Angular momentum in quantum mechanics

📚 Angular momentum is key in understanding many physical situations, perhaps most importantly the behaviour of particles moving in central potentials, like an electron in a hydrogen atom. In this video we upgrade classical angular momentum to a quantum mechanical operator, and we explore some of its properties. We also learn that angular momentum in quantum mechanics is not restricted to *orbital* angular momentum and introduce the idea of *spin* angular momentum, which has no classical analogue.

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Director and writer: BM
Producer and designer: MC

[qbtc]

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[dutonic]

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[ashutoshyadav]

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[vaibhavshukla]

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[josem]

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[juliaifrank]

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[anantbadal]

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[debankurbasak]

Great explanation. Keep posting more videos

[arunenquiry]

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I am looking forward to your future videos on other areas of Physics. Keep up the good work!

[noelwass]

This is such a clear exposition and wonderfully explained of the angular momentum in quantum mechanics. I want to go through this more carefully so that I understand all the quantum mechanical calculations. This definitely helps me understand spin better. You show in the video how orbital angular momentum is defined in quantum mechanics and then show calculations of all the commutation relations. Then spin angular momentum is defined as a triple of three Hermitian operators that satisfy these same commutation relations. This is very nicely explained.

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[satishayila]

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[Bhagya]

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[DrDeuteron]

I didn't really understand angular-momentum until a I wrote some python to split rank "N", for 0 to 16, into irreducible representations. It took about 3 years as a hobby, but it started with N, say "3", partitioned it into 3, 2+1, 1+1+1, ....then made those into young tableaux, then normal young diagrams, of which there are 1, 2, and 1, from those 4 diagrams, you get representations of the permutation group on three indices (ijk), and then make some kind of product (I forgot what it was called) that permutes ijk into stuff, e.g. for 3=1+1+1 is +ijk, +jki, +kij, -kji, -jik, -ikj so add those up on the tensor and get L=0 M=0, which is the famed Levi-Civitta thing. It's scalar. for 3=3, all the signs are + and you get the total symmetric traceless pure rank-3 tensor, with, wait for it, 7 degrees of freedom because its L=3, M=-3, -2, -1, 0, 1, 2, 3 (breaking out the M's requires Clebsch-Gordanry, and is hard), The middle ones have mixed symmetry, but they ALL look just like Spherical Harmonics in cartesian coordinates made out of not x, y, z coordinates, but x, y, z unit vectors in outer-product mode.
So what I learned is that angular momentum is conserved b/c space is isotropic, and scalars, vectors, tensor, or |l, m>, |j, m> are things that can be rotated but are also eigenvectors of Z rotations, and then to go from a cartesian dyads to spherical tensors is the same as CG going from |j, m>|j', m'> --> |J, M> + |J', M'> + ...., and because rotations don't commute, angular momentum operators don't commute, and the math is easy to understand but hard to do.

*Oh btw: said tensor are classical systems (e.g. the rank-4 generalized Hooke's tensor), but in the spherical basis they behave EXACTLY like quantum systems with L-= 4, 3, 2, 1, 0. (though I suspect the odd L are all zero for that example).*

[emelieanes]

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[zoubirbecer]

Pure and clear presentation, great work. Both presenters are really excellent . They kept the same behaviour and the two face looks similar, I am right.?

[paulbk]

It will be absolutely awesome to hear from you on statistical mechanics too.