Diracs belt trick, Topology, and Spin ½ particles NoahExplainsPhysics 2021-08-23 | ANSWERS TO FREQUENTLY ASKED QUESTIONS:https://scholar.harvard.edu/files/noahmiller/files/dirac_belt_trick_faq.pdfThis 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 spin 1/2 particles, such as electrons.I created the 3D animations using Three.js/CCapture.js, and the math animations with Manim Community v0.8.0.00:00 Introduction4:14 The space of rotations9:40 Paths through the space of rotations18:48 Group theory & the fundamental group31:30 Quantum spin and SU(2) 39:31 SU(2) as the double cover of SO(3)48:26 Bringing it all together52:22 Tying up loose endsMusic by Vincent RubinettiDownload the music on Bandcamp:vincerubinetti.bandcamp.com/album/the-music-of-3blue1brownStream the music on Spotify:open.spotify.com/album/1dVyjwS8FBqXhRunaG5W5uLargo From Concerto No 5Exzel Music Publishing (freemusicpublicdomain.com)Licensed under Creative Commons: By Attribution 3.0http://creativecommons.org/licenses/by/3.0
Why does False imply True in logic? NoahExplainsPhysics 2020-10-11 | [High School Level] - In this video I discuss something in symbolic logic that always bothered me: why does "false" imply "true"? I look at the question from a variety of perspectives in order to get an answer.
Quantum Spin (8) - Stern Gerlach, Measurement, & Entanglement NoahExplainsPhysics 2020-10-10 | [Undergraduate Level] - In my final video in my series on quantum spin, I discuss how quantum measurement works, using the Stern Gerlach set up as my basic example. (This video builds heavily off of the last video, in which I discuss particle motion and the influence of magnetic fields.) I then zoom out a bit, and discuss how quantum measurement works in general, and how entanglement is related. It turns out that a quantum measurement will entangle the degree of freedom you are trying to measure with some other degree of freedom. I close out the video by discussing quantum measurement collapse and decoherence. Essentially, after a measurement is performed, the measurement device will become entangled with the surrounding environment, which will prevent you from "seeing" the superposition state of the final entangled measured state. (This is closely related to Schrodinger's cat paradox, which is why you don't see the cat in a superposition of an alive and dead state, just one or the other.)
Quantum Spin (7) - Wavefunctions & Motion in Magnetic Field NoahExplainsPhysics 2020-10-04 | [Undergraduate Level] - In this video we finally discuss how particles can "move around" in space. We discuss Gaussian wavepackets and how they roughly approximate the notion of a classical particle with a position and a velocity. We then take a look at the Pauli Equation for the motion of a spin 1/2 particle in an external magnetic field and get a rough sense for how it works. The video concludes on a slightly different note, with a derivation of the expression for the force of a varying magnetic field on a magnetic moment. The contents of this video should allow one to understand how the Stern Gerlach experiment works.
Quantum Spin (6) - H Is Self Adjoint & The Consequences Thereof NoahExplainsPhysics 2020-03-16 | [Undergraduate Level] - In this video, I define what it means for a matrix to be self adjoint, show that our Hamiltonian is self adjoint, and show what the consequences of this fact are, including the conservation of total probability and the fact that all energies are real numbers. I also discuss the uniqueness of time evolution in quantum mechanics, and wrap up with an extended discussion on the conservation of energy and Noether's theorem in the context of Schrödinger's equation.All in all, I had a great time making this video, discussed many deep mathematical facts I've been wanting to for a while, and hopefully will show you the richness of the relationship between physics and mathematics in this simple context.
Quantum Spin (5) - Schrödingers Equation & Matrix Exponentiation NoahExplainsPhysics 2020-01-21 | [Undergraduate Level] - In this video I explain how to solve Schrödinger's Equation for a Spin 1/2 particle in an external magnetic field. I give three methods of solving it:1 - just solve it2 - find the eigenvectors of the Hamiltonian3 - matrix exponentiationOf the three methods, the third is the most interesting. I end the video by discussing the evolution of the Bloch vector in a magnetic field, and discuss what it means to spatially rotate a quantum spin state.
