On the conection between the Liouville equation and the Schrodinger equation

Abstract

We derive a classical Schrodinger type equation from the classical Liouville equation in phase space. The derivation is based on a Wigner type Fourier transform of the classical phase space probability distribution, which depends on an arbitrary constant $\alpha$ with dimension of action. In order to achieve this goal two requirements are necessary: 1) It is assumed that the classical probability amplitude $\Psi(x,t)$ can be expanded in a complete set of functions $\Phi_n(x)$ defined in the configuration space; 2) the classical phase space distribution $W(x,p,t)$ obeys the Liouville equation and is a real function of the position, the momentum and the time. We show that the constant $\alpha$ appearing in the Fourier transform of the classical phase space distribution, and also in the classical Schrodinger type equation, has its origin in the spectral distribution of the vacuum zero-point radiation, and is identified with the Planck's constant $\hbar$.