# Extra term in Klein-Gordon probability current?

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?

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