Rotational symmetry | Articles containing proofs | Lie groups | Matrices
In mathematical physics and mathematics, the Pauli matrices are a set of three 2 × 2 complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (σ), they are occasionally denoted by tau (τ) when used in connection with isospin symmetries. These matrices are named after the physicist Wolfgang Pauli. In quantum mechanics, they occur in the Pauli equation which takes into account the interaction of the spin of a particle with an external electromagnetic field. They also represent the interaction states of two polarization filters for horizontal/vertical polarization, 45 degree polarization (right/left), and circular polarization (right/left). Each Pauli matrix is Hermitian, and together with the identity matrix I (sometimes considered as the zeroth Pauli matrix σ0), the Pauli matrices form a basis for the real vector space of 2 × 2 Hermitian matrices. This means that any 2 × 2 Hermitian matrix can be written in a unique way as a linear combination of Pauli matrices, with all coefficients being real numbers. Hermitian operators represent observables in quantum mechanics, so the Pauli matrices span the space of observables of the complex 2-dimensional Hilbert space. In the context of Pauli's work, σk represents the observable corresponding to spin along the kth coordinate axis in three-dimensional Euclidean space The Pauli matrices (after multiplication by i to make them anti-Hermitian) also generate transformations in the sense of Lie algebras: the matrices iσ1, iσ2, iσ3 form a basis for the real Lie algebra , which exponentiates to the special unitary group SU(2). The algebra generated by the three matrices σ1, σ2, σ3 is isomorphic to the Clifford algebra of , and the (unital associative) algebra generated by iσ1, iσ2, iσ3 is effectively identical (isomorphic) to that of quaternions. (Wikipedia).
Pauli matrices vs. su(2) basis vs. quaternions
In this video we discuss Pauli matrices as base for hermitean 2x2 complex matrices, as relevant for modeling observables in quantum theory - but also for quantum mechanics, as demonstrated. You can find the text used in this video here: https://gist.github.com/Nikolaj-K/103f07367c116b64b56
From playlist Physics
An introduction to how the jacobian matrix represents what a multivariable function looks like locally, as a linear transformation.
From playlist Multivariable calculus
Linear Algebra for Computer Scientists. 12. Introducing the Matrix
This computer science video is one of a series of lessons about linear algebra for computer scientists. This video introduces the concept of a matrix. A matrix is a rectangular or square, two dimensional array of numbers, symbols, or expressions. A matrix is also classed a second order
From playlist Linear Algebra for Computer Scientists
[Lesson 13] QED Prerequisites - The Pauli Spin Matrices...from scratch!
The purpose of this video is to motivate the Pauli Spin matrices from first principles. We will use these matrices a lot during the study of QED and it is critical that every aspect of their design and origin is well understood. This video begins by describing what it means to "rotate a sp
From playlist QED- Prerequisite Topics
The Atom C2 The Pauli Exclusion Principle
The Pauli exclusion principle.
From playlist Physics - The Atom
Complex Matrices ( An intuitive visualization )
Complex Matrices are not given enough credit for what they do and even when they are used its often introduced as an foreign entity. This video was made to shed light on such a misinterpreted topic. Timestamps 00:00 - Introduction 00:11 - Matrix 00:45 - Complex Number 02:50 - Complex Ma
From playlist Summer of Math Exposition Youtube Videos
The Atom C1 The Pauli Exclusion Principle
The Pauli exclusion principle.
From playlist Physics - The Atom
5. Linear Algebra: Vector Spaces and Operators
MIT 8.05 Quantum Physics II, Fall 2013 View the complete course: http://ocw.mit.edu/8-05F13 Instructor: Barton Zwiebach In this lecture, the professor talked about vector spaces and dimensionality. License: Creative Commons BY-NC-SA More information at http://ocw.mit.edu/terms More cours
From playlist 8.05 Quantum Physics II - Prof. Barton Zwiebach
What is a matrix? Free ebook http://tinyurl.com/EngMathYT
From playlist Intro to Matrices
Lie Groups and Lie Algebras: Lesson 28 - SU(2) from su(2)
Lie Groups and Lie Algebras: Lesson 28 - SU(2) from su(2) In this lecture we discover the basis of the su(2) Lie algebra and exponentiate a typical element of su(2) to discover a parameterization of the Lie group SU(2). Please consider supporting this channel via Patreon: https://www.p
From playlist Lie Groups and Lie Algebras
The Atom C3 The Pauli Exclusion Principle
The Pauli exclusion principle.
From playlist Physics - The Atom
Advanced Quantum Mechanics Lecture 4
(October 14, 2013) Building on the previous discussion of atomic energy levels, Leonard Susskind demonstrates the origin of the concept of electron spin and the exclusion principle. Originally presented by the Stanford Continuing Studies Program. Stanford University: http://www.stanford
From playlist Lecture Collection | Advanced Quantum Mechanics
Richard Kerner - Geometry, Matter and Physics
We show how the fundamental statistical properties of quantum fields combined with the superposition principle lead to continuous symmetries including the $SL(2,\mathbb C)$ group and the internal symmetry groups $SU(2)$ and $SU(3)$. The exact colour symmetry is related to ternary $\mathbb
From playlist Combinatorics and Arithmetic for Physics: special days
Quantum Mechanics 12a - Dirac Equation I
When quantum mechanics and relativity are combined to describe the electron the result is the Dirac equation, presented in 1928. This equation predicts electron spin and the existence of anti-matter.
From playlist Quantum Mechanics
Dirac's belt trick, Topology, and Spin ½ particles
ANSWERS TO FREQUENTLY ASKED QUESTIONS: https://scholar.harvard.edu/files/noahmiller/files/dirac_belt_trick_faq.pdf This is my submission to 3Blue1Brown's "Summer of Math Exposition 1" #SoME1. In this video, I explain what Dirac's famous belt trick has to do with the topology of rotating s
From playlist Summer of Math Exposition Youtube Videos
Richard Kerner - Unifying Colour SU(3) with Z3-Graded Lorentz-Poincaré Algebra
A generalization of Dirac’s equation is presented, incorporating the three-valued colour variable in a way which makes it intertwine with the Lorentz transformations. We show how the Lorentz-Poincaré group must be extended to accomodate both SU(3) and the Lorentz transformations. Both symm
From playlist Combinatorics and Arithmetic for Physics: 02-03 December 2020
You've Heard of SPIN - But How Is it Encoded in the Math of Quantum Physics? Parth G
The concept of Spin is hard, but the mathematics is actually quite simple! In this video I wanted to take a look at how we build up our mathematical representation (or at least one of them) of quantum mechanical spin. To do this, we'll start by looking at the spin of an electron, and unde
From playlist Quantum Physics by Parth G
PreCalculus - Matrices & Matrix Applications (1 of 33) What is a Matrix? 1
Visit http://ilectureonline.com for more math and science lectures! In this video I will define what is a matrix and its rows and columns. Next video in the Matrices series can be seen at: http://youtu.be/YTV7ei1hyJI
From playlist Michel van Biezen: PRECALCULUS 12 - MATRICES
L4.3 The Pauli equation for the electron in an electromagnetic field
MIT 8.06 Quantum Physics III, Spring 2018 Instructor: Barton Zwiebach View the complete course: https://ocw.mit.edu/8-06S18 YouTube Playlist: https://www.youtube.com/playlist?list=PLUl4u3cNGP60Zcz8LnCDFI8RPqRhJbb4L L4.3 The Pauli equation for the electron in an electromagnetic field Lice
From playlist MIT 8.06 Quantum Physics III, Spring 2018