The reduced Planck constant $\hbar$ occurs in multiple places in physics. For example, the spin operator of a spin-$\frac12$ particle is given by $$\bf{J}=\frac{\hbar}{2}\bf{\sigma}$$ where $\bf{\sigma}$ is a vector of dimensionless matrices. Also, the Schrödinger equation states that $$i\hbar \frac{\delta}{\delta t}|\psi(t)\rangle=\hat{H}|\psi(t)\rangle$$
In both places, we needed something with unit $Js$ which is the unit of $\hbar$. But it could also have been another number with the unit $Js$. I would like to know if these two appearances of $\hbar $ are related? I think that they can both be derived by measurements. But is it also possible to start reasoning from one (like the $\hbar$ Schrödinger Equation) and prove the other (like the $\hbar$ in the spin operator? Or can both be derived from a single, more general formula?
