The Schr\"odinger equation, the zero-point electromagnetic radiation and   the photoelectric effect
  A Schr\"odinger type equation for a mathematical probability amplitude {\psi}(x,t), is derived from the generalized phase space Liouville equation valid for the motion of a microscopic particle, with mass M, moving in a potential V(x). The particle phase space probability density is denoted W(x,p,t) and the entire system is immersed in the "vacuum" zero-point electromagnetic radiation . We show that the generalized Liouville equation is reduced to a non-quantized Liouville equation in the equilibrium limit where the small radiative corrections cancel each other approximately. Our derivation will be based on a simple Fourier transform of the non-quantized phase space probability distribution W(x,p,t). For convenience, we introduce in this Fourier transform an auxiliary constant {\alpha}, with dimension of action, and an auxiliary coordinate denoted by y. We shall prove that {\alpha} is equal to the Planck's constant present in the momentum operator of the Schr\"odinger equation. Moreover, we shall show that this momentum operator is deeply related with the ubiquitous zero-point electromagnetic radiation. It is also important to say that we do not assume that the mathematical amplitude {\psi}(x,t) is a de Broglie matter-wave, in other words, the wave-particle duality hypothesis is not used within our work. The implications of our study for the standard interpretation of the photoelectric effect is discussed by considering the main characteristics of the phenomenon. We also mention, briefly, the effects of the zero-point radiation in the tunnelling phenomenon and in the Compton's effect. 
H. M. Fran\c{c}a, A. Kamimura and G. A. Barreto
quant-ph