In quantum mechanics, certain observables like spin and energy in bound systems are quantized, meaning they can only take on discrete values. But others, like the position or momentum of a free particle, seem to vary continuously.
What determines whether an observable is quantized or not? Is this purely a result of boundary conditions, or is there a deeper mathematical reason related to the properties of the corresponding operators?
I'm looking for both an intuitive explanation and the formal reason from the perspective of quantum theory.
