Hamiltonian mechanics | Mathematical quantization | Symplectic geometry
The phase-space formulation of quantum mechanics places the position and momentum variables on equal footing in phase space. In contrast, the Schrödinger picture uses the position or momentum representations (see also position and momentum space). The two key features of the phase-space formulation are that the quantum state is described by a quasiprobability distribution (instead of a wave function, state vector, or density matrix) and operator multiplication is replaced by a star product. The theory was fully developed by Hilbrand Groenewold in 1946 in his PhD thesis, and independently by Joe Moyal, each building on earlier ideas by Hermann Weyl and Eugene Wigner. The chief advantage of the phase-space formulation is that it makes quantum mechanics appear as similar to Hamiltonian mechanics as possible by avoiding the operator formalism, thereby "'freeing' the quantization of the 'burden' of the Hilbert space". This formulation is statistical in nature and offers logical connections between quantum mechanics and classical statistical mechanics, enabling a natural comparison between the two (see classical limit). Quantum mechanics in phase space is often favored in certain quantum optics applications (see optical phase space), or in the study of decoherence and a range of specialized technical problems, though otherwise the formalism is less commonly employed in practical situations. The conceptual ideas underlying the development of quantum mechanics in phase space have branched into mathematical offshoots such as Kontsevich's deformation-quantization (see Kontsevich quantization formula) and noncommutative geometry. (Wikipedia).
State Space Models, Part 1: Creation and Analysis
Get a Free Trial: https://goo.gl/C2Y9A5 Get Pricing Info: https://goo.gl/kDvGHt Ready to Buy: https://goo.gl/vsIeA5 Create and analyze state-space models using MATLAB® and Control System Toolbox™. State-space models are commonly used for representing linear time-invariant (LTI) systems.
From playlist Control System Design and Analysis
Phase space representation of billiards interpolating between a circle and a hexagon
In this simulation, I wanted to see what happens when you continuously deform the boundary of a billiard from a circle to a regular hexagon. The billiard in a circle has very regular dynamics (the technical work is "integrable"), because a given trajectory always hits the boundary with the
From playlist Particles in billiards
Special Relativity: 2 - Spacetime Diagrams
An introduction to spacetime diagrams which are a valuable tool used to understand special relativity. The second in a series on special and general relativity. Let us know what you think of these videos by filling out our short survey at http://tinyurl.com/astronomy-pulsar. Thank you!
From playlist Special Relativity
A unary phase diagram is a diagram that shows the phases of matter (solid, liquid, gas) for a single component, such as water. We show the triple point, boiling point, melting point etc. These are also explained for 1 atm as well as lower pressures corresponding to higher altitudes.
From playlist Materials Sciences 101 - Introduction to Materials Science & Engineering 2020
Now we know about vector spaces, so it's time to learn how to form something called a basis for that vector space. This is a set of linearly independent vectors that can be used as building blocks to make any other vector in the space. Let's take a closer look at this, as well as the dimen
From playlist Mathematics (All Of It)
Phase diagrams tell us where to find specific microstructures
Phase diagrams are like treasure maps in that they tell us what conditions we need to follow to achieve desired microstructures or phases.
From playlist Materials Sciences 101 - Introduction to Materials Science & Engineering 2020
Phasers locked on target: Phase evolution in the quantum hexagonal billiard
This simulation shows the same solutions of the Schrödinger equation in a hexagonal domain as the videos https://youtu.be/8WTIjRWjG1o and https://youtu.be/OJGTXVK3lpk but with a different representation. The hue represents the phase (or argument) of the wave function, while the luminosity
From playlist Billiards in polygons
Natalia Tronko: Exact conservation laws for gyrokinetic Vlasov-Poisson equations
Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual Mathematics Library: http://library.cirm-math.fr. And discover all its functionalities: - Chapter markers and keywords to watch the parts of your choice in the video - Videos enriched with abstracts, b
From playlist SPECIAL 7th European congress of Mathematics Berlin 2016.
