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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! :)

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    $\begingroup$ Are you aware that in classical mechanics the eccentricity of planetary orbits depends on angular momentum? $\endgroup$
    – Ghoster
    Apr 26, 2023 at 0:41
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    $\begingroup$ This has been the topic of several similar questions. Did you read some of the posts under the "Related" section on the right side of the page? Like physics.stackexchange.com/questions/288468/…. $\endgroup$ Apr 26, 2023 at 2:20
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    $\begingroup$ It's not clear to me what you mean by "connected with the shape of the orbitals." Can you clarify this before I attempt to answer? To my eye, the "shape" of a wavefunction is just as much affected by $n$ as it is by $l$. However, occasionally one comes across plots of the spherical harmonics claiming to somehow be plots of the entire hydrogen wavefunction - which is wrong and misleading. Unfortunately we just can't nicely plot a function of three variables which outputs a complex number, so we often plot some kind of simplification which removes a lot of detail. $\endgroup$
    – AXensen
    Apr 26, 2023 at 13:24
  • $\begingroup$ I'm just going to plug this very nice program for viewing the hydrogenic orbitals that you might find useful: falstad.com/qmatom. Using it you can clearly see how all the quantum numbers affect the shape of the orbital (if you switch to the "Complex Orbitals" mode). $\endgroup$
    – JSorngard
    Apr 26, 2023 at 14:17

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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.

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  • $\begingroup$ I was hoping that, since you already typed up a nice answer, you would have written the one more statement that "superposing different eigenfunctions with the same overall L value (or different) can give vastly different looking orbitals, which is important in, say, molecular bonding" $\endgroup$ Apr 26, 2023 at 6:27
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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.

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