Fokker-Planck to Schrodinger Equation Transformation

In order to obtain the transformation of Fokker-Planck equation into Schrodinger equation one starts employing the autoadjoint operator introduced in Eq. (5.277), which along (5.276) becomes  

the diffusion factor ( x ) = D will be considered as a constant of the nonequilibrium dynamic process. With this identification in the stationary (5.269) one yields the equation  

With this, the autoadjoint operator expression form above, Eq. (5.297), can be successively developed (Gardiner, 1994 Risken, 1984)  

The obtained expression permits the direct analogy with the Hamiltonian operator specific to the Schrodinger equation  

This correspondence stays at the basis of the transformation of the Fokker-Planck equation into the associated Schrodinger equation. Through the corresponding relations (5.299) and (5.301) one has in a direct way the transformation relation  

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