While messing around with the Schrödinger equation on paper, I found an interesting, elegant way of expressing it. Let [tex]P[/tex] be the probability density [tex]|\Psi |^2[/tex], and let [tex]\vec Q[/tex] be a real-valued vector field. [tex]\vec F[/tex] is a vector field describing the forces acting on the system when in a given configuration. Then,(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\frac{\partial P}{\partial t}=-\nabla \cdot \vec Q[/tex]

[tex]\frac{\partial \vec Q}{\partial t}=\frac{P\vec F}m-\frac{\vec Q\nabla \cdot \vec Q + \frac{\hbar^2}{4m^2}[\nabla,P]\Delta P}P[/tex]

Sorry if this looks obvious, but I haven't seen this mentioned in any book. Hopefully, my calculations are correct. It's evident that [tex]\vec Q[/tex] is a velocity density. The first equation just says that the probability density decreases as the wave function expands about a point. The term [tex]-\frac{\vec Q}P \nabla \cdot \vec Q[/tex] represents the flow of velocity density in the direction of the velocity itself. However, I'm unsure of how to physically interpret the last term, which is a strange looking one. Does it simply mean that the probability density tends to accelerate away from concentrations of probability density?

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# Interesting equivalent to Schrödinger equation

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