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 $α$ with dimension of action. In order to achieve this goal two requirements are necessary: 1) It is assumed that the classical probability amplitude $Ψ(x,t)$ can be expanded in a complete set of functions $Φ_n(x)$ 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 $α$ 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 $\hbar$.
Quantum simulation of Liouville equation in geometrical optics with partial transmission and reflection via Schrödingerization
School of Mathematical Sciences, Shanghai Jiao Tong University（上海交通大学数学科学学院）；Institute of Natural Sciences, Shanghai Jiao Tong University（上海交通大学自然科学研究院）；Ministry of Education Key Laboratory in Scientific and Engineering Computing, Shanghai Jiao Tong University（上海交通大学教育部科学和工程计算重点实验室）
