Inference of Schrodinger Equation from Classical Wave Mechanics[1]

A localized oscillatory point charge $q$ generates in a one- dimensional box electromagnetic waves which for potential field $V=0$ may be generally described by monochromatic plane waves $\{ \varphi_i=C_K e^{i (KX-\Omega T + \alpha_i)} \}$ of angular frequency $\Omega$, wavevector $K=\Omega/c$, and initial phases $\{\alpha_i\}$, traveling at the velocity of light $c$. $q$ 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, $q=-e$ amd $\hbar\Omega =511$ 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 $v$, then $\{\varphi_i'\}$ are Doppler effected and form a total wave $\psi'={\mit \Phi} {\mit \Psi}$, with ${\mit \Psi}= C \sin (K_d X)e^{i \Omega_d T}$ enveloping a beat wave and identifiable as de Broglie wave of angular frequency $\Omega_d= \Omega (v/c)^2$, and ${\mit \Phi}$ being an undisplaced monochromatic wave. Using $\psi'$ in CWE (see [1]2004b for incorporation of $V\ne 0$), gives upon decomposition a separate equation describing the particle dynamics, $[-\frac{ \hbar^2 }{2 M } \frac{\partial ^2} {\partial X ^2}+V]{\mit \Psi}(X,T) =i\hbar\frac{\partial {\mit \Psi} (X,T) }{\partial T}$, which is equivalent to Schr\"odinger's equation. \quad [1] J. X. Zheng-Johansson and P-I. Johansson, arXiv:Physics/0411134 (2004a); "Unification of Classical, Quantum and Relativistic Mechanics and of the Four Forces", (Nova Science, 2004b).