Klein Gordon equation, probability density

[opous]

[SOLVED] Klein Gordon equation, probability density

Homework Statement

Use the Klein-Gordon Equation to show that

$$\partial_{\mu}j^{\mu} = 0$$

Homework Equations

KG:

$$\left(\frac{\partial^{2}}{\partial t^{2}} - \nabla^{2} + m^{2}\right) \phi = (\partial_{\mu}\partial^{\mu} + m^{2}) \phi = 0$$

j:

$$j^{\mu} = \frac{i}{2m} \left[\phi^{*}(\partial^{\mu} \phi) - (\partial^{\mu} \phi^{*}) \phi]\right$$

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?

 

[Dick]

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.