# Scaling of electron wavefunction near the nucleus

As is shown in e.g. Quantum Mechanics: A New Introduction by Paffuti and Konishi, the wavefunction $$\psi$$ of an electron (in the Hartree-Fock approximation) in a general atom with atomic number $$Z$$ scales such that $$|\psi(a_Z)|^2 a_Z^3\sim 1/Z^2$$ where $$a_Z=a_0/Z$$ with $$a_0$$ the Bohr radius.

I was wondering if this could be explained in a classical picture. From Kepler's third law, the time spent by the electron around $$r=a_Z$$ should be proportional to $$a_Z^{3/2}$$, while that spent by the electron far from the nucleus should be proportional to just $$a_0^{3/2}$$. Hence, the probability should be $$|\psi(a_Z)|^2 a_Z^3\sim a_Z^{3/2}/a_0^{3/2}=Z^{-3/2}$$ which is close but not quite right. What is wrong with this logic?

The problem was that $$a_Z$$ is not the only thing dependent on $$Z$$ in Kepler's third law, and the proportionality constant of the force is needed as well. In particular, for a central force $$F=k/r^2$$, we find $$T^2\propto R^3/k$$ for period $$T$$ and some average radius $$R$$ (here $$a_Z$$). Close to the nucleus, $$k$$ is also proportional to $$Z$$, so we instead find that the time spent by the electron around $$r=a_Z$$ is proportional to $$Z^{-2}$$, which gives the expected result.