Klein Gordon equation, probability density

  1. [SOLVED] Klein Gordon equation, probability density

    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)² terms to complete the KG equation and hence show the desired expression is zero. Can anyone see what I'm missing here?
     
  2. jcsd
  3. Dick

    Dick 25,893
    Science Advisor
    Homework Helper

    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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