Planck's constant ($ h = 6.63 \times 10^{-34}~{\rm J \cdot s} $ ) represents the smallest amount of action that is possible in a system.
Is this value simply a lower bound, or do all dynamic systems have action equal to an integer multiple of $h$?
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2$\begingroup$ Given that $h$ is often used in combination with other factors (such as frequency), I don't agree with your first sentence. Then, that makes the question not make much sense. $\endgroup$– Jon CusterCommented Oct 19, 2016 at 19:34
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$\begingroup$ Possible duplicate: physics.stackexchange.com/q/44965/2451 and links therein. $\endgroup$– Qmechanic ♦Commented Oct 19, 2016 at 21:01
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$\begingroup$ @JonCuster, sorry but I don't agree with you. Although it is true that it is most often used in conjunction with other physical constants, I don't think that prevents it from having a physical meaning all by itself. Think about it, the speed of light is often used in combination with other factors, but it still has a physical meaning. $\endgroup$– psitaeCommented Oct 31, 2016 at 16:27
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2 Answers
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It is not the smallest action possible in a system. It is the scale of quantum fluctuations in the action away from the classical path. Look into the path integral formulation of quantum mechanics; it is that formulation that leads us to conclude that the classical path is an extremum of the action - because that is the path with the fewest oscillations, and the farther from that path the more the oscillations will tend to cancel each other, averaging out.
answered Oct 19, 2016 at 19:58
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Dimensions of Planck's constant are $$[\hbar] =\frac{ M L^2 }{ T }$$ In natural units we simply set it to $1$. Its physical significance is that it reflects the fact that we are too slow to directly experience relativistic effects in everyday life and too big to directly experience quantum effects. In a very deep sense, that's all there is to it.
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$\begingroup$ it is interesting that those are the units of angular momentum. $\endgroup$– anna vCommented Oct 20, 2016 at 6:34
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1$\begingroup$ Angular momentum has the same units as action. It's kind of like how torque and work both have units of Newton-meters, even thought they represent different physical things. I don't think this manner of dimensional analysis can be conclusive. $\endgroup$– psitaeCommented Oct 31, 2016 at 16:24
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1$\begingroup$ It simply refers to the fact that at the fundamental level $L=T=1/M$. We choose to differentiate between those dimensions, so the dimensionality and the value of $\hbar$ simply reflect that choice. It's human-made. Relative dimensions between other quantities, on the other hand, are not arbitrary. But they could all be expressed as one single dimension exponentiated to a rational power. $\endgroup$– user20250Commented Oct 31, 2016 at 17:29