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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:

  1. Change $|u|^ 2$ into $u^* u$.
  2. Bring the derivative inside the integral.
  3. Apply the product rule.
  4. 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

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    Please define your notation. $u$ isn't the wavefunction, is it? Or if it is, then don't you want the integral to equal 1 rather than 0? As written, your equation would imply $u(x,0)=0$ for all $x$.2012-09-01
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    u is the wave function. This isn't homework, but extra practice. Also, in the absolute value it should be absolute value(u)^2 dx2012-09-02
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    OK, and the integral equals 1, which makes more sense. This expresses the fact that the total probability of finding the electron somewhere must be 1. Physically, this isn't something specific to the hydrogen atom, so this suggests that mathematically, your method of solution shouldn't depend on any of the details of the specific potential that applies in this case. It's also true in any number of dimensions, so you might want to warm up by doing it in one dimension.2012-09-02
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    So in broad strokes: (1) Change $|u|^2$ into $u^*u$. (2) Bring the derivative inside the integral. (3) Apply the product rule. (4) Apply the Schrodinger equation and try to show that the result is zero. This should be easier in one dimension. Then you can try to generalize to three.2012-09-02
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    What is the equation for time evolution? I mean how does $u$ evolve in time?2012-09-02
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    @timur: It evolves according to the Schrodinger equation.2012-09-02
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    @Ben: I guess so, but I want to see exactly what the OP is talking about.2012-09-02
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    @Ben, your hint about using Schrodinger seems logical to me2012-09-02
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    @mary: "and to bring the derivative inside the integral, isn't dx already inside?" dx isn't a derivative, it's like the width of the little rectangles in the Riemann sum. You said in the question that you wanted to differentiate with respect to time and prove that the derivative was zero, so write down that derivative *of* your integral.2012-09-02
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    Still not understanding it very well w.r.t time?2012-09-03
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    mary: one of the ways to show that an expression is constant with respect to some parameter is to show that its derivative with respect to that parameter is identically 0. (This is true by the definition of the derivative - can you see why?) In your case, this means that $d/dt (\int |u|^2 dx)=0$. Now, a general theorem says that if $t$ and $x$ are different variables, then $d/dt(\int f(x,t) dx) = \int (df(x,t)/dt) dx$ (plus or minus boundary conditions on the integral that are irrelevant here. Does that make more sense?2012-09-03
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    Thanks Steven, it makes some more sense2012-09-04

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