According to quantum mechanics, any quantum angular momentum is quantized in units of $\hbar$. Does it mean that the angular momentum of the ceiling fan (due to its rotation) is quantized? If yes, what does it physically mean? Does it mean that it cannot rotate with arbitrary speed?
Answer
Yes, the angular momentum of a ceiling fan is quantized. This means that when the ceiling fan speeds up, it is actually jumping from one speed to another. However, the size of these jumps is so small--because Planck's constant $h$ is so small--that the difference between two allowed speeds is immeasurably small.
This is similar to how a thrown baseball's position is uncertain because of Heisenberg's Uncertainty Principle. Same as the ceiling fan, the smallness of $h$ makes the size of the uncertainty immeasurably small.
There are macroscopic systems where quantized angular momentum can be observed. When liquid helium is cooled enough to become a superfluid (~2 Kelvin), it will not rotate if the container is rotated slowly. If the containers rotation is slowly sped up, there will be a certain speed where a little whirlpool suddenly appears. The liquid helium has gained one unit of angular momentum. As the container continues to speed up, more of these quantum vortices appear, each one containing a quantized unit of angular momentum.
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