3 B A FIG.1:Maupertuis'Principle B.Stationary Schradinger equation The parallelism between light and matter can go further Light: wave nature omitted geometric optics wave nature can not be omitted new mechanics namely quantum mechanics Quantum Mechanics二Wave Optics☐ Light wave geometric optics Fermat's principle wave propagati Matter wav particle dynam Principle of least action wave propagation I'resently unknown Now conide light wave propagation in a non-homogenousmedium B light path=n(问d Fermat's principle n=0 For a particle moving inapotential field V(),the principle of least action reads iV2mia=iV2me-v阿as=0 The corresponding light wave quation (Helmholtzequation)is (（-)保利-0 3 A B FIG. 1: Maupertuis’ Principle. B. Stationary Schr¨odinger equation The parallelism between light and matter can go further Light: wave nature omitted geometric optics wave nature can not be omitted wave optics Matter: wave nature omitted particle dynamics wave nature can not be omitted a new mechanics, namely quantum mechanics Particle Dynamics ⇐⇒ Geometric Optics ⇓ ⇓ Quantum Mechanics ? ⇐⇒ Wave Optics Here we make comparison between light propagation of monochromatic light wave and wave propagation of monochromatic matter wave Light wave geometric optics Fermat’s principle wave propagation Helmholtz equation Matter wave particle dynamics Principle of least action wave propagation Presently unknown Now consider light wave propagation in a non-homogeneous medium light path = ∫ B A n(⃗r)ds Fermat’s principle δ ∫ B A n(⃗r)ds = 0 For a particle moving in a potential field V (⃗r), the principle of least action reads δ ∫ B A √ 2mT ds = δ ∫ B A √ 2m(E − V (⃗r))ds = 0 The corresponding light wave equation (Helmholtz equation) is ( ∇2 − 1 c 2 ∂ 2 ∂t2 ) u (⃗r, t) = 0