# How does the application of Schrödinger equation to model a particle in a box explain the origin of degeneracy of atomic orbitals?

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?

• No, that would be too much of a stretch. – Ivan Neretin Apr 5 at 12:11
• 1D or 2D box? 1D has no degeneracy – Andrew Apr 5 at 13:12
• 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 5 at 13:31
• I was referring to 1D. Ok understood. So how would you guys go about answering the question? – ChemDude Apr 5 at 15:33
• I have to ask, what is the source of this question? Curiosity or homework or ...? – Buck Thorn Apr 5 at 18:13

## 1 Answer

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?.