Inference of Schr\"odinger's Equation from Classical Wave Mechanics[1]

A localized oscillatory point charge $q$ generates in a one-dimensional box electromagnetic waves which may be generally described by monochromatic plane waves $\{\varphi_i=C_K e^{i(KX- \Omega T+ \alpha_i)}\}$ of angular frequency $\Omega$, wavevector $K=\Omega/c$, velocity (of light) $c$, and initial phases $\{\alpha_i\}$. $q$ and $\{\varphi_i\}$ as a whole is here taken as a particle, which total energy ${\sf E}$ and mass $M$ are given by the basic equations ${\sf E}=\hbar \Omega=M c^2 $, $2\pi\hbar$ being Planck constant. (For example, $q=-e$ and $M=511$ keV give an electron.) $\{\varphi_i\}$ as incident and reflected and those from the charge as reflected in the box superimpose into a total wave $\psi=\sum \varphi_i$ that, as with $\varphi_i$, obeys the classical wave equation (CWE): $c^2 \frac{d^2\psi}{d X^2}= \frac{d^2\psi}{d T^2}$. If now the particle is traveling at velocity $v$, in a potential field $V=0 $ here (see Ref. 2004b for $V\ne 0$), then $\{\varphi_i'\}$ are Doppler effected and form a total wave $\psi'={\mit\Phi} {\mit\Psi}$, with ${\mit\Psi}= C \sin(K_d X)e^{i \Omega_d T}$ being the envelope about a beat wave and identifiable as de Broglie wave of angular frequency $\Omega_d= \Omega (v/c)^2$, and ${\mit \Phi}$ an undisplaced monochromatic wave. Using $\psi'$ in CWE gives upon decomposition a separate equation describing the particle dynamics, $-\frac{\hbar^2}{2M} \frac{\partial^2 {\mit\Psi}(X,T)}{\partial X^2}=i\hbar\frac {\partial {\mit\Psi}(X,T)}{\partial T}$, which is equivalent to Schr\"odinger's equation. \\[0cm] [1] J. X. Zheng-Johansson and P-I. Johansson, arXiv:Physics/0411134 (2004a); "Unification of Classical, Quantum and Relativistic Mechanics and the Four Forces" (in printing, 2004b).