I am a physics bachelor student and currently learning quantum mechanics. In my course we derived the wave function for the hydrogen atom.
I know that the quantum number L is connected with the shape of the orbital but is there a deeper intuitive explanation for that or do I just have to accept that one can just see it by plotting the probability distribution? Is it coincidence that the angular momentum quantum number is connected with the shape of the orbitals?
Thanks in advance! :)
is there a deeper intuitive explanation
No. Or Yes. It depends on what you mean by "deeper" and "intuitive." Anyways, I will ignore this probably unanswerable part of the question.
do I just have to accept that one can just see it by plotting the probability distribution?
The orbital is the probability amplitude and the absolute square of the orbital is the probability density.
So, to "see" the shape of the orbital you can plot the probability density. The probability density is convenient since it is real. If you want to plot the actual orbital you may need a couple plots (one for the real part and one for the imaginary part).
For example, in a hydrogenic atom the orbital looks like: $$ \psi_{n\ell m}(r,\theta,\phi) = R_n(r)Y_{\ell m}(\theta, \phi) \propto R_n(r)P_{\ell m}(\cos(\theta))e^{im\phi}\;, $$ where $P_{\ell m}$ is an associated Legendre function.
The probability density looks like: $$ |\psi(r,\theta,\phi)|^2 = \rho(r,\theta) \propto {\left(R_n(r)P_{\ell m}(\cos(\theta))\right)}^2 $$
So if you want to "see" what this looks like at fixed $r$, you could plot the associated Legendre function.
Is it coincidence that the angular momentum quantum number is connected with the shape of the orbitals?
It is not a coincident, it is a fact.
By the way, the shape also depends on the principal quantum number, just not the angular part of the shape.
And if we are talking about the probability amplitude rather than the density, then the shape also depends on the azimuthal quantum number.
To simplify it, you can assume quantum numbers as a type of residential address given for electrons. So it is more like a convention (or think of it as set of values used to describe particular positions in space of electron orbits) to avoid confusion while describing it. I don't understand the intuitive part of your question.
$l$ or the azimuthal quantum number tells us about the region where we are most likely to find the electron.
Generally speaking, and as you already know there are $4$ types of orbitals S, P, D and F.
$l = 0$ which means S orbital
$l = 1$ means P orbital
$l = 2$ means D orbital
$l = 3$ means F orbital
And not only that $l$ also represents the number of angular nodes. And it was found that the maximum number of angular nodes and the maximum number of angular nodes for S orbital $= 0$ , for P orbital $= 2$ similarly for D and F.
Hence the convention
Hope it helps.
