Planck's constant $\hbar$ appears in the Schrodinger equation:
$$i\hbar \frac{d|\psi\rangle}{dt}\ = \hat{H}|\psi\rangle$$
which implies for stationary states,
$$|\psi(x,t)\rangle=e^{-iE_n/\hbar}|\psi(x,t=0)\rangle \, .$$
What would you say is the role of $\hbar$ or is it worth considering the other formulations of quantum mechanics, like the Heisenberg equation of motion (where the operators carry the time evolution:
$$i\hbar \frac{d\hat{A}(t)}{dt} = [\hat{A}(t),H] + i\hbar\frac{\partial\hat{A}(t)}{\partial t} \, ?$$
Or is it worth considering this from the standpoint of the Path Integral Formulation where $\hbar$ is used to work out the probability amplitude of a specific path:
$$e^{iS/\hbar}.$$
I understand that $\hbar$ is often used to make equations dimensionally consistent and that it is often set to 1 when dealing with complicated equations. It is also sometimes stated that $\hbar \to 0$ is the "classical limit". I do not understand this.
