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

Lecture

Postulates of Quantum Mechanics

Description

This lecture covers the postulates of Quantum Mechanics, focusing on the state of a system as a complex-valued vector in a Hilbert space. It delves into linear operators, eigenvalues, eigenvectors, and the concept of completeness. The lecture also discusses Dirac notation, inner products, adjoint operators, and the 'bra-ket' notation. It emphasizes the importance of self-adjoint operators and provides mathematical notations for describing physical states.

Official source

Ontological neighbourhood

Mathematics

Analysis: Functional analysis

Algebra: Linear algebra

Related concepts (32)

Eigenvalues and eigenvectors

In linear algebra, an eigenvector (ˈaɪgənˌvɛktər) or characteristic vector of a linear transformation is a nonzero vector that changes at most by a constant factor when that linear transformation is applied to it. The corresponding eigenvalue, often represented by , is the multiplying factor. Geometrically, a transformation matrix rotates, stretches, or shears the vectors it acts upon. The eigenvectors for a linear transformation matrix are the set of vectors that are only stretched, with no rotation or shear.

Eigendecomposition of a matrix

In linear algebra, eigendecomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors. Only diagonalizable matrices can be factorized in this way. When the matrix being factorized is a normal or real symmetric matrix, the decomposition is called "spectral decomposition", derived from the spectral theorem. Eigenvalue, eigenvector and eigenspace A (nonzero) vector v of dimension N is an eigenvector of a square N × N matrix A if it satisfies a linear equation of the form for some scalar λ.

Unbounded operator

In mathematics, more specifically functional analysis and operator theory, the notion of unbounded operator provides an abstract framework for dealing with differential operators, unbounded observables in quantum mechanics, and other cases. The term "unbounded operator" can be misleading, since "unbounded" should sometimes be understood as "not necessarily bounded"; "operator" should be understood as "linear operator" (as in the case of "bounded operator"); the domain of the operator is a linear subspace, not necessarily the whole space; this linear subspace is not necessarily closed; often (but not always) it is assumed to be dense; in the special case of a bounded operator, still, the domain is usually assumed to be the whole space.

Generalized eigenvector

In linear algebra, a generalized eigenvector of an matrix is a vector which satisfies certain criteria which are more relaxed than those for an (ordinary) eigenvector. Let be an -dimensional vector space and let be the matrix representation of a linear map from to with respect to some ordered basis. There may not always exist a full set of linearly independent eigenvectors of that form a complete basis for . That is, the matrix may not be diagonalizable.

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.

Related lectures (45)

Postulates of Quantum Mechanics

Explains the postulates of Quantum Mechanics, focusing on self-adjoint operators and mathematical notation.

Linear Operators: Basis Transformation and Eigenvalues

Explores basis transformation, eigenvalues, and linear operators in inner product spaces, emphasizing their significance in Quantum Mechanics.

Linear Operators: Quantum Mechanics and Linear Algebra

Explores the role of linear operators in Quantum Mechanics and linear algebra, emphasizing eigenvalues and basis transformations.

Measurement: Quantum State and Observables

Discusses measurement in Quantum Mechanics, focusing on state vectors, observables, eigenvalues, and probabilities.

Crash Course on Quantum Mechanics