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Klein Gordon equation, probability density 
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#1
Jan308, 11:36 AM

P: 16

1. The problem statement, all variables and given/known data
Use the KleinGordon Equation to show that [tex]\partial_{\mu}j^{\mu} = 0[/tex] 2. Relevant equations KG: [tex]\left(\frac{\partial^{2}}{\partial t^{2}}  \nabla^{2} + m^{2}\right) \phi = (\partial_{\mu}\partial^{\mu} + m^{2}) \phi = 0[/tex] j: [tex]j^{\mu} = \frac{i}{2m} \left[\phi^{*}(\partial^{\mu} \phi)  (\partial^{\mu} \phi^{*}) \phi]\right[/tex] 3. The attempt at a solution OK, so I've expanded j out, taking the differentials and have the following for my components: time component: [tex]\frac{i}{2m} \left[\phi^{*} \frac{\partial^{2} \phi}{\partial t^{2}}  \phi \frac{\partial^{2} \phi^{*}}{\partial t^{2}}\right][/tex] space components: [tex]\frac{i}{2m} \left[ \phi^{*} \frac{\partial^{2} \phi}{\partial r^{2}} + \phi \frac{\partial^{2} \phi^{*}}{\partial r^{2}}\right][/tex] (where I've used r = (x,y,z) ) Putting them in this form: [tex]\frac{i \phi^{*}}{2m} \left[\frac{\partial^{2} \phi}{\partial t^{2}}  \frac{\partial^{2} \phi}{\partial r^{2}}\right]  \frac{i \phi}{2m} \left[\frac{\partial^{2} \phi^{*}}{\partial t^{2}}  \frac{\partial^{2} \phi^{*}}{\partial r^{2}}\right][/tex] Which looks slightly more like the KG equation. However, my trouble is with the mass terms, I have no (mass)² terms to complete the KG equation and hence show the desired expression is zero. Can anyone see what I'm missing here? 


#2
Jan308, 01:56 PM

Sci Advisor
HW Helper
Thanks
P: 25,251

Don't expand out components, just compute [tex]\partial_{\mu}j^{\mu}[/tex] expanding the partial using the product rule. You can reduce parts using the KleinGordon equation and then they and the other parts will cancel.



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