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# AKT-140110 Video

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The Schroedinger equation and path integrals.

# Week of... Notes and Links
: Course introduction, knots and Reidemeister moves, knot colourings.
Tricolourability without Diagrams
: The Gauss linking number combinatorially and as an integral.
: The Schroedinger equation and path integrals.
Friday Introduction (the quantum pendulum)
2 Jan 13 Homework Assignment 1.
: The Kauffman bracket and the Jones polynomial.
: Euler-Lagrange problems, Gaussian integration, volumes of spheres.
3 Jan 20 Homework Assignment 2.
: The definition of finite-type and some examples.
: The self-linking number and framings.
: Integrating a polynomial times a Gaussian.
Class Photo.
4 Jan 27 Homework Assignment 3.
: Chord diagrams and weight systems.
: Swaddling maps and framings, general configuration space integrals.
: Some analysis of $d^{-1}$.
5 Feb 3 Homework Assignment 4.
: 4T, the Fundamental Theorem and universal finite type invariants.
The Fulton-MacPherson Compactification (PDF).
: The Fulton-MacPherson Compactification, Part I.
: More on pushforwards, $d^{-1}$, and $d^\ast$.
6 Feb 10 Homework Assignment 5.
: The bracket-rise theorem and the invariance principle.
: The Fulton-MacPherson Compactification, Part II.
: Gauge fixing, the beginning of Feynman diagrams.
7 Feb 24 : A review of Lie algebras.
: Graph cohomology and $\Omega_{dR}^\ast(\Gamma)$.
: More on Feynman diagrams, beginning of gauge theory.
From Gaussian Integration to Feynman Diagrams (PDF).
8 Mar 3 Homework Assignment 6 (PDF)
: Lie algebraic weight systems.
: Graph cohomology and the construction of $Z_0$.
Graph Cohomology and Configuration Space Integrals (PDF)
: Gauge invariance, Chern-Simons, holonomies.
Mar 9 is the last day to drop this class.
9 Mar 10 Homework Assignment 7 (PDF)
: The $gl(N)$ weight system.
: The universal property, hidden faces.
: Insolubility of the quintic, naive expectations for CS perturbation theory.
10 Mar 17 Homework Assignment 8 (PDF)
: $W_{\mathfrak g}\colon{\mathcal A}(\uparrow)\to{\mathcal U}({\mathfrak g})$ and PBW.
: The anomaly.
Gaussian Integration, Determinants, Feynman Diagrams (PDF).
11 Mar 24 Homework Assignment 9 (PDF)
: ${\mathcal A}$ is a bi-algebra.
: Understanding and fixing the anomaly.
Friday: class cancelled.
12 Mar 31 Monday, Wednesday: class cancelled.
: A Monday class: back to expansions.
E Apr 7 : A Friday class on what we mostly didn't have time to do.

Dror's Notebook
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0:01:35  What is Quantum Mechanics?

Quantum mechanics is a tool or theory in physics that is used to describe the state of the world or a system at the subatomic level. The theory depends on states of the system, observables, measurements and evolution evolution of the system. These are used in building a mathematical model to study quantum mechanics.

In classical mechanics, Newton's laws are used to study physical systems whose behaviour are visible to the human eye. These laws do not apply to systems with quantum effect (this is an effect that cannot be described by classical physics) such as quantum entanglement, radioactive decay and wave particle duality. Such systems are called quantum systems (examples are atoms, molecules, particle on a pendulum etc). Schroedinger equation is one of the tools used in describing such systems.

Schroedinger equation is an equation that describes a systems that has significant of quantum effect. It is a partial differential equation whose solution is a wave function $\psi(x, t)$. This is used to study the evolution of the system as time goes by.

0:01:35 [add] Handout view 2: Title and abstract
0:02:27 [add] Handout view 3: History
0:02:55 [add] Handout view 4: The Question
0:08:10 [add] Handout view 5: More generally
0:08:33 [add] Handout view 6: e^H
0:10:24 [add] Handout view 7: The Trotter Formula
0:13:19 [add] Handout view 8: Use Trotter
0:14:01 [add] Handout view 9: e^V
0:14:10 [add] Handout view 8: Use Trotter
0:14:59 [add] Handout view 9: e^V
0:15:37 [add] Handout view 10: e^Delta
0:22:31 [add] Handout view 11: e^Delta proof
0:24:44  In the proof of Lemma 3.3, I am okay up to this point.
0:25:22 [add] Handout view 12: Trotter
0:25:46 [add] Handout view 13: Iterated convolutions
0:26:12 [add] Handout view 13: Iterated convolutions
0:26:18 [add] Handout view 12: Trotter
0:26:40 [add] Handout view 13: Iterated convolutions
0:26:53 [add] Handout view 12: Trotter
0:27:02 [add] Handout view 13: Iterated convolutions
0:27:44 [add] Handout view 12: Trotter
0:28:02 [add] Handout view 13: Iterated convolutions
0:29:16 [add] Handout view 14: Discrete path integral
0:29:33 [add] Handout view 13: Iterated convolutions
0:29:44 [add] Handout view 14: Discrete path integral
0:30:46 [add] Handout view 15: A path
0:31:52 [add] Handout view 14: Discrete path integral
0:32:59 [add] Handout view 16: Path integrals
0:34:28 [add] Handout view 15: A path
0:34:40 [add] Handout view 16: Path integrals
0:35:09 [add] Handout view 17: The Action
0:35:11 [add] Handout view 16: Path integrals
0:35:34 [add] Handout view 17: The Action
0:35:38  Lagrangian Mechanics is a tool used in studying motions in Classical Mechanics and it was introduced by Joseph-Louis Lagrange in 1788. An important concept in Lagragian Mechanics is that the equations of motion of Classical Mechanics can be based on a variational principle, namely, that along a path describing classical motion the action integral assumes a minimal value (Hamiltonian Principle of Least Action)

The action integral is given by $S[x(t)] = \int^{t_1}_{t_0} dt \mathcal{L}(x(t),x^\prime(t),t)$, where $\mathcal{L}(x(t),x^\prime(t),t) = \frac12 {x^\prime(t)}^2-U(x(t))$ is called the Lagrangian.

0:35:55  I have noticed that the d$x$ always comes before the integrand. Any reason for this or it just notation?
0:36:14 [add] Handout view 18: Semi-classical
0:39:54  I found this link about the Lagrangian: [1], and it talks about The Principle of Least Action which says that the path that has the minimum action is the one satisfying Newton's law for a conservative system which is the famous $F = ma$, where $m$ is the mass of the system and $a$ is the acceleration of the system with respect to the total force on the system.