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So ive been encountering several exercises where we have a particle decay (into two other particles) and we're asked to calculate the, say, CM orbital angular momentum of the outgoing 2 particles (based on our knowledge of spins of the particles in question and angular momenta algebra). This is a straightforward exercise to solve but I am left with the question of why do we assume angular momentum is quantized in the first place?

Take the particle before it decays. There is no mention of external forces we assume it's a free particle. It has quantized spin, sure, but a free particle has no (to the best of my knowledge) quantized orbital angular momentum, so how do we compute $J$ (i.e. total angular momentum) in this case? can we even talk about $J$ at all? The same applies to the outgoing particles, why do we assume they have quantized orbital angular momentum? All I see is two free particles which, again, have no quantized orbital angular momentum.

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    $\begingroup$ Free electrons can have orbital angular momentum. See this article here en.wikipedia.org/wiki/… and this post on PSE physics.stackexchange.com/q/414758 $\endgroup$
    – joseph h
    Sep 25, 2020 at 10:20
  • $\begingroup$ Well, that's a surprise. So is orbital AM always quantized? $\endgroup$
    – Leonid
    Sep 25, 2020 at 10:49
  • $\begingroup$ In the realm of quantum scales, yes. $\endgroup$
    – joseph h
    Sep 26, 2020 at 0:26
  • $\begingroup$ @Leonid quantization arises when there are potentials with a solution to the quantum mechanical state that gives specific angular momentum wavefunctions $\endgroup$
    – anna v
    Sep 26, 2020 at 4:39

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Angular momentum conservation is one of the strict axiomatic laws that covers all of classical and quantum physics.

Quantization arises because of the quantum mechanical equations that are needed to describe the microcosm of particles, and quantization arises when there are potentials that have to be imposed on the particle. To begin with it was necessary in order to model the stability of atoms: electrons do not fall on the nucleus and neutralize the atom, see this.

A free particle has a specific spin. The particles from the decay also have a specific spin, which is by construction quantized, not continuous. Conservation of angular momentum imposes quantization to the products, in order for the initial particle spin (angular momentum) to be conserved. Spin for particles had to be invented in order to have conservation of angular momentum at the quantum level too. See this answer.

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