# 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. Commented Apr 5, 2021 at 12:11
• 1D or 2D box? 1D has no degeneracy Commented Apr 5, 2021 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. Commented Apr 5, 2021 at 13:31
• I was referring to 1D. Ok understood. So how would you guys go about answering the question? Commented Apr 5, 2021 at 15:33
• I have to ask, what is the source of this question? Curiosity or homework or ...? Commented Apr 5, 2021 at 18:13

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).