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.

  • $\begingroup$ Hi Justin, welcome to PSE. I am sure you will get an answer but from my level of knowledge, I can only say that during a drop from a higher to a lower level, the electron can bounce all over the place in terms of states briefly occupied. Apologies if you are already know this, or your question has been misunderstood by me, $\endgroup$
    – user179430
    Commented Dec 30, 2017 at 14:15

1 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:

1s density

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:

1s P(r)

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

  • $\begingroup$ Thanks, John, perfect answer! Quick follow up: can you explain the significance of the nodes between the local maxima for a given n (e.g. 2 nodes on the radial probability function for n = 3)? I believe it has something to do with the mode of the electron energy level, but my intuition for it is just that: intuition that lacks analytical rigor. Any guidance you can provide would be greatly appreciated. Thanks again! $\endgroup$
    – Justin
    Commented Dec 30, 2017 at 18:14
  • $\begingroup$ @Justin I think the best explanation of the nodal structure I can give is my answer to Electron shells in atoms: What causes them to exist as they do? $\endgroup$ Commented Dec 30, 2017 at 19:16

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