This only makes sense if the photon exists, there is no classical analogue, because the energy of classical waves depends only on their amplitude, not frequency.

Experiments that suggest this:

Proportionality factor in the Planck-Einstein relation between light energy and frequency.

And analogously for matter, appears in the de Broglie relations relating momentum and frequency. Also appears in the Schrödinger equation, basically as a consequence/cause of the de Broglie relations most likely.

Intuitively, the Planck constant determines at what length scale do quantum effects start to show up for a given energy scale. It is because the Plank constant is very small that we don't perceive quantum effects on everyday energy/length/time scales. On the $lim_{h→0}$, quantum mechanics disappears entirely.

A very direct way of thinking about it is to think about what would happen in a double-slit experiment. TODO think more clearly what happens there.

Defined exactly in the 2019 redefinition of the SI base units to:

$6.62607015×10_{−34}J⋅s$

Appears in the Schrödinger equation.

Equals the quantum of angular momentum in the Bohr model.