Schrödinger applied Hamilton’s optico-mechanical analogy to develop his wave mechanics for subatomic particles. ^{[ 67 ] : xi} Consequently, wave solutions to the Schrödinger equation share many properties with results of light wave optics. In particular, Kirchhoff’s diffraction formula works well for electron optics ^{[ 29 ] : 745} and for atomic optics. ^{[ 68 ]} The approximation works well as long as the electric fields change more slowly than the de Broglie wavelength. Macroscopic apparatus fulfill this condition; slow electrons moving in solids do not.