Bound-preserving numerical solutions of variable density two-phase flows
Date and Time: Thursday, November 11, 12:00pm Eastern time zone Speaker: Beatrice Riviere, Rice University Abstract: Modeling pore-scale flows modeling is important for many applications relevant to energy and environment. Phase-field models are popular models because they implicitly tra
From playlist SIAM Geosciences Webinar Series
Matrix Models, Gauge-Gravity Duality, and Simulations on the Lattice (Lecture 2) by Georg Bergner
NONPERTURBATIVE AND NUMERICAL APPROACHES TO QUANTUM GRAVITY, STRING THEORY AND HOLOGRAPHY (HYBRID) ORGANIZERS: David Berenstein (University of California, Santa Barbara, USA), Simon Catterall (Syracuse University, USA), Masanori Hanada (University of Surrey, UK), Anosh Joseph (IISER Mohal
From playlist NUMSTRING 2022
Martin Vohralík: Adaptive inexact Newton methods and their application to multi-phase flows
Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual Mathematics Library: http://library.cirm-math.fr. And discover all its functionalities: - Chapter markers and keywords to watch the parts of your choice in the video - Videos enriched with abstracts, b
From playlist Numerical Analysis and Scientific Computing
Progress and Prospects of Lattice Supersymmetry by David Schaich
PROGRAM Nonperturbative and Numerical Approaches to Quantum Gravity, String Theory and Holography (ONLINE) ORGANIZERS: David Berenstein (UCSB), Simon Catterall (Syracuse University), Masanori Hanada (University of Surrey), Anosh Joseph (IISER, Mohali), Jun Nishimura (KEK Japan), David Sc
From playlist Nonperturbative and Numerical Approaches to Quantum Gravity, String Theory and Holography (Online)
Felix Otto - 23 September 2016
Otto, Felix "The thresholding scheme for mean curvature flow"
From playlist A Mathematical Tribute to Ennio De Giorgi
Qubit Regularization of Asymptotic Freedom by Shailesh Chandrasekharan
PROGRAM Nonperturbative and Numerical Approaches to Quantum Gravity, String Theory and Holography (ONLINE) ORGANIZERS: David Berenstein (UCSB), Simon Catterall (Syracuse University), Masanori Hanada (University of Surrey), Anosh Joseph (IISER, Mohali), Jun Nishimura (KEK Japan), David Sc
From playlist Nonperturbative and Numerical Approaches to Quantum Gravity, String Theory and Holography (Online)
Lattice Supersymmetric Field Theories (Lecture 3) by David Schaich
PROGRAM NONPERTURBATIVE AND NUMERICAL APPROACHES TO QUANTUM GRAVITY, STRING THEORY AND HOLOGRAPHY (HYBRID) ORGANIZERS: David Berenstein (University of California, Santa Barbara, USA), Simon Catterall (Syracuse University, USA), Masanori Hanada (University of Surrey, UK), Anosh Joseph (II
From playlist NUMSTRING 2022
Lecture 6 | Modern Physics: Classical Mechanics (Stanford)
Lecture 6 of Leonard Susskind's Modern Physics course concentrating on Classical Mechanics. Recorded November 19, 2007 at Stanford University. This Stanford Continuing Studies course is the first of a six-quarter sequence of classes exploring the essential theoretical foundations of mo
From playlist Course | Modern Physics: Classical Mechanics
Quantum Ergodicity for the Uninitiated - Zeev Rudnick
Zeev Rudnick Tel Aviv University; Member, School of Mathematics October 26, 2015 https://www.math.ias.edu/seminars/abstract?event=47561 A key result in spectral theory linking classical and quantum mechanics is the Quantum Ergodicity theorem, which states that in a system in which the cl
From playlist Members Seminar
Jose Antonio Font - Numerical analysis: binary neutron stars - IPAM at UCLA
Recorded 21 September 2021. Jose Antonio Font of the University of Valencia presents "Numerical analysis: binary neutron stars" at IPAM's Mathematical and Computational Challenges in the Era of Gravitational Wave Astronomy Tutorial. Abstract: Merging binary neutron stars are among the str
From playlist Tutorials: Math & Computational Challenges in the Era of Gravitational Wave Astronomy
Phase separation in the Allen-Cahn equation, with a start in slow motion
Like the video https://youtu.be/t1swj0QJUTw this simulation shows a solution of the Allen-Cahn equation for phase separation, on a rectangular domain with periodic boundary conditions, starting from a random initial configuration. Time has been slowed down at the beginning, to give a bette
From playlist Reaction-diffusion equations