One of the best explanations of orbital angular momentum for the electron comes from Dirac himself. At around 39:30 of this youtube video (you will need headphones, but it is well worth it), Dirac talks about the non-commutation of operators, how quantum mechanics is more general then classical mechanics and how quantum mechanics can use any functions to give equations of motion for any hamiltonian variables.(adsbygoogle = window.adsbygoogle || []).push({});

He then talks in detail about electron spin and specifically how the 3 spin axis variables s1, s2, and s3 can vary in time and are non-commuting (ie. turn on one axis followed by a turn on a different axis depends on the order you do the turns). He points out that the 3 components of spin s1, s2, and s3 satisfy the same conditions as orbital angular momentum.

It is important to remember he is talking about an intrinsic property of the electron, not its relationship with the nucleus (ie. the electron is not orbiting anything). The only classical analogue I can think of to an electrons orbital angular momentum as a precession of the spin axis of a spinning particle. The spin of the particle (left or right) leads to the spin quantum number (spin up or down), the precession of the spin axis (around an imaginary real axis) provides the orbital angular momentum quantum number. Precession of the spin axis of a spinning particle not under the influence of an outside force (gravity, magnetic, or anything) would map out some pretty weird orbitals.

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# I What is an electron's orbital angular momentum?

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