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

Lecture

Addition of Angular Momenta

Description

This lecture covers the addition of angular momenta, focusing on finding eigenvalues and eigenvectors of the total angular momentum. It explores the concepts of eigenvalues and eigenvectors, as well as the notation and calculations involved in determining the total angular momentum.

Official source

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.

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.

Eigenvalue algorithm

In numerical analysis, one of the most important problems is designing efficient and stable algorithms for finding the eigenvalues of a matrix. These eigenvalue algorithms may also find eigenvectors. Eigenvalues and eigenvectors and Generalized eigenvector Given an n × n square matrix A of real or complex numbers, an eigenvalue λ and its associated generalized eigenvector v are a pair obeying the relation where v is a nonzero n × 1 column vector, I is the n × n identity matrix, k is a positive integer, and both λ and v are allowed to be complex even when A is real.

Quantum mechanics

Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, quantum field theory, quantum technology, and quantum information science. Classical physics, the collection of theories that existed before the advent of quantum mechanics, describes many aspects of nature at an ordinary (macroscopic) scale, but is not sufficient for describing them at small (atomic and subatomic) scales.

Related lectures (51)

Orbital Angular Momentum

Explores orbital angular momentum in quantum mechanics, including rotation operators and eigenvalues.

Spherical Harmonics

Explores spherical harmonics, discussing their mathematical properties and significance in quantum mechanics.

Postulates of Quantum Mechanics

Explores the postulates of Quantum Mechanics, emphasizing measurement, operators, eigenvalues, eigenvectors, and state transformation.

Basis of Nuclear Magnetization MOOC: Fundamentals of Biomedical Imaging: Magnetic Resonance Imaging (MRI)

Explores the classical and quantum-mechanical basis of nuclear magnetization and the quantization of angular momentum and energy levels.

Vector Operators and Wigner-Eckart Theorem