# [SOLVED] Klein Gordon equation, probability density

by opous
Tags: density, equation, gordon, klein, probability, solved
 P: 16 1. The problem statement, all variables and given/known data Use the Klein-Gordon Equation to show that $$\partial_{\mu}j^{\mu} = 0$$ 2. Relevant equations KG: $$\left(\frac{\partial^{2}}{\partial t^{2}} - \nabla^{2} + m^{2}\right) \phi = (\partial_{\mu}\partial^{\mu} + m^{2}) \phi = 0$$ j: $$j^{\mu} = \frac{i}{2m} \left[\phi^{*}(\partial^{\mu} \phi) - (\partial^{\mu} \phi^{*}) \phi]\right$$ 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: $$\frac{i}{2m} \left[\phi^{*} \frac{\partial^{2} \phi}{\partial t^{2}} - \phi \frac{\partial^{2} \phi^{*}}{\partial t^{2}}\right]$$ space components: $$\frac{i}{2m} \left[- \phi^{*} \frac{\partial^{2} \phi}{\partial r^{2}} + \phi \frac{\partial^{2} \phi^{*}}{\partial r^{2}}\right]$$ (where I've used r = (x,y,z) ) Putting them in this form: $$\frac{i \phi^{*}}{2m} \left[\frac{\partial^{2} \phi}{\partial t^{2}} - \frac{\partial^{2} \phi}{\partial r^{2}}\right] - \frac{i \phi}{2m} \left[\frac{\partial^{2} \phi^{*}}{\partial t^{2}} - \frac{\partial^{2} \phi^{*}}{\partial r^{2}}\right]$$ 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?
 Sci Advisor HW Helper Thanks P: 24,989 Don't expand out components, just compute $$\partial_{\mu}j^{\mu}$$ expanding the partial using the product rule. You can reduce parts using the Klein-Gordon equation and then they and the other parts will cancel.

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