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I'm trying to find the density and probability currents that satisfy the continuity equation. My lecture notes give the following construct:

$$ i\hbar(\phi^*(\partial_t^2\phi) - (\partial_t^2\phi^*)\phi = \phi^*(\nabla^2\phi - m^2\phi)-\phi(\nabla^2\phi^*-m^2\phi^*)$$

And although they haven't stated where this comes from, I have a feeling they got here by substituting $\phi^*\phi$ into the Klein-Gordon Equation.

$$(\partial_\mu\partial^\mu +m^2)\phi^*\phi = 0 $$

Where $\phi$ is proportial to $e^{i(\mathbf{k}.\mathbf{r}-\omega t)}$.

But just taking the time derivative: $\partial_t^2(\phi^*\phi) \propto (\partial_t^2\phi)\phi^*-\phi(\partial_t^2\phi^*) + 2(\partial_t\phi)(\partial_t\phi^*)$:

We see that we have a middle term $2(\partial_t\phi)(\partial_t\phi^*)$. How does this middle term vanish?

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$2(\partial_t\phi)(\partial_t\phi^*) \propto e^{i(\mathbf{k}.\mathbf{r}-\omega t)}e^{-i(\mathbf{k}.\mathbf{r}-\omega t)} = constant$

Hence the middle terms of the spatial and time derivatives don't affect the probability currents or the probability density.

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