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http://www.nt.ntnu.no/users/jensoa/E-FY1006-31mai2012.pdf

Please open the link and go to page 11, problem 3.

It appears, after all, I understand nothing when it comes to the wave function of Hydrogen like atoms. So I kindly ask you to answer some questions I got:

1) "A selection of these atoms will then be left in an ensemble described by the wave function ##\psi(r,\theta,\phi)## " - can somebody please translate this? What, precisely, is an ensemble?

2) What does it mean when the angular function ##Y(\phi,\theta)## is a linear combination of other ##Y_{lm}(\phi,\theta)## functions? Like, in the problem, "##Y(\phi,\theta) = Y_{p_x} n_x + Y_{p_y} n_y + Y_{p_z} n_z ##", meaning ##Y## is a combination of a bunch of p-orbital states. How can this be? Surely an electron can only be in one state, one orbital, after measurement?

Or does this have something to do with the ensemble stuff?

3) "##Y = \sqrt{\frac{3}{4 \pi}} \hat{n} \cdot \hat{r}##" - Why are they multiplying the direction vector with a normal vector? How does this become the angular function?

Thank you.

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# Spherical harmonics of Hydrogen-like atoms

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