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Continuity Equation

  1. Nov 18, 2009 #1
    Hi. I have a new one!

    Prove that if [tex] V \left(\stackrel{\rightarrow}{r} , t \right) [/tex] is complex the continuity equation becomes [tex] \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} [/tex]

    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 [tex]\Psi \stackrel{\rightarrow}{r},t \right)[/tex] is square integrable

    [tex] \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} [/tex]

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