I understand mathematically that the ladder operator increases the z-component of angular momentum (Lz) by ℏ. However, I don't quite grasp the physical meaning behind why the specific structure of the ladder operator, L+=Lx+iLy, causes this increase in Lz. While I follow the derivation, the underlying intuition remains unclear.
Furthermore, is it impossible for an operator to exist that changes Lz by an amount smaller than ℏ while maintaining the total magnitude of angular momentum (L2)?
Some explanations suggest that the ℏ unit arises to satisfy the standing wave condition on a circular path. But as we know, electron spin (which has half-integer values) does not satisfy this spatial standing wave condition. If the result is strictly derived from the standing wave condition, shouldn't half-integer z-component angular momentum be impossible?
I am confused about how the ladder operator fundamentally guarantees that the z-component of angular momentum changes only by the minimum unit of ℏ.