# Continuity Equation

1. Nov 18, 2009

### Tales Roberto

Hi. I have a new one!

Prove that if $$V \left(\stackrel{\rightarrow}{r} , t \right)$$ is complex the continuity equation becomes $$\frac{\partial}{\partial t}P \left(\stackrel{\rightarrow}{r},t \right)+\nabla \stackrel{\rightarrow}{j} \left(\stackrel{\rightarrow}{r},t \right) = \frac{2}{h} \int \right[Im \ V \left(\stackrel{\rightarrow}{r},t \right) \right] P \left(\stackrel{\rightarrow}{r},t \right) d\stackrel{\rightarrow}{r}$$

so that the addition of an imaginary part of the potential describes the presence of sources if Im V > 0 or sinks if Im V < 0. Show that if the wave function is $$\Psi \stackrel{\rightarrow}{r},t \right)$$ is square integrable

$$\frac{\partial}{\partial t} \int P \left(\stackrel{\rightarrow}{r},t \right) d\stackrel{\rightarrow}{r}= \frac{2}{h} \int \right[Im \ V \left(\stackrel{\rightarrow}{r},t \right) \right] P \left(\stackrel{\rightarrow}{r},t \right) d\stackrel{\rightarrow}{r}$$

This first is easy and i can prove it, now the second part! However i need to go now, later i return to continue!