If discrete/quantized observables result from a combination of the wave nature of matter and boundary value conditions, then how does spin arise?
From reading this post, it seems to me that it's accurate to say that quantization refers to describing particles as waves, and then discreteness naturally comes from that through boundary value conditions (or maybe more accurately confinement). To be clear, I use the terms quantization and discreteness mostly interchangeably except for emphasis.
For example, in an infinite square well, the energy levels $E_n$ are quantized as a result of the boundary conditions on each side of the well. Also, in context of the hydrogen atom or just orbital angular momentum, if a particle is restricted to a ring, then quantization of the energy values and eigenfunctions result from the boundary/periodicity condition $\psi(\theta) = \psi(\theta + 2\pi)$.
I have more of an undergraduate understanding of quantum mechanics, but I've heard of the idea that spin is a relativistic effect resulting from the Dirac equation and spinors. I think that the idea that spin must be quantized because of boundary value conditions since energy and orbital angular momentum are too is a fallacious, but I figured there might be something to it. Spin is also different from energy and orbital angular momentum in that it's intrinsic, and less of a result of the particular system being analyzed like energy and orbital angular momentum are. Maybe this is nonsense :p
Edit: Upon realizing that there two terms to the right hand side of the idea that discreteness is waves + boundary value conditions, it may also be a good or better question to ask, how does spin arise of the wave nature of matter? We could also take the view that energy and orbital angular momentum values and eigenstates result from nature of waves too (just along with boundary value conditions), so perhaps it may be good to consider the same about spin too.
