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Is the particle in a box concept analogous to an electron in an orbital? If so, can we apply the equation for the allowed energies of a particle in a box

$$E = \frac{h^2n^2}{8mL^2}$$

to the allowed energies of electrons in orbitals? Are the orbitals degenerate because this equation only includes the quantum number $n$ and not the others and therefore all orbitals in a subshell will roughly be degenerate?

Finally, if this equation can be used to explain the allowed energies of orbitals/electrons, what would $L$ be in this instance?

andselisk
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ChemDude
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    No, that would be too much of a stretch. – Ivan Neretin Apr 05 '21 at 12:11
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    1D or 2D box? 1D has no degeneracy – Andrew Apr 05 '21 at 13:12
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    Electrons in orbitals do not have spatial constrains like particles in a box. There is no sharp cut off of probability density. Particular borders of orbital 3D shapes are just arbitrary convention of probability density threshold and the residual integral probability threshold. – Poutnik Apr 05 '21 at 13:31
  • I was referring to 1D. Ok understood. So how would you guys go about answering the question? – ChemDude Apr 05 '21 at 15:33
  • I have to ask, what is the source of this question? Curiosity or homework or ...? – Buck Thorn Apr 05 '21 at 18:13
  • This question is on an assignment I was given for university level chemistry. "How does the application of the Schrodinger equation to model system such as particle in a box help us understand the origin of the degeneracy of atomic orbitals". I have been racking my brain all week and can't get it. – ChemDude Apr 06 '21 at 18:31
  • My answer to another post tries to explain a link between the problems: https://chemistry.stackexchange.com/questions/149738/schr%c3%b6dinger-equation-and-degeneracy-of-atomic-orbitals/149741#149741 This seems unlikely to be a good question for intro chem though. That's why I asked why you were posting this. – Buck Thorn Apr 13 '21 at 08:40

1 Answers1

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Is the particle in a box concept analogous to an electron in an orbital?

No, but there is an analogy to the $\pi$ systems of dye molecules, and (for the 3D box) to the band structure of nanodots. Some physical chemistry courses have a lab that explore this relationship, e.g. at Saarland university (sorry, in German, but the figures and the math are universally understandable, I hope).

Finally, if this equation can be used to explain the allowed energies of orbitals/electrons, what would L be in this instance?

It would be the length of the conjugated system (for the dyes) or the radius (for the nanoparticles)

Are the orbitals degenerate because this equation only includes the quantum number n and not the others and therefore all orbitals in a subshell will roughly be degenerate?

For a 1D case with degeneracy, you can explore the particle in a ring, Why do solutions of electron in a box (and in a ring) predict coefficients for LCAO (linear combination of atomic orbitals) in 1D systems?.

Karsten
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