explores characteristic polynomials, similarity of matrices, and eigenvalues in linear transformations.