The Relation between Classical Mechanics and Wave Mechanics

Abstract

Introducing a dimensional unit h into classical mechanics, we make classical wave functions v'p;: exp (iS1/h) -vp; exp (iS2/h) for a one-particle system in the configuration space where vp;: exp (iS1/h) corresponds to the incident path (or one half) of an orbit and v'P.; exp (iSJh) to the outgoing (or another half), and p's and S's are densities and actions respectively. These classical wave functions are compared with the corresponding wave functions of solutions of the Schriidinger equation for three one-particle systems, the scattering by a Coulomb potential, the linear harmonic oscillator and an electron bound by a Coulomb potential. From the essential correspondence between them it is concluded that the Schriidinger wave function describes the steady, causal but a little imaginary motion of the one-particle system as v'/l exp (iS/h) in the configuration space where p is a density and S an action, and that the states which the wave equation describes exist continuously (not discrete) and the (discrete) stationary states are the ones where Vp exp (iS/h) is uniquely determined throughout an orbit. The Dirac equation shows that an electron has four characteristics besides its momentum at a position. The scalar wave equation of optics (or the electromagnetic wave equation) can be considered as the Schriidinger (or Dirac) equation for the motion of a photon, a particle, in a potential. A quantum-electrodynamical wave equation which is composed of the Dirac equation with a Coulomb potential and the electromagnetic wave equation is constructed for the steady motion of a one-electron system without recource to the so-called second-quantization method. It gives finite energy shifts. A line shift is calculated: JE(2S1J2->2P1J2)~709 Mc/s.