Suppose an empty universe with the exception of a single hydrogen atom (1 proton, 1 electron). The electron may be in its ground state or it may be excited a certain number of levels. Suppose it is at level $n$.

Pick some reasonable measure of size, e.g., the average distance of the electron to the proton or the radius of the sphere surrounding the proton which would, with sufficiently high probability (e.g., $1 - 10^{-20}$), contain the electron. If some other definition of size is easier to compute, I'd be happy to use that instead.

How does this size grow with $n$?

I was inspired by this question about the Bekestein bound, in particular Jerry Schirmer's suggestion to store the information in a single atom. I've seen this idea thrown around before, but never properly analyzed it. In particular, the bound seems to place (weak) limits on just how close the electron could be to the proton in a highly excited state, though perhaps the high escape chance of the electron plays havoc with the bound.

Suppose an empty universe with the exception of a single hydrogen atom (1 proton, 1 electron). The electron may be in its ground state or it may be excited a certain number of levels. Suppose it is at level $n$.

Pick some reasonable measure of size, e.g., the average distance of the electron to the proton or the radius of the sphere surrounding the proton which would, with sufficiently high probability (e.g., $1 - 10^{-20}$), contain the electron. If some other definition of size is easier to compute, I'd be happy to use that instead.

How does this size grow with $n$?

I was inspired by this question about the Bekestein bound, in particular Jerry Schirmer's suggestion to store the information in a single atom. I've seen this idea thrown around before, but never properly analyzed it. In particular, the bound seems to place (weak) limits on just how close the electron could be to the proton in a highly excited state, though perhaps the high escape chance of the electron plays havoc with the bound.