Theorems in quantum mechanics | Hilbert space
Wigner's theorem, proved by Eugene Wigner in 1931, is a cornerstone of the mathematical formulation of quantum mechanics. The theorem specifies how physical symmetries such as rotations, translations, and CPT are represented on the Hilbert space of states. The physical states in a quantum theory are represented by unit vectors in Hilbert space up to a phase factor, i.e. by the complex line or ray the vector spans. In addition, by the Born rule the absolute value of the unit vectors inner product, or equivalently the cosine squared of the angle between the lines the vectors span, corresponds to the transition probability. Ray space, in mathematics known as projective Hilbert space, is the space of all unit vectors in Hilbert space up to the equivalence relation of differing by a phase factor. By Wigner's theorem, any transformation of ray space that preserves the absolute value of the inner products can be represented by a unitary or antiunitary transformation of Hilbert space, which is unique up to a phase factor. As a consequence, the representation of a symmetry group on ray space can be lifted to a projective representation or sometimes even an ordinary representation on Hilbert space. (Wikipedia).
Projection Theorem | Special Case of the Wigner–Eckart Theorem
The projection theorem is a special case of the Wigner–Eckart theorem, which generally involves spherical tensor operators. If we consider one example of a spherical tensor operator, a rank-1 spherical tensor, we can derive a powerful theorem, which states that expectation values of vector
From playlist Quantum Mechanics, Quantum Field Theory
Multivariable Calculus | The Squeeze Theorem
We calculate a limit using a multivariable version of the squeeze theorem. http://www.michael-penn.net http://www.randolphcollege.edu/mathematics/
From playlist Multivariable Calculus
Divergence Theorem. In this video, I give an example of the divergence theorem, also known as the Gauss-Green theorem, which helps us simplify surface integrals tremendously. It's, in my opinion, the most important theorem in multivariable calculus. It is also extremely useful in physics,
From playlist Vector Calculus
Spherical Tensor Operators | Wigner D-Matrices | Clebsch–Gordan & Wigner–Eckart
In this video, we will explain spherical tensor operators. They are defined like this: A spherical tensor operator T^(k)_q with rank k is a collection of 2k+1 operators that are numbered by the index q, which transform under rotations in the same way as spherical harmonics do. They are als
From playlist Quantum Mechanics, Quantum Field Theory
In this video, I present another example of Stokes theorem, this time using it to calculate the line integral of a vector field. It is a very useful theorem that arises a lot in physics, for example in Maxwell's equations. Other Stokes Example: https://youtu.be/-fYbBSiqvUw Yet another Sto
From playlist Vector Calculus
Geometry and Topology in Quantum Mechanics - Mathematical Properties by N. Mukunda
DISCUSSION MEETING GEOMETRIC PHASES IN OPTICS AND TOPOLOGICAL MATTER ORGANIZERS: Subhro Bhattacharjee, Joseph Samuel and Supurna Sinha DATE: 21 January 2020 to 24 January 2020 VENUE: Madhava Lecture Hall, ICTS, Bangalore This is a joint ICTS-RRI Discussion Meeting on the geometric pha
From playlist Geometric Phases in Optics and Topological Matter 2020
Quantum chaos, random matrices and statistical physics (Lecture 03) by Arul Lakshminarayan
ORGANIZERS: Abhishek Dhar and Sanjib Sabhapandit DATE: 27 June 2018 to 13 July 2018 VENUE: Ramanujan Lecture Hall, ICTS Bangalore This advanced level school is the ninth in the series. This is a pedagogical school, aimed at bridging the gap between masters-level courses and topics in
From playlist Bangalore School on Statistical Physics - IX (2018)
Multivariable Calculus | Differentiable implies continuous.
