Derivation of the Schr\"odinger equation from classical stochastic dynamics

• URL: http://arxiv.org/abs/2307.06461v1
• Date: Wed, 12 Jul 2023 21:24:54 GMT
• Title: Derivation of the Schr\"odinger equation from classical stochastic dynamics
• Authors: M\'ario J. de Oliveira
• Abstract summary: The wave function $phi$ is assumed to be a complex time dependent random variable. The Schr"odinger equation follows from the Liouville equation.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: From classical stochastic equations of motion we derive the quantum Schr\"odinger equation. The derivation is carried out by assuming that the real and imaginary parts of the wave function $\phi$ are proportional to the coordinates and momenta associated to the degrees of freedom of an underlying classical system. The wave function $\phi$ is assumed to be a complex time dependent random variable that obeys a stochastic equation of motion that preserves the norm of $\phi$. The quantum Liouville equation is obtained by considering that the stochastic part of the equation of motion changes the phase of $\phi$ but not its absolute value. The Schr\"odinger equation follows from the Liouville equation. The wave function $\psi$ obeying the Schr\"odinger equation is related to the stochastic wave function by $|\psi|^2=\langle|\phi|^2\rangle$.

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