MATHPH In the literature: open

Atomic radius bounded independently of nuclear charge

In plain words

Heavy atoms are about the same size as light ones, roughly 1e-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.