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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?

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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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    $\begingroup$ This means that when the ceiling fan speeds up, it is actually jumping from one speed to another - not necessarily: the fan could very well be in a smoothly varying superposition of different Eigenstates of the angular momentum observable $\endgroup$ – Christoph Aug 5 '17 at 20:07
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    $\begingroup$ @Christoph Since superpositions are not observable, every time you measure the ceiling fan's speed, you'll get an eigenvalue. So, even if the transitions are smooth, the measurements will be discontinuous. $\endgroup$ – Mark H Aug 5 '17 at 21:16
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    $\begingroup$ but speeding up is a continuous process that increases the contribution of Eigenstates of higher velocity and not some sort of discontinuous jumping between states $\endgroup$ – Christoph Aug 5 '17 at 21:57
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    $\begingroup$ Alas, you beat me to the quantum vortices. Ultracool stuff, in both literal and exact sense. $\endgroup$ – Thorsten S. Aug 6 '17 at 0:45
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    $\begingroup$ Is the fan small enough that the angular momentum is well-defined? It's not a rigid object. $\endgroup$ – MSalters Aug 7 '17 at 13:50
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This is like asking if water is quantized. Of course! : the basic unit of water is the $H_2O$ molecule. Naturally when you water your plants don't think of water as "quantized": you think of it as a continuous fluid, because the basic unit is so small that it makes no practical sense to count the water molecules needed to properly dilute your fertilizer. If anything the number of molecules is not exactly constant either (due to evaporation, humidity in the air etc).

Likewise the basic unit of angular momentum is so small and common objects have so many of these basic units that it's not practical to keep track of angular momentum this way. Moreover, there are air molecules hitting the fan and transferring angular momentum, and other outside factors that make this number of units not exactly constant in everyday situations.

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Just to conjecture a contrarian opinion, no, you can't say that the angular momentum of a macroscopic fan is quantized by any quantum-mechanical stretch of the imagination.

A macroscopic fan (or any other object) is a hode-podge, extremely decoherent composite of many, many (on the order of, say, $6\times10^{23}$) microscopic (sub)systems. As such, there is no single quantum–mechanical observable corresponding to its angular momentum.

What you observe instead as the fan's angular momentum is just an ensemble average from many, many microscopic interactions between the fan and your measuring apparatus.

For example, suppose you rolled a single die (one of two dice) $6\times10^{23}$ times, and averaged all those outcomes. Each roll is "quantized", with possible "eigenvalues" $1,2,3,4,5,6$, so to speak. But the average will be something like $3.nnn\ldots$ with some $24$ digits after the decimal point. Hardly quantized.

And the macroscopic fan situation is even worse. Although the fan as a whole is a decoherent hodge-podge composite, there are no doubt many microscopic localized regions containing entangled superpositions of neighboring particles, just by happenstance. And then, calculating probabilities for their stochastic outcomes -- comprising an "epistemic mixture" (the dice analogy) of "ontological superpositions" -- gets even goofier.

And if that's not bad enough, the fan material likely has some crystalline/lattice/etc structure, making the situation just that much more worse, so to speak, e.g., https://en.wikipedia.org/wiki/Crystal_momentum although I have no idea what some corresponding angular momentum calculation would look like.

So one way or another, it's very, very likely that somewhere along the line, the overall system transitions from a discrete to a continuous spectrum. Thus, to repeat, the answer to your question is no (like the sign says, "Don't even think about parking here!":)

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    $\begingroup$ "Hardly quantized." does not mean "not quantized". Actually, it means the exact opposite thing. $\endgroup$ – AccidentalFourierTransform Aug 5 '17 at 10:57
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    $\begingroup$ @AccidentalFourierTransform Please take it in context, where "hardly quantized" is clearly meant as a vernacular figure of speech. What I mean to literally say there is that it's not quantized. But I see no ambiguity leaving it as is. You have any actual physical objection to what I'm saying? $\endgroup$ – John Forkosh Aug 5 '17 at 11:00
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    $\begingroup$ @AccidentalFourierTransform Oh, wait... I just read your profile. Okay, go ahead, take it out of context:) $\endgroup$ – John Forkosh Aug 5 '17 at 11:12
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Yes, the angular momentum of the ceiling fan is quantized, in a technical sense. But in such a big system, with a macroscopic angular momentum, the possible values are so close together that it's indistinguishable from a continuum.

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  • $\begingroup$ Thanks. What would it imply for the motion of the ceiling fan if the value of $\hbar$ were large? In other words, how would we, in that case, perceive the discreteness of angular momenta? $\endgroup$ – mithusengupta123 Aug 5 '17 at 7:57
  • $\begingroup$ @mithusengupta123 The quantization would be more obvious, as the momentum levels would be farther apart. But it would have to get quite large indeed for you to see macroscopic effects. $\endgroup$ – probably_someone Aug 5 '17 at 7:58
  • $\begingroup$ I'm trying to ask what peculiarity would I observe in its motion? Let's assume it's quite large (just for a matter of understanding). $\endgroup$ – mithusengupta123 Aug 5 '17 at 8:01
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    $\begingroup$ @mithusengupta123 You would only be able to get the fan to change speeds by adding a specific amount of angular momentum. Keep in mind that changing $\hbar$ changes the way everything works, not just the fan, so there's no simple explanation beyond that. $\endgroup$ – probably_someone Aug 5 '17 at 8:04
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    $\begingroup$ @mithusengupta123 If ℏ were large then life as we know it would be impossible, and no one would be around to observe ceiling fans. The anthropic principle applies to ceiling fans as much as to anything else. $\endgroup$ – Mike Scott Aug 6 '17 at 9:27

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