For the hydrogen atom, if $\int |u|^2 ~dx = 1,$ at $t = 0$, I am trying to show that this is true at all later times.
What I need help is with differentiating the integral with respect to $t$, and taking care about the solution being complex valued. Except that my notation is getting me mixed up. I think this might get me there.
Following Ben's hint, here is what I have:
- Change $|u|^ 2$ into $u^* u$.
- Bring the derivative inside the integral.
- Apply the product rule.
- Apply the Schrödinger equation and try to show that the result is zero.
$\int u^* u ~dx = 1 $
and to bring the derivative inside the integral, isn't $dx$ already inside?
From Schrödinger equation I have:
$-i\hslash u_t = \sum_{i=1}^n \frac{\hslash^2}{2m_i}(u_{x_i x_i} + u_{y_i y_i} + u_{z_i z_i}) + V(x_1,\ldots,z_n)u$
for $n$ particles and the potential would $V$ depend on all of the $3n$ coordinates.
I'm not sure how to extend it to even 2 dimensions with the notation below