Photon energy is proportional to its frequency:
or with common weird variables:
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 , 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:
Appears in the Schrödinger equation.
Equals the quantum of angular momentum in the Bohr model.