/sci/ - Science & Math

>>12529987
This video is extremely basic, but I find it beautiful and am sad that they don't do this anymore

https://youtu.be/Di-jAhrAXOY?t=510
(especially from 8:35 onwards)

Similarly, this
https://youtu.be/dBH-Id8VC3U

>Would it be accurate to simple define what a Hamilitonian operator is as "The only operator that commutes with every other operator"?
No, e.g. the the ladder operators of a quantum harmonic oscillator fulfill an eigenstate-like equation w.r.t. adjoint action of the Hamoltonian of that model
[math]H = c·N+d·E[/math], with [math]N = a^T·a[/math] and [math][N,a^T]=a^T[/math], i.e. H doesn't commute with [math]a^T[/math]

For a collection of observables the form a Lie algebra, consider the Casimir operators
https://en.wikipedia.org/wiki/Casimir_element
for something along your lines. But I don't know to what extend the theory extends to Jordan algebras.

>Basically, given a system with some symmetries, the hamiltonian is the only observable whos measurements wont be affected by any symmetry transformation
That's weaker than what you said before - but yes, symmetry also means H commutes with the generator of the unitary transformation.

>So the most symmetric of all observables=literal definition of a hamiltonian.
You can scale and extend by anything in the center and you're still having all of le symmetries, so that doesn't seem to pin it down. Why wouldn't you characterize it with respect to the Heisenberg equation?

I'd say the Hamiltonian is a (possibly uncountable) list of frequencies measurable in a certain base and, in the an act of modeling, it's usually not given in diagonal form.

>>12423727
>>12424254
This guy >>12424153 was right, but he understated his point and I think you're confused to "what a Lie algebra" is. The definition of a Lie algebra, to quote the first line of the Wikipedia article is
>In mathematics, a Lie algebra is a vector space together with an operation called the Lie bracket, an alternating bilinear map, that satisfies the Jacobi identity.
The point is that the given operation - and indeed the only given operation! - is the Lie bracket. I.e. the "Lie bracket" [A,B] is apriori not A·B-B·A.

Here's a clip on the Axioms
https://youtu.be/cCbn5x0z7pA

There are then rather technical result that savior the situation, in that it turns out that there's isomorphisms between Lie algebras and algebras with multiplications "·" that indeed makes it true that Lie algebras are "of the form A·B-B·A".
https://en.wikipedia.org/wiki/Poincar%C3%A9%E2%80%93Birkhoff%E2%80%93Witt_theorem