We prove the classic result that if a function is differentiable, then it is continuous. To start, we prove this for a two variable function and then repeat for an n-variable function. http://www.michael-penn.net https://www.researchgate.net/profile/Michael_Penn5 http://www.randolphcolleg
From playlist Multivariable Calculus
Gergely Harcos - A glimpse at arithmetic quantum chaos
slides for this talk: https://www.msri.org/workshops/801/schedules/21771/documents/2987/assets/27969 Introductory Workshop: Analytic Number Theory February 06, 2017 - February 10, 2017 February 07, 2017 (11:00 AM PST - 12:00 PM PST) Speaker(s): Gergely Harcos (Central European Universit
From playlist Number Theory
The Campbell-Baker-Hausdorff and Dynkin formula and its finite nature
In this video explain, implement and numerically validate all the nice formulas popping up from math behind the theorem of Campbell, Baker, Hausdorff and Dynkin, usually a.k.a. Baker-Campbell-Hausdorff formula. Here's the TeX and python code: https://gist.github.com/Nikolaj-K/8e9a345e4c932
From playlist Algebra
Wigner–Eckart Theorem | Clebsch-Gordan & Spherical Tensor Operators
In this video, we will explain the Wigner-Eckart theorem in theory and then explicitly show how to use it to solve a problem. This theorem is closely related to Clebsch-Gordan coefficients and spherical tensor operators. 𝗥𝗲𝗳𝗲𝗿𝗲𝗻𝗰𝗲𝘀: [1] Particle Data Group, "The Review of Particle Phy
From playlist Mathematical Physics
Random Matrix Theory and its Applications by Satya Majumdar ( Lecture 2 )
PROGRAM BANGALORE SCHOOL ON STATISTICAL PHYSICS - X ORGANIZERS : Abhishek Dhar and Sanjib Sabhapandit DATE : 17 June 2019 to 28 June 2019 VENUE : Ramanujan Lecture Hall, ICTS Bangalore This advanced level school is the tenth in the series. This is a pedagogical school, aimed at bridgin
From playlist Bangalore School on Statistical Physics - X (2019)
Dyson Brownian motion, free fermions and connections to Random Matrix Theory by Gregory Schehr
PROGRAM : BANGALORE SCHOOL ON STATISTICAL PHYSICS - XII (ONLINE) ORGANIZERS : Abhishek Dhar (ICTS-TIFR, Bengaluru) and Sanjib Sabhapandit (RRI, Bengaluru) DATE : 28 June 2021 to 09 July 2021 VENUE : Online Due to the ongoing COVID-19 pandemic, the school will be conducted through online
From playlist Bangalore School on Statistical Physics - XII (ONLINE) 2021
Multivariable Calculus | Differentiability
We give the definition of differentiability for a multivariable function and provide a few examples. http://www.michael-penn.net https://www.researchgate.net/profile/Michael_Penn5 http://www.randolphcollege.edu/mathematics/
From playlist Multivariable Calculus
Irreducibility and the Schoenemann-Eisenstein criterion | Famous Math Probs 20b | N J Wildberger
In the context of defining and computing the cyclotomic polynumbers (or polynomials), we consider irreducibility. Gauss's lemma connects irreducibility over the integers to irreducibility over the rational numbers. Then we describe T. Schoenemann's irreducibility criterion, which uses some
From playlist Famous Math Problems
Pierre Youssef: Outliers in sparse Wigner matrices
Given a Wigner matrix with centered bounded entries, we study the effect of sparsity on the extreme eigenvalues. More precisely, multiplying the entries by independent Bernoulli variables with parameter pn, we show that as pn decreases, outliers start emerging in the semi-circular law whic
From playlist Workshop: High dimensional measures: geometric and probabilistic aspects
Benson Au: "Finite-rank perturbations of random band matrices via infinitesimal free probability"
Asymptotic Algebraic Combinatorics 2020 "Finite-rank perturbations of random band matrices via infinitesimal free probability" Benson Au - University of California, San Diego (UCSD) Abstract: Free probability provides a unifying framework for studying random multi-matrix models in the la
From playlist Asymptotic Algebraic Combinatorics 2020
Spatio-temporal correlations across the melting of 2D Wigner molecules by Amit Ghosal
29 May 2017 to 02 June 2017 VENUE: Ramanujan Lecture Hall, ICTS Bangalore This program aims to bring together people working on classical and quantum systems with disorder and interactions. The extensive exploration, through experiments, simulations and model calculations, of growing cor
From playlist Correlation and Disorder in Classical and Quantum Systems
Introduction to additive combinatorics lecture 10.8 --- A weak form of Freiman's theorem
In this short video I explain how the proof of Freiman's theorem for subsets of Z differs from the proof given earlier for subsets of F_p^N. The answer is not very much: the main differences are due to the fact that cyclic groups of prime order do not have lots of subgroups, so one has to
From playlist Introduction to Additive Combinatorics (Cambridge Part III course)