A challenge post! 😉
Quantum physics becomes important when the magnitude called action is of order of Planck constant (an action itself!). Action is quantized. It is a much more essential quantization than that of energy or angular momentum (action itself the latter!). Whenever 
you have quantum effects.
Exercise 1. Compute the action from an antenna with power 
and frequency 
. Is it quantum?
Exercise 2. A pocket clock. It uses a device of size 
and mass 
to get times with precision of 
. Compute its action. Is it quantum?
Exercise 3. Compute the action for a single atomic nucleus. Typical energies are about 1 MeV and distances about 1 fm (
). Binding energy per nucleon is about 
, mass of the nucleus can be taken as the proton mass. Compute the action. Is it quantum?
Exercise 4. Compute the action for an electron in the shell of the hydrogen atom. Is it quantum?
Exercise 5. Compute the so-called Fermi energy in the case of potassium (metal). The atomic weight is about 39 g/mol, and its atom density 0.86 g per cubic centimeter. Suppose a single electron per atom. Compute its action. Is it quantum?
Remark:
![\[\left[\hbar\right]=ML^2T^{-1}=\mbox{Energy}\cdot \mbox{Time}=\mbox{Momentum x Length}=\mbox{angular momentum x angle}\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-7f794ca261fbda0e8cb02dbe6d2076aa_l3.png "Rendered by QuickLaTeX.com")
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