Ciro Santilli @cirosantilli 40

In some contexts, we want to observe what happens for a given fixed magnetic field strength on a specific plate (thus $t$ and $n$ are also fixed).

In those cases, it can be useful to talk about the "Hall resistance" defined as:So note that it is not a "regular resistance", it just has the same dimensions, and is more usefully understood as a proportionality constant for the voltage given an input $I_x$ current:

This notion can be useful because everything else being equal, if we increase the current $I_x$, then $V_y$ also increases proportionally, making this a way to talk about the voltage in a current independent manner.

And this is particularly the case for the quantum Hall effect, where $R_{xy}$ is constant for wide ranges of applied magnetic field and TODO presumably the height $t$ can be made to a single molecular layer with chemical vapor deposition of the like, and if therefore fixed.

Equivalent to Lagrangian mechanics but formulated in a different way.

Motivation: Lagrangian vs Hamiltonian.

TODO understand original historical motivation, www.youtube.com/watch?v=SZXHoWwBcDc says it is from optics.

Intuitively, the Hamiltonian is the total energy of the system in terms of arbitrary parameters, a bit like Lagrangian mechanics.

Bibliography:

First we hand-wave some intuition. Then we prove. That's the way to teach.

Head of the theoretical division at the Los Alamos Laboratory during the Manhattan Project.

Richard Feynman was working under him there, and was promoted to team lead by him because Richard impressed Hans.

He was also the person under which Freeman Dyson was originally under when he moved from the United Kingdom to the United States.

And Hans also impressed Feynman, both were problem solvers, and liked solving mental arithmetic and numerical analysis.

This relationship is what brought Feynman to Cornell University after World War II, Hans' institution, which is where Feynman did the main part of his Nobel prize winning work on quantum electrodynamics.

Hans must have been the perfect PhD advisor. He's always smiling, and he seemed so approachable. And he was incredibly capable, notably in his calculation skills, which were much more important in those pre-computer days.

1-dimensional heat equation example with Dirichlet boundary condition

Cody'sLab had a nice 5 video series on making it at home! But the United States Government asked him to take it down as suggested at Video "What's Been Going On With Cody'sLab? by Cody'sLab (2019)" at youtu.be/x1mv0vwb08Y?t=84.

Here's a copy online as of 2020: www.youtube.com/watch?v=bCXB6BdMh9Y

The point of these is that they are good for transfection apparently.

WTF is wrong with that family???

Show up in the solution of the quantum harmonic oscillator after separation of variables leading into the time-independent Schrödinger equation, much like solving partial differential equations with the Fourier series.

I.e.: they are both:

This is the possibly infinite dimensional version of a Hermitian matrix, since linear operators are the possibly infinite dimensional version of matrices.

There's a catch though: now we don't have explicit matrix indices here however in general, the generalized definition is shown at: en.wikipedia.org/w/index.php?title=Hermitian_adjoint&oldid=1032475701#Definition_for_bounded_operators_between_Hilbert_spaces