Obtaining the Schrodinger Equation from a Fourier Series of Potential?

Mathematically, it is possible to write a potential V(x) in classical mechanics as a Fourier series. It is not guaranteed, however, that this series have any physical meaning. In this note, we argue that it has meaning if one thinks of the potential as an ensemble of different momentum sources, each with a different weight. Then, placing a particle in such a potential could yield an ensemble of momenta f(p). This ensemble W(x) at this point is not associated with a Fourier series, but it is an equilibrium ensemble and can also be thought of as a potential, it seems, which has different weights associated with different momenta. If this is the case, this ensemble can also be written as a Fourier series W(x) = sum on p  f(p) exp(ipx). Now, the ensemble W(x) created by the potential V(x) in the first place, can interact with the potential again and again. This can be described by writing V(x)W(x). This combination should return the original ensemble W(x) up to a constant if one has an equilibrium situation. Now V(x)=Energy-average Kinetic energy which leads to the Schrodinger equation.