
#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: 25,170

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.



Register to reply 
Related Discussions  
klein gordon equation  Quantum Physics  9  
KleinGordon equation for electromagnetic field?  Quantum Physics  27  
[SOLVED] KleinGordon Causality calculation  Advanced Physics Homework  5  
KleinGordon Equation  Advanced Physics Homework  14  
klein gordon equation  Quantum Physics  2 