Timeline for Is there a good mathematical explanation for why orbital lengths in the periodic table are perfect squares doubled?
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28 events
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| Oct 17, 2023 at 8:46 | comment | added | mathoverflowUser | Thanks for linking to your question. I have to learn more physics, to understand what you have written. | |
| Mar 29, 2022 at 17:43 | comment | added | Dan Romik | @CarloBeenakker I posted a follow-up question, see here. | |
| Mar 25, 2022 at 7:11 | comment | added | Dan Romik | @CarloBeenakker great idea. It might take me a couple of days but I’ll work on it soon. | |
| Mar 25, 2022 at 7:09 | comment | added | Carlo Beenakker | @DanRomik -- that is a great question, why not ask it (as a separate question); this particular question has an accepted answer, so it's done. | |
| Mar 25, 2022 at 6:43 | comment | added | Dan Romik | … I feel like there is still a big gap in my understanding of where orbital shells come from, mathematically, or indeed whether the notion of “orbital shell” even makes sense as a well-defined mathematical concept if one takes Schrodinger’s equation as the basic model for a multi-electron atom. I hope you or someone else could address this issue in more detail. Specifically, I’m hoping for an answer that starts from the (many-electron) Schrodinger’s equation and, possibly after several steps of making simplifying assumptions (which are precisely stated), ends up with the series 2, 8, 8, 18, … | |
| Mar 25, 2022 at 6:41 | comment | added | Dan Romik | Thanks, that’s helpful. Having now read a bit more about the subject, I think from a mathematician’s point of view it’s really important to understand that these many-electron wave functions expanded as Slater determinants are not actually solutions to the Schrodinger’s equation for a multi-electron atom, but only approximations. In fact the whole orbital shell model seems to be only a useful approximation to a much more complex picture. So while your answer provides some good insight about OP’s question, … | |
| Mar 24, 2022 at 6:57 | comment | added | Carlo Beenakker | @DanRomik -- the quantum numbers $n,l,m$ refer to a basis set of single-electron wave functions. The many-electron wave function is expanded in this basis set. There are many techniques to do this, see en.wikipedia.org/wiki/Configuration_interaction | |
| Mar 23, 2022 at 6:21 | comment | added | Dan Romik | Can you clarify what you mean exactly by “electronic wave functions”? Are they solutions to Schrödinger’s equation for a multi-electron atom? And if so, I was under the impression that the Hamiltonian eigenfunctions for this system do not have an analytic description for anything heavier than a helium atom, so how is the principal quantum number even defined? | |
| Mar 22, 2022 at 14:54 | comment | added | Michael Engelhardt | @Joe - The fact that in a (self-)interfering wave there are regions of destructive interference does not imply that the wave is somehow impeded from traveling and hence that its position is more constrained. That's not how how waves work, that's not how the definition of the uncertainty $\Delta x$ works, and one can also then not conclude that $\Delta p$ is increased - that's not how the uncertainty principle works. | |
| Mar 22, 2022 at 12:13 | comment | added | Jojo | @MichaelEngelhardt what do you not like about the upvoted answer? | |
| Mar 22, 2022 at 9:20 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
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| Mar 22, 2022 at 7:15 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
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| Mar 22, 2022 at 5:34 | comment | added | Michael Engelhardt | Just to be clear, in that physics.stackexchange post, the answer with the many upvotes seems rather suspect to me, whereas the (accepted) answer with currently only one upvote makes sense. | |
| Mar 21, 2022 at 20:44 | comment | added | Jojo | Oh yeah thanks that answer gives a good explanation | |
| Mar 21, 2022 at 19:46 | comment | added | Carlo Beenakker | for a qualitative explanation of the statement that the energy increases with increasing number of nodes, see physics.stackexchange.com/q/186140 --- and you're right, spin-orbit coupling does not affect the ordering of the levels, I was referring to the fact that the energy of the hydrogen atom depends on $n$ and $l$ separately once you include spin-orbit coupling. | |
| Mar 21, 2022 at 19:36 | comment | added | Jojo | I wondered if you were making some statement in general for bound states in QM, or if you were making a statement about the hydrogen atom only. I think it does hold including the spin orbit interaction because this is subleading and the statement is only approximate? | |
| Mar 21, 2022 at 19:27 | comment | added | Carlo Beenakker | That the energy eigenvalue depends only on the principal quantum number $n$ is exact for the hydrogen atom, so for a single electron in the $1/r$ potential. Even there, it no longer holds if we include spin-orbit interaction. For more than a single electron the energy depends on both $n$ and the angular momentum quantum number $l$ separately. | |
| Mar 21, 2022 at 18:44 | comment | added | Jojo | "Wave functions with the same number of nodes have approximately the same energy." is this an observation based on the hydrogen atom and other systems? Or is there some general proof of this? (presumably for systems with a finite number of nodes, are they always bound states?) | |
| Mar 21, 2022 at 16:38 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
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| Mar 21, 2022 at 15:37 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
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| Mar 21, 2022 at 14:28 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
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| Mar 21, 2022 at 14:15 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
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| Mar 21, 2022 at 14:02 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
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| Mar 21, 2022 at 13:56 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
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| Mar 21, 2022 at 13:25 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
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| Mar 21, 2022 at 12:32 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
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| Mar 21, 2022 at 12:23 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
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| Mar 21, 2022 at 12:14 | history | answered | Carlo Beenakker | CC BY-SA 4.0 |