Show that the Klein-Gordon field satisfies the continuity equation


 * 1) Find the equations that describe the Klein-Gordon field: \(\rho = \frac{i\hbar}{2mc^2}\left(\phi^*\partial_t\phi - \phi\partial_t\phi^*\right)\) \(\mathbf{j} = -\frac{i\hbar}{2m}\left(\phi^*\nabla\phi - \phi\nabla\phi^*\right)\)
 * 2) Substitute these into the continuity equation, \(\partial_t\rho + \nabla\cdot\mathbf{j} = 0\): \(\partial_t\rho + \nabla\cdot\mathbf{j} = \frac{i\hbar}{2mc^2}\left(\phi^*\partial_t^2\phi - \phi\partial_t^2\phi^*\right) - \frac{i\hbar}{2m}\left(\phi^*\nabla^2\phi - \phi\nabla^2\phi^*\right)\)
 * 3) Simplify: \(\partial_t\rho + \nabla\cdot\mathbf{j} = \frac{i\hbar}{2mc^2}\left(\phi^*\left(\partial_t^2 - c^2\nabla^2\right)\phi - \phi\left(\partial_t^2 - c^2\nabla^2\right)\phi^*\right)\)
 * 4) Substitute in the Klein-Gordon equation \(m^2c^4\phi = \hbar^2c^2\nabla^2\phi - \hbar^2\partial_t^2\phi\): \(\partial_t\rho + \nabla\cdot\mathbf{j} = \frac{i\hbar}{2mc^2}\left(\phi^*\left(-\frac{m^2c^4}{\hbar}\right)\phi - \phi\left(-\frac{m^2c^4}{\hbar}\right)\phi^*\right) = 0\)