# Calculating with quantum numbers and shape of nodal shells

I'm attending in a chemistry course now, and in the beginning we scratch the surface of the quantum physics, which have led me to some problems. Mostly I understand all what we get until now, but there some blind spots, where I don't understand some things. Now I have two major problems, one with some calculations with quantum numbers, and now with the nodal shells and I hope somebody could explain these to me (who have never learned this part of physics before).

So the first problem was with some exercising. I've understood the questions and had the approximate way to solve them, but the solution by the teacher was a little bit surprising, and it left me in doubt and confused. Here is the tasks and the solution by the teacher:

I've thought that every orbital could take one electron, so in the (a) I've "calculated" 3 of them, but there is 6 in her solution. Before that she doesn't told us about how many electron could be on one orbit, but she said that in the (b), so okay... there are 2 electrons on each orbit.. in this case at least (hydrogen atom).. For the (b) I've put again my bad 3 electrons because of the lack of information from earlier. But if there was a surprise in (a), I was (and still) totally confused by the solution of (c).. where comes the 5??

Okay, next one. This time there is no calculation, but some basic knowledge (lack of mine). It's about the orbit's naming and it's "transformation". So firstly we checked the s orbital (1s, 2s, etc) which has a sphere shaped "appearance". Second the p orbital, where are 3 orientations (-1,0,1) and on the illustration, the signed areas were laid on the axes. But in the next page on the d orbital, she said one of the fig.s (which one I highlighted with green) is not correct, because the orbital areas should have been in between the axes, not on them.

And it caused some problem to me again. Why the orbital areas are between the axes when one orbital earlier we(/she) put them on the axes without problem? Furthermore there is an other orbital version which is the $d_{x^2-y^2}$ and she also doesn't tell why this is looks like that (areas placed on the axes).

I know these are the very basics and I would like to understand every aspects of this topic (partly because it is very interesting, but mainly because I don't want to fall behind because of "some" misunderstanding), so I hope and I would be very grateful if somebody could make me some explanation of these, or just give me a reliable site or book name where these topics are explained well (for someone like me, who just started this first time).

For the first part: Each orbital can hold two electrons with opposite spin states (this is the exclusion principle). That is why the answers to (a) and (b) are doubled. However, in case (c) the $m_s=1/2$ condition demands that the electron(s) be only spin-up (i.e. no spin-down states) so you don't double them.
Regarding the figures: the wavefunction includes a factor that's shown in the subscript. So for $d_{xy}$ the wavefunction is proportional to $xy$. Therefore, along, say, the $x$ axis, $y=0$ and therefore the wavefunction is 0. Likewise, for $d_{x^2-y^2}$, the wavefunction is zero when $|x|=|y|$ (i.e. the two diagonals).