Wave functions, operators, de Broglie, Compton, spin, and the physical meaning of quantum formulas
Photon energy and kinetic energy share one structure: momentum times delivery velocity. In Planck units both reduce the Planck energy by a wavelength ratio — and, for matter, a velocity ratio. A photon and an electron with the same wavelength carry the same momentum but deliver different energies.
Classical p = mv and quantum p = ħ/ƛ are one formula. Rest mass supplies the momentum scale of the Compton wavelength; velocity rescales it to the de Broglie wavelength; the product is inverse wavelength — for matter and radiation alike.
Einstein’s equation can be read factor by factor: one c forms the momentum scale set by the particle’s Compton wavelength, the other is the universal speed of delivery. In universal (Planck-ratio) units the equation needs no constant at all — rest energy and rest mass are the same dimensionless number.
The Planck constant appears in every quantum formula. But what is it, physically? When expressed in Planck units, ℏ reveals itself as a composite of three recognizable quantities — the Planck length, the Planck mass, and the speed of light — and the formulas of quantum mechanics become statements about ratios to the Planck scale.