Atomic radius bounded independently of nuclear charge
In plain words
Heavy atoms are about the same size as light ones, roughly $10^{-8}$ cm across. Prove from the many-electron Schrodinger equation that the size of a neutral atom does not grow with Z.
Precise statement
For a neutral ground state $\psi$ of $H_{Z,Z}$ with one-electron density $\rho$, define $R(Z)$ as the radius with $\int_{|x|>R(Z)}\rho(x)\,dx=1$. Prove $c a_0\le R(Z)\le C a_0$ with $a_0=\hbar^2/(m e^2)$ and constants c, C independent of Z, or show R(Z) grows or shrinks without bound.
What would settle it
Z-independent upper and lower bounds on R(Z) for the neutral many-body ground state.
Status in the literature
Proved in Hartree-Fock theory (Solovej 2003); open for the Schrodinger Hamiltonian.
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