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I found here that the Planck constant is defined as an exact number: $6.626 070 15\times10^{−34}\ \mathrm{J/Hz}$. How could this be done? Shouldn't it be a quantity with uncertainty measured by experiments?

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    $\begingroup$ Why are you asking this about Planck’s constant and not also about the speed of light and the electron charge? Your link shows that all three are “exact”. $\endgroup$
    – G. Smith
    Oct 10 '20 at 4:10
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    $\begingroup$ These arguments don’t make sense. For example, Planck’s constant is the universal constant of QM in the same way that the speed of light is the universal constant of SR. $\endgroup$
    – G. Smith
    Oct 10 '20 at 4:36
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    $\begingroup$ Planck constant has exact value 2Pi. $\endgroup$ Oct 10 '20 at 13:36
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    $\begingroup$ Possible duplicate: Why was the Planck constant h fixed to be exactly 6.62607015×10−34Js and not some other value? $\endgroup$ Oct 10 '20 at 14:20
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    $\begingroup$ @G. Smith it was a joke about natural units ... $\endgroup$ Oct 10 '20 at 16:12
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Planck's constant relates two different types of quantities, namely energy and frequency. That means it is a conversion factor which converts the units of quantities from one form to another. If the units of these two quantities are separately defined, then one can use measurements to determine the value of the conversion factor. That value would then have some uncertainty due to the experimental conditions. That is what has been done before. However, recently it was decided to define the units of one of the quantities in terms of the other, by setting the conversion factor (Planck's constant) to a fixed value without uncertainty. It came about by the redefinition of the kilogram. Now it does not have any uncertainty anymore. The same thing was done for the speed of light some time ago.

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  • $\begingroup$ Does 1983 fall under "...not too long ago..."? $\endgroup$
    – notovny
    Oct 10 '20 at 4:32
  • $\begingroup$ This answer seems to ignore the usage of Planck's constant outside the equation $E=h\nu$. $\endgroup$
    – Sandejo
    Oct 10 '20 at 4:39
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    $\begingroup$ @Sandejo the question didn't ask about them. The energy-frequency thing was only brought up because it helped answer the question, I'm guessing. $\endgroup$
    – PyRulez
    Oct 10 '20 at 13:47
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    $\begingroup$ @PyRulez I realise that. I just thought it would be worth noting to people reading this answer that Planck's constant is seen in other places, especially since the way the answers is worded suggests that it's only used to relate energy and frequency. $\endgroup$
    – Sandejo
    Oct 10 '20 at 16:51
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    $\begingroup$ @notovny I guess that it depends on your age. It is "not too long ago" for me as I was not only alive but old enough to understand it. $\endgroup$
    – badjohn
    Oct 12 '20 at 12:03
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Before May 2019, Planck's constant was not defined by an exact value and instead was measured experimentally to be $6.626069934(89)\times10^{−34}\ \mathrm{J\cdot s}$. However, it is worth noting what we mean in saying that this constant has a certain numerical value when expressed in certain units. In essence, when we measure a physical quantity, we are comparing to the value of some constant that has been declared as a standard, i.e. a unit.

When Planck's constant was measured experimentally, this meant comparing to the old value of the joule-second, which was, in part, defined based off the mass of a lump of metal in a vault in France. In other words, the quantity would change if the mass of the International Prototype of the Kilogram were to change. Because of this, it was generally recognised that it was not ideal to define units based on artefacts, that it is better to define units based on physical constants. However, up until recently, there wasn't a good way to define the unit of mass based a physical constant.

What changed recently was the development of the Kibble balance, which made it possible to measure Planck's constant with sufficient precision to define it to be an exact value. Now, you may be wondering how the uncertainty goes away, since measurements always have uncertainties. The answer is that this uncertainty gets shifted to the calibration of devices that make measurements in the units defined by Planck's constant, namely the kilogram. In other words, whenever you measure the mass of something in kilograms, you are indirectly comparing the mass to Planck's constant (combined with some other constants to get the dimensions right), and the uncertainty in Planck's constant propagates to the calibration of your balance.

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  • $\begingroup$ A useful follow up to this would be to ask why Planck's constant is fixed in the SI rather than some other constant (such as the electron mass), but the answer to that is beyond my knowledge. $\endgroup$
    – Sandejo
    Oct 10 '20 at 5:05
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    $\begingroup$ An important part of "measur[ing] Plan's constant with sufficient precision to define it to be an exact value" was that the measurements of Plan's constant were becoming more reliable than the measurements we could make of the lump of metal in a French vault. We would not have switched if it had made things leess accurate. $\endgroup$
    – Cort Ammon
    Oct 10 '20 at 5:42
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    $\begingroup$ @Sandejo The reason is that we don't have means to directly compare, in a sufficiently accurate way, macroscopic masses, like those employed in everyday's life, with microscopic masses like those of elementary particles. $\endgroup$ Oct 10 '20 at 20:17
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    $\begingroup$ @FilipMilovanović Note that the value was not chosen arbitrarily, but equal to the best estimate of the Planck constant that we had at the moment of the revision of the SI, as to ensure the compatibility, within the state-of-the-art uncertainties, of the mass measurements before and after the revision. $\endgroup$ Oct 10 '20 at 20:21
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    $\begingroup$ @FilipMilovanović Basically yes. As Massimo Ortolano pointed out, an important element of the choice was that the major methods of measuring Plank's constant were in sufficient agreement to the number of sig figs chosen, but within that agreed upon range, the actual choice of number was arbitrary. At the time it was chosen, there were no measurements made which a) disagreed and b) were sufficiently precise that the disagreement could not be treated simply as a measurement error. $\endgroup$
    – Cort Ammon
    Oct 10 '20 at 22:06
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This comes down to how units are defined. If you look at the definition of SI units, in particular the one for the kilogram:

Interim (1889): The mass of a small squat cylinder of ≈47 cubic centimetres of platinum-iridium alloy kept in the International Burueau of Weights and Measures (BIPM), Pavillon de Breteuil, France. Also, in practice, any of numerous official replicas of it.

This was how the kilogram was defined in the past. Note this is obviously undesirable. There's exactly one small squat cylinder of platinum-iridium alloy that qualifies as the definition. Not only is that intrinsically problematic (the replicas are not "official" so different people can end up with different kilograms), there are other problems: For example solids undergo sublimation and become gas. This process is extremely slow for solid metals, but the rate is still not zero. How is the kilogram to be defined then? Do we also have to specify the year?

The solution to this was to define the kilogram in terms of the Planck constant. Now that the Planck constant has an exact value, if its value "shifts" slightly, it's the value of the kilogram that actually shifts.

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