# The uncertainty principle and the quantization of electron momentum in orbitals - Why are larger energy gaps considered more uncertain?

I am a chemist trying to understand how the uncertainty principle can be applied to the mechanics of atomic and molecular orbitals in order to have better intuitions about stability and reactivity. While it seems like this should be useful tool for quickly assessing stability and reactivity, I am struggling to wrap my head around it. I'll do my best to explain my confusion but if you want to skip straight to the question just jump to the last paragraph or two.

In spectroscopy, it is well known that the electronic transitions of electrons get smaller as the size of the orbital increases, leading to effects like larger conjugated pi systems absorbing progressively longer wavelengths of light and smaller quantum dots emitting higher energy light.

And of course, the Particle in a Box model, which chemists are exposed to in Physical Chemistry courses, explains why: because the smaller your box is $$Δx$$, the larger $$ΔE$$ will be for the momentum $$Δp$$ of a wave. And this is where the uncertainty Principle seems to come in. I'll quote the Wikipedia article on Particle in a Box:

"It can be shown that the uncertainty in the position of the particle is proportional to the width of the box.[9] Thus, the uncertainty in momentum is roughly inversely proportional to the width of the box.[8]"

The Particle in a Box model seems to be saying that the reason electronic energy level gaps are inversely proportional to the size of atomic and molecular orbitals is because of the uncertainty principle. And, at first, I get it: as $$Δx$$ decreases, $$Δp$$ must increase.. except, when I think about this more, I don't get it. Here's why:

$$Δp$$ here refers to the gaps between the energy levels for electronic transitions (the principle quantum number $$n$$ is changing), which means that as the orbital gets larger the electronic transitions get smaller, closer together, and this is what is happening as the momentum is becoming more certain. But how is it possible for the momentum to be more certain as the energy gaps get smaller? If you take this to the extreme, then the energy gaps get close to zero and the momentum of the electron would be all over the place from tiny energy fluctuations.. it would be uncertain. And you can take the opposite extreme.. if the gaps get infinitely large, then you know for certain that the momentum of the electron will not change because it would need a relatively large amount of energy to jump the gap.. but the uncertainty principle seems to be saying that a really big gap would mean a lot of uncertainty. You can even see this relationship between momentum uncertainty and the energy gap play out in Boltzmann statistics, which can be used to tell you the population (or probability) of electrons at each energy level based on the temperature and the size of the gap.. the smaller the gap is, the more spread out electrons will be among the energy levels and vice versa.

How do large quantum energy gaps correlate with more uncertainty? How do smaller quantum energy gaps correlate with less uncertainty? It seems like there must be some conceptual intuition here which makes sense out of this. Is it because the larger the gaps are, the more the momentum will be spread throughout the gap? But that seems like it contradicts the whole notion of quantization.

EDIT: According to this derivation of momentum uncertainty in PIAB, $$\sigma_p = \frac{\bar h \pi n}{\ell}$$, so uncertainty of momentum is proportional to $$n$$ and inversely proportional to $$\ell$$, so the uncertainty of momentum increases as $$n$$ increases but is also decreases as the size of orbitals increase with $$n$$. I dunno if this helps, don't have time to think about it yet, gotta go to bed.

Lets start here, the hydrogen atom orbits, which are really orbitals, probability loci.

$$λ=h/p$$ $$p=h/λ$$ so dhe higher the n the smaller the $$p$$

I would express the Heisenberg uncertainty $$dpdx>h/4π$$ not with respect to the gaps between n levels, but take the momentum of the n level itself.

In this link there is the orbital for level n=1

I think the HUP is the effective envelope for the uncertainty in the position of the electron in this simple example.