As I understand it, the value of Planck's constant $h$ was originally introduced, in some sense, as a fitting parameter to understand blackbody radiation and the quantization of photons. In that sense, one can think of it as simply the proportionality constant between the energy and frequency of a photon.
However, we also have other relations such as the de-Broglie wave relation which use $h$, and some at a deeper theoretical level like in Schrodinger's wave equation, the commutation relation $[x,p] = i\hbar$, and the quantum path integral in the path integral formulation of quantum mechanics or quantum field theory.
Why is $\hbar$, which is essentially a proportionality constant related to light, so deeply tied to and seemingly universal in quantum theory? To be concrete, why does the value of $\hbar$ for the photon have to be the same as that of the de-Broglie wave relation, and why is $\hbar$ the same for all massive particles? Why does it have to be the same as that in Schrodinger's wave equation or the weight in the path integral or in the commutation relation? Why does classical theory generically stop being relevant on action scales close to $\hbar$? Qualitatively speaking, it doesn't feel like quantum theory intrinsically restricts $\hbar$ to be the same in all these cases, in the same way that general relativity identifies inertial mass with gravitational mass. Is it simply black magic that all of these are related, or is there something deeper? Or does it have something to do purely with the observables we can measure being deeply tied to the photon in some way?
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