Quantum Spin (4) - Classical Dynamics in Magnetic Field NoahExplainsPhysics 2020-01-06 | [High School Level] - (MISTAKE: From 43:09 - 55:08 all of the cosines should be sines. Thank you to N.H for pointing this out!) I did not expect this video to get so long! In this video I go through what happens to a spinning electron in classical physics. I explain Larmor Precession and derive the formula for the torque and potential energy of an electron in an external magnetic field. I then use the classical formula for energy to motivate Schrödinger's equation for a spin 1/2 particle. I then give a little discussion on the g-factor and quantum field theory.
Quantum Spin (3) - The Bloch Sphere NoahExplainsPhysics 2019-08-19 | [Undergraduate Level] - In my third video on quantum spin, I discuss how all quantum spin states can be thought of as vectors on the Bloch Sphere. Each quantum spin state is the eigenstate of a spin measurement in some direction.
Quantum Spin (2) - Pauli Matrices NoahExplainsPhysics 2019-08-16 | [Undergraduate Level] - An introduction to the Pauli spin matrices in quantum mechanics. I discuss the importance of the eigenvectors and eigenvalues of these matrices, as well as the outer product. I also provide an introduction to matrix multiplication. This is the second video in my series on quantum spin.
Quantum Spin (1) - Introduction NoahExplainsPhysics 2019-08-12 | [High School Level] - An introduction to spin 1/2 particles. I discuss states, bra/ket notation, measurements, probabilities, and quantum measurement collapse.
Noethers Theorem Explained (Part 6/6) - Energy and Time Translations NoahExplainsPhysics 2019-01-10 | [Undergraduate Level] - In this video I show that energy is conserved when the laws of physics are symmetric under a time translation.
Noethers Theorem Explained (Part 5/6) - Total Derivatives NoahExplainsPhysics 2019-01-09 | [Undergraduate Level] - In this video I show why transformations which change the Lagrangian by a total derivative truly should be called "symmetries." (These transformations take a solution to the equations of motion to another solution to the equations of motion.) I also discuss the relevance of total derivatives to the Noether procedure. This is all so that we can discuss time translational symmetry in the next video
Noethers Theorem Explained (Part 4/6) - Proof NoahExplainsPhysics 2019-01-09 | [Undergraduate Level] - In this video I finally prove Noether's theorem in general. I show that when you have a symmetry, you have a conservation law! I also show why when you don't have a symmetry, you don't have a conservation law!
Noethers Theorem Explained (Part 3/6) - Angular Momentum and Rotations NoahExplainsPhysics 2019-01-09 | [Undergraduate Level] - In this video I prove that total angular momentum is conserved when a system has a rotational symmetry. In part 4 of my series I give a general proof for why symmetries give conservation laws. In this video I am just working out a specific example.
Noethers Theorem Explained (Part 2/6) - Momentum and Spatial Translations NoahExplainsPhysics 2019-01-09 | [Undergraduate Level] - In this video I prove that total linear momentum is conserved when a system has a spatial translational symmetry. In part 4 of my series I give a general proof for why symmetries give conservation laws. In this video I am just working out a specific example.
Noethers Theorem Explained (Part 1/6) - Introduction NoahExplainsPhysics 2019-01-08 | [Undergraduate Level] - In this video I state of Noether's theorem and discuss symmetries in general. The only prerequisite is Lagrangian Mechanics.
The Euler Lagrange Equation and Lagrangian Mechanics NoahExplainsPhysics 2018-08-12 | [Undergraduate Level] In this video I prove the Euler Lagrange equation and discuss its applications. Its a sequel to my previous video on the principle of least action.
The Principle of Least Action NoahExplainsPhysics 2018-08-09 | [High School Level] In this video I explain the principle of least action, and show that it is equivalent to the regular way of doing things.
