Intro to Spinors 4: Algebraic theory of Spin

In this video I talk about the algebraic theory of spin. We start out by defining some commutation relations and from there find out what the spin operators have to be. Small mistake at the end, “Sz” should be the Pauli spin matrix sigma z. Other than that, hope you guys enjoyed this video!

P.S…This is a topic that is still a bit elusive to me, so if some parts seem a bit hand wavy or aberrant, sorry! I’m generally good with spinors, but the spin operators still seem unmotivated and lacking physical justification to me. Other videos on spin will be much more clearer since this is the only video I’ll do on the spin matrices (for now at least 😂)

[KipIngram]

I think you're doing a great job. It's fairly clear you're in the beginning stages of your own training, so I think the effectiveness with which you're feeding it back is admirable.

So far as the angular momentum stuff goes, I think the best way to get at it is to solve Schrodinger's equation in a spherically symmetric potential (the hydrogen atom is the prototypical case). You can write the equation and then separate it and solve using spherical harmonics, and your m and another parameter usually called l (lower case L) pop out - they can only take on certain values which each correspond to a spherical harmonic. This doesn't pick up the "quantum" spin - you have to add the possibility of two cases there as an "add on." But the process captures m (often called n) and lower-L quite nicely. This whole process of course just gives us the classical "electron orbitals" (s, p, d, f, ...) that you see pictures of all over the net.

Sometimes people get confused by this treatment of the atom, thinking it implies that there's some "special" coordinate system that we're orienting the atom in. But there's not - you actually some spherical coordinate system to use when you draw the problem, and the solution that arises is in coordinate system. You could have chosen anything. The solution process produces results that correspond to the chosen system, regardless of what you pick. This corresponds directly to the fact that you can only get "up" or "down" in the SG experiment, regardless of how you orient your magnet.

Keep it up - I think you're doing great. I know a lot of this stuff fairly well, but still with some fuzziness and you've managed to touch on a couple of things that were quite helpful to me.

[jackiewhitt]

Love your vids, because I can understand them thanks to your explanations

[nachiketakumar]

Wow nice explained, please give a full lecture series of 1hour on QFT, really love your work 👍👍👍👍

[FermionPhysics]

About Sz, we would use Sz|up>=h/2|up> and from there get the matrix elements for Sz (analogous to how we did S^2)