Bohmian quantum mechanics revisited

Abstract

By expressing the Schr\"odinger wave function in the form $\psi=Re^{iS/\hbar}$, where $R$ and $S$ are real functions, we have shown that the expectation value of $S$ is conserved. The amplitude of the wave ($R$) is found to satisfy the Schr\"odinger equation while the phase ($S$) is related to the energy conservation. Besides the quantum potential that depends on $R$, \emph{viz.}, $V_Q=-\frac{\hbar^2}{2m}\frac{\nabla^2R}{R}$\,, we have obtained a phase potential $V_S=-\frac{S\nabla^2S}{m}$ that depends on the phase $S$ derivative. The phase force is a dissipative force. The quantum potential may be attributed to the interaction between the two subfields $S$ and $R$ comprising the quantum particle. This results in splitting (creation/annihilation) of these subfields, each having a mass $mc^2$ with an internal frequency of $2mc^2/\hbar$, satisfying the original wave equation and endowing the particle its quantum nature. The mass of one subfield reflects the interaction with the other subfield. If in Bohmian ansatz $R$ satisfies the Klein-Gordon equation, then $S$ must satisfies the wave equation. Conversely, if $R$ satisfies the wave equation, then $S$ yields the Einstein relativistic energy momentum equation.