Inference of Schr\"{o}dinger Equation from Classical-Mechanics Solution*

We set up the classical wave equation for a particle formed of an oscillatory massless charge, traveling at velocity $v$ in a potential $V(X)$ in a one-d box along $X$ axis, and its electromagnetic waves $\{\varphi_p^j \}$ (as virtual or ``hidden'' processes) as: $[c^2-\frac{V}{m}] \frac{\partial^2 \psi}{\partial X^2} = \frac {\partial^2\psi}{\partial T^2}$ (1). Where $\psi=\sum \varphi^j_p$; $p=$ {\it incident} or {\it reflected}, $j=\dagger $ or $ \ddagger$ for $\angle{[c,v]}=0 $ or $\pi$, $c$ velocity of light, $M=m\sqrt{1-(v/c)^2}=\frac{h {\mit\Omega}}{2\pi c^2}$ the particle's rest mass, $\frac{{\mit\Omega}}{2\pi}$ wave frequency for $v=0$, and $h$ Planck constant. For $V=$const, Eq (1) has the plane wave solutions: $\{ \varphi_p^j=C_1 e^{i(K^j X-{{\mit\Omega}}^j T )} \}$; $K^j(j={\mbox{$\tiny{{\dagger \brace \ddagger}}$}})=\frac{K}{1 \mp v/c}$ is a Doppler-displaced wavevector; ${{\mit\Omega}} ^j=K^j c$. From $\sum\varphi_p^j$, we get a standing beat, or de Broglie phase wave for the particle total motion: $\psi =4C_1\cos(KX)e^{i({\mit\Omega}+\frac{{{\mit\Omega}}_d}{2}) T } {\mit\Psi}$. Where ${\mit\Psi}=C\sin(K_d X)e^{-i\frac{{{\mit\Omega}}_d}{2} T} $ describes the particle motion, and $K_d= \sqrt{(K^{\mbox{\tiny${\dagger}$}}-K)(K-K^{\mbox {\tiny${\ddagger}$}})} =(\frac{v}{c})K$ the de Broglie wavevector; ${{\mit\Omega}} _d=vK_d $. For $V$ varying, we get similarly a $\psi$ and ${\mit\Psi}$ from sums of partial plane waves from all of infinitesimal $(X_i, X_i+\Delta X)$. We can in turn subtract (1) by itself but with $v=0$, getting an equation for ${\mit \Psi} $: $[-\frac{\hbar^2}{2M} \frac{\partial ^2}{\partial X ^2} +V(X)] {\mit \Psi}=i\hbar\frac{\partial {\mit \Psi}}{ \partial T}$, which is equivalent to the Schr\"{o}dinger equation. (The so represented QM invites not the so-called EPR paradox.) \quad