On the Schrödinger-Langevin Equation

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Abstract
It is shown that the Heisenberg‐Langevin equation can be used to derive a Schrödinger equation for a Brownian particle interacting with a thermal environment. The equation derived is iℏ(∂ψ/∂ t)=−(ℏ2/2m)∇2ψ+Vψ+VRψ+[(ℏ f/2im) ln(ψ/ψ*)+W(t)]ψ(r, t),          W(t)=−(ℏ f/2im)∫ψ* ln(ψ/ψ*)ψ dr, where f is the friction constant and VR is a random potential.