I How to derive orbital angular momentum of spins in quantum mechanics?

[geelpheels]

TL;DR Summary

How to derive orbital angular momentum of spins in quantum mechanics?

How to derive orbital angular momentum of spins in quantum mechanics?
Make up for some lost things in Griffiths QM 3rd Ed. Equation (4.134) in Griffiths QM 3rd Ed. was given but no explanation. Please explain how to derive this equation in detail or provide references that by reading them one can understand how it is derived.
Thanks in advance.

 

[TensorCalculus]

Assuming you mean this equation: $[S_i,S_j]=i\hbar\epsilon_{ijk}S_{k}$
Some quick searching on google (thanks @fresh_42 for the google pdf searching tips - they're helping, a lot), I found this, which seems like it uses the classical $\vec L = \vec r \times \vec p$ to derive the equation, and I was able to follow it pretty easily despite not knowing nearly enough QM to actually use/apply the equation: https://www2.ph.ed.ac.uk/~ldeldebb/docs/QM/lect8.pdf

This one I don't understand - but just from a quick skim over it appears to be more thorough than the first source. It starts off with a similar approach but then seems to use Hamiltonians (?). Maybe it is just fully proving everything. When the first source said "it can be proven that [...]" I just took it for granted, but maybe if you want something more thorough and have the maths/physics knowledge to understand this, then this one is the way to go: https://people.ee.duke.edu/~jungsang/ECE590_01/LectureNotes1.pdf

(maybe I am not the best person to answer this question though. I have never seen this equation before in my life, or at least if I have I don't remember it. My physics is not this level. Just trying to help out.)

 

[div_grad]

Make up for some lost things in Griffiths QM 3rd Ed. Equation (4.134) in Griffiths QM 3rd Ed. was given but no explanation.

In the footnote just before these equations, Griffiths explains that he will take the commutation relations $[S_x, S_y]=i\hbar S_z$ (etc.) for spin operators as postulates of the theory, meaning he will not derive these relations but just adopt them as they are. In this same footnote, he suggests another QM textbook in which apparently you can find the derivation of these commutation relations. So you can check that out first.

Physically speaking, one of the experimental proofs that spin $\mathbf{S}$ is a form of angular momentum (and hence that it should satisfy the same commutation relations that the orbital angular momentum $\mathbf{L}$ does) is the so-called Einstein-de Haas effect (https://en.wikipedia.org/wiki/Einstein–de_Haas_effect) in which magnetizing a body causes it to rotate.