Crater Property in Two-Particle Bound States: When and Why

Crater has shown that, for two particles (with masses $m_1$ and $m_2$) in a Coulombic bound state, the charge distribution is equal to the sum of the two charge distributions obtained by taking $m_1\to\infty$ and $m_2\to\infty$ respectively, while keeping the same Coulombic potential. We provide a simple scaling criterion to determine whether an arbitrary Hamiltonian possesses this property. In particular we show that, for a Coulombic system, fine structure corrections preserve this Crater property while two-particle relativistic corrections and/or hyperfine corrections may destroy it.