I'm going through Tao's book on nonlinear dispersive equations. In dealing with the Schrodinger equation $$ iu_t + \frac{1}{2}\Delta u = 0, $$ he defines the stress-energy tensor in standard coordinates by $$ T_{00} = |u|^2, \ \ T_{j0} = T_{0j} = \text{Im}(\bar{u}\partial_{x_j}u). $$ (Other components of $T$ are also given, but not really needed for my question.) He then says that the following conservation law follows by a direct computation: $\partial_t T_{00} + \partial_{x_j}T_{0j} = 0$ (here we are summing from $j = 1$ to $d$).
I have been working on this but am very stuck. From the definition of $T$ and the equation $u$ satisfies, we get \begin{align*} \partial_t T_{00} &= 2uu_t = iu\Delta u \\ \partial_{x_j} T_{0j} &= \text{Im}(\partial_{x_j}\bar{u}\partial_{x_j}u + \bar{u}\partial_{x_jx_j}u) = \text{Im}(\bar{u}\Delta u). \end{align*} The last equality follows since the derivative commutes with the conjugate, and $\bar{z}z = |z|^2$ is real. But then, writing $u = u_R + i u_I$ in terms of its real and imaginary components, we have $$ u\Delta u = u_R\Delta u_R - u_I \Delta u_I + i(u_R \Delta u_I + u_I \Delta u_R). $$ Since $T$ is real, its derivatives are real-valued, and thus $iu\Delta u \in \mathbb{R}$. Therefore the real part of $u\Delta u$ is zero, and we're left with $$ \partial_t T_{00} = iu\Delta u = -u_R \Delta u_I - u_I \Delta u_R. $$ But $\text{Im}(\bar{u}\Delta u) = u_R \Delta u_I - u_I \Delta u_R$. Then the sum $\partial_t T_{00} + \partial_{x_j}T_{0j}$ is $-2u_I \Delta u_R$, not zero, as desired.
If anyone could point me in the right direction, it would be greatly appreciated. I cannot see where I've gone wrong.
This is how you can derive the conservation law
\begin{align} \partial_t T_{00} &= \partial_t \lvert u \rvert^{2} \\ &= (\partial_t u)u^{*} + (\partial_t u^{*})u \quad (1) \\ &= \frac{i [ (\Delta u) u^* - (\Delta u^{*}) u ]}{2} \quad (2) \\ &= \frac{i \partial_{x_i} [ (\partial_{x_i} u)u^{*} - u(\partial_{x_i} u^{*})] }{2} \quad (3) \\ &= -\partial_{x_i}[\text{Im}(u^{*} \partial_{x_i} u)] \quad (4) \\ &= -\partial_{x_i}T_{i,0} \end{align}
where we used
(1) the Leibniz rule
(2) the differential equation and its conjugate
(3) the fact that partial parts $\partial u \partial u^*$ cancel. Repeated indexes are summed (Einstein notation)
(4) $z-z^*=2i \text{Im}(z)$ for every complex $z$