A few weeks ago I became frustrated with all the resources I was reading about angular momentum in QM. I was looking for a fundamental understanding of why quantum systems were modeled as they were. Not simply accepting the theories. Angular momentum was irking me because I had this idea that the ladder operators were using themselves to prove quantization. That if we approach angular momentum with the ladder operators it just works. This wasn't good enough for me, so I kept looking deeper. Why is angular momentum actually quantized? I kept digging and digging and became almost hopeless and exasperated as I began thinking I needed to master group theory and lie algebras to claw my way to the bottom of this well of endless questions. Then I had the silliest realization. It somehow took weeks to dawn on me: angular momentum is quantized because energy is quantized. Angular momentum is literally just a form of kinetic energy. And if that parameter is not quantized, then energy is no longer quantized. A non-infinite divisibility of total energy demands a model that quantizes angular momentum. After weeks of digging through math papers I could hardly read and watching hours of lectures, to realize how simple the answer was all along was an experience that made me reflect on my approach to learning. This idea that I must delve all the way to the bottom of scientific knowledge and slowly build my way back up from quantum physics to chemistry to biology; it almost seems like I'm pulling a Descartes, trying to throw everything away and build a world from "I think therefore I am". Physics can become overwhelming when I have the notion that I must always know "why". But accepting that I won't know why is equally outrageous. So in the end I'm not really sure. This comment is more of a thought log as I continue to learn and grow. But I get the feeling that all physicists confront this notion eventually.

The explanation is so elegant. This series has helped me to better understand quantum mechanics. I am really grateful for this. Have you posted the video on hydrogen atom and spin angular momentum? I'm looking forward to it!

I think it is somewhat confusing to use an ket when computing the wavefunction here. The position bra associated with the angular momentum states should only depend on the angles theta and phi. A more correct way would be to have a tensor product of a bra of . I don’t think many textbooks get this right, but you can consult J. Math. Phys. 62, 072102 (2021) for a proper description of how to develop the separation of variables, based on the translation operator in cartesian versus spherical coordinates.

At 1:10 I think there is a small error. There should be a (h/2π) factor multiplied to the RHS of the equation for [Ji,Jj].

thank you. This series is helping me a lot

Very well explained. Thank you

Thanks for this❤ lect.

Really nice content...

Thank you! Gee your videos are so clear, but they make one think. Don't you find it strange that a particle can have an intrinsic quality (quantum number), spin, which actually can "add" to a quantity it picks up from its behavior in space (orbital angular momentum), with the combination determining how it reacts to, e.g., a B field? S and L are two quantities with entirely different origins (apparently, as far as we know), but they are somehow the same in how they interact with the world. (I can just picture they guys doing the Stern Gerlach experiment looking at the results and thinking "OMG" over and over again at different layers as all the implications sank in.) I guess in a sense mass is maybe similar -- sort of an intrinsic energy that adds to energy picked up from extrinsic circumstances in determining how a particle behaves. But I gather there are other quantum numbers, such as electric charge, where there is no way a fundamental particle can pick up extra charge from extrinsic circumstances. Somehow angular momentum seems special . . . Anyway, obviously I need to learn a lot more. Thank you again; you guys are great!

The condition used to determine integral m is actually incorrect. Many textbooks get this wrong. The wavefunction need not be periodic. It is the probability density and the probability current that must be continuous, not the wavefunction. This is discussed well in both Green’s textbook and Ballentine’s. It is true, of course that m must be an integer, but periodicity of the wavefunction is not the reason why. It is more fundamental than that. And periodicity of the wavefunction is actually not required. This problem is closely related to the particle in a box, when you make the box a circle. There, you can pick any boundary condition desired at the endpoints, and all is well. But for angular momentum, one does require m to be integral for other reasons. For similar reasons, there are no pure angle operators in quantum mechanics, but cosine or sine of these angles are well defined, because these can be determined in terms of the Cartesian operators. Just thought you should know.

Where is the spin angular momentum video? Is it available?

Hi, as you have mentioned in the video to checkout the video on spin, I can't find it on ur channel? Can you please provide the link?

6:31 I'm pretty confused. In another video, the one where you talk about representations, I think, the wave function was the coefficient that multiplied the eigenstate, while here, the wave function is the eigenstate itself.

Where is the video on spin?

I couldn't find video on spin angular momentum on channel u were referring to in this video...??

Please make a video on Hydrogen atom. Thanks!

Hello professor. As I'm totally new in this subject I don't know this as much. I just read the position and momentum representations of state vectors and followed your videos. What I understand is the inner product of r with state vector psi ......that means gives us the so called wave function. But here you took the Inner product of r with eigenket |l, m> that is and called it wave function. But |psi> and |l, m> are not same right? |l, m> is just eigenkets of angular momentum operator just like |p> are eigenkets of momentum operator. So I can't understand that..... I know this is a weird question but as I said I'm new in this subject so don't know as much. If you help me a bit that would be great for me.....

Hai prof, why do these observables quantise their values? Who is prohibiting them to take any value they like? I understood the theory(with minimum uncertainty) but couldnt figure it out this. SE may be the reason, but do we have any other reasons Regards

Cool

Why do we take the third component of J, and J² as compatible observables?

I've heard before that "spin 1/2 means that you have to turn a particle two times to get back where you started". This derivation gives a glimpse as to why this explanation might have come about. If you let phi be [0,4*pi], m can be a half-integer