Radial Probability Function of 3s Orbital

Question

I am an older student returning to the field, trying to reteach myself the basics and have run into a problem interpreting a pattern I see.

In an introductory book explaining atomic orbitals of a hydrogen atom, it shows the radial probability functions for the 1s, 2s, and 3s subshells. Each one peaks (i.e. has a global maximum) at the radius expected given the subshell in question.

HOWEVER, for each n>1, there are n-1 smaller peaks (i.e. local maxima) at the approximate radii of the n-1 subshells before it (i.e. for the 3s orbital there are smaller peaks at the approximate locations for the 1s and 2s orbitals). To my amateur eyes, that suggests a probability of finding a 3s electron in either the 2s or 1s subshell.

This probability obviously makes sense in the weird world of quantum mechanics, but can anyone provide a more specific answer? Does a more specific answer for why a 3s electron might be found hovering around a sphere of radius approximately equal to the 1s or 2s subshell, or is quantum weirdness as far as it goes?

Again, I'm returning to the subject after many years, so my knowledge is spotty.

Answer 1

What you have discovered is that the atomic orbitals $1s$, $2s$, $3s$, etc all overlap with each other. That is, at any particular distance from the nucleus there will (in general) be a non-zero probability of finding electrons from all the occupied orbitals in the atom.

The electron orbitals are not precisely defined shells that nest inside each other like a set of Russian dolls. They are more like fuzzy blobs that all overlap with each other. The electrons don't (to borrow your phrase) hover around a sphere of radius of whatever. The electrons are spread out over the whole orbital.

Finally there is a subtlety I should mention. If you plot the $1s$ electron density it looks like this:

Note that the maximum is at the nucleus i.e. $r=0$. the plots you are describing show the probability of find the electron at some distance $r$ from the nucleus, and this is the density multiplied by the volume of a shell of radius $r$ i.e. it is:

$$ P(r)dr = \psi^2(r) 4 \pi r^2 dr $$

It's that extra factor of $r^2$ that produces the maximum to produce a graph that looks like:

So there is no shell of $1s$ electrons at the peak shown in the graph.