Three dimensional current and conservation
Three-dimensional case. Now, in the future homework, you will be doing the equivalent of this calculation here with the Laplacians-- it's not complicated-- so that you will derive with the current is. And the current must be a very similar formula as this one. And indeed, I'll just write it here. The current is h bar over m, the imaginary part of psi star. And instead of ddx, you expect the gradient of psi. That is the current for the probability in three dimensions. And the analog of this equation, d rho dt plus dj dx equals 0, is d rho dt plus divergence of j is equal to 0. That is current conservation. Perhaps you do remember that from your study of electromagnetism. That's how Maxwell discovered the displacement current when he tried to figure out how everything was compatible with current conservation. Anyway, that argument I'll do in a second so that it will become clearer. So one last thing here-- it's something also-- you can check the units he...