What's My Quantum Homework on This Week

Today I go over the topics that this weeks quantum homework is on, without going over specific problems (university policy). Mostly just talk about commutators tbh.

[AndrewDotsonvideos]

I have been informed by world experts in string theory and theoretical topological group theory that 6*5 is 30, not 35

[cantonlittle]

Heisenberg would win. Also, can't wait till the Quantum videos! Keep being awesome Andrew!

[ultrafinesharpie]

Unitary matrices are closed under composition, contain an inverse, have an identity, and are associative; i.e., they form a group (of transformations, w.r.t composition if you like). These properties are quite useful in QM dynamics. In particular, modeling the evolution of a quantum system with unitary matrices ensures that the system is symmetric w.r.t time, which is exactly what we want physically (if we decide to rewind a movie, and play it again, we'd better get the same sequence of events). I think this is a nice way of explaining the usefulness of unitary matrices for QM. It really only requires an understanding of general algebraic operations (not really necessary to point out that these form a group). Good luck!

[SalimonuDavid]

Just arrived at college. Excited to start learning this stuff. Thanks for being a role model, Andrew.

[JaxzanProditor]

I’m glad you’re taking the time to explain out commutators—as someone just starting out in undergrad physics and math I appreciate the look ahead!

[KyleKabasares_PhD]

As for the fight...
Schrodinger would turn his back on Heisenberg, dramatically increasing Heisenberg's uncertainty in position space, take the Fourier transform, and then beat him up in momentum space.

[michaelastwood]

I would explain unitary matrices in terms of rotations. For some unitary operator U and a 2d vector x, Ux rotates the vector x around the complex unit "sphere" while preserving the length of x. You can visualize this in 3d but you have to remember that this the complex unit sphere is a 4-sphere.

The real version of these are orthonormal matrices. Those are much easier to visualize and you can think about extending orthonormal matrices into the complex numbers

[__donez__]

A way that unitary operators were explained to me last week in my solid state class was this: When you have real-space vectors, you can find a 3x3 matrix that will rotate the vector to another direction. Unitary operators are kinda like rotation "matrices" for Hilbert space vectors (kets). Of course, there's a lot more that you can do with unitary operators, but that basic idea of "rotating kets" helped me understand their purpose a bit more.

[tem_althor]

What do you do in the background when you do homework?
Also you've kidnapped Schrodinger's cat so you want Heisenberg to beat him

[finthechat]

I wouldn't know who won until I observed it.

[QuanticSniperTGL]

'Unitary' means 'reversible'. So if you apply a unitary matrix on something it will affect this thing, but you can 'undo' it by applying the conjugate matrix corresponding to the first one.

[tatjanagobold]

Challenge accepted! Let's take any matrix U. Now transpose that matrix and if there are any complex values in the matrix, switch their sign (complex conjugate). If you now multiply the modified matrix ny the old one, you get the unity matrix, where there are all 1s across the main diagonal!

[iriswillbeheresoon]

Unitary matrices preserve angles because when the elements are real they are matrices in the orthogonal group, and so they correspond to rotations about an axis, which preserves angles between vectors #continuummechanics

[drandrewsanchez]

Yo this is cool and all but did you just say 5*6=35? 🤔 1:14

[zoltankurti]

Heisenberg, eassily. The only way to represent spin and operations related to them is via vectors and matrices. Hesienberg generalises more easily.

[Gokuyen]

Apparently this boi solved the god flipping riemann hippo!! I almost jumped outta my goddamn seat during class when I saw this. Name is Michael Atiya, and if he is trolling, well damn

[davidhoopsfan]

your bit on commutators was pretty interesting. andrew did you enjoy taking linear algebra? cuz that's what you were talking about reminded of

[roygreem]

Schrodinger would beat and not beat Heisnberg simultaneously and win since Heisnberg would either miss a really good punch, or land a really weak one.

[oak]

Man I just had a quiz that I believe I didn't perform well on. The professor gave us a spin state problem where the ket was composed of a |+z> and |+x> and he basically asked for the probabilities for +- h/2 for z and x but the fact that the main basis were not the same, being z and x, completely thew me off because all the questions i studied where all either in z basis or x basis only :( good luck in your homework, dude, i just wanted to complain somewhere lol

[andrej]

It's uncertain who would win in the fight