How is Planck's constant relevant in quantum mechanics?

Question

Planck’s constant relates the frequency of a photon to its energy but how does that relate to other quantum particles? For example in the Schrödinger equation it is used according to my book “to correct the units”. Is that true? And why Planck’s constant and not 1?

Answer 1

It is true that Planck's constant was originally understood as the ratio between the photon's energy and its frequency, but later it was found that it is much deeper.

Planck's constant is actually a measure of the granularity of quantum mechanics. For example, you will see (or have seen) that the position and momentum operators respect $$[\hat{x},\hat{p}] = i \hbar,$$ where $\hbar = \frac{h}{2\pi}$. In this sense, $\hbar$ tells us how position and momentum fail to behave classically (in which case they would commute), and hence it leads to the quantum behavior. $\hbar$ sets the scale in which quantum effects start to be meaningful.

Another way of understanding this is that $\hbar$ is necessary not only to fix the units, but also to have agreement with experiment in the Schrödinger equation. We can use the results of the Schrödinger equation to compute, for example, the energies of the hydrogen atom and use those results to fix what is the constant occurring in the Schrödinger equation. In this sense, the properties of photons end up being a corollary, rather than a definition.

Finally, notice that if we used 1, that would depend on the system of units. In natural units, we indeed take $\hbar = 1$, but this value is simply not true in the SI.