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.
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.