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Klein Gordon equation, probability density

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opous
#1
Jan3-08, 11:36 AM
P: 16
1. The problem statement, all variables and given/known data

Use the Klein-Gordon 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) terms to complete the KG equation and hence show the desired expression is zero. Can anyone see what I'm missing here?
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Dick
#2
Jan3-08, 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 Klein-Gordon equation and then they and the other parts will cancel.


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