{"record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"f73abfcc28ea138909dd7e9294e24a5c8ccf9323bd02453613c05821b5862228","created":"2026-10-03T07:18:06Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011"],"salt":"c006783f6dfcfe9c0867e4b3c9fd4b659759714f034bcdf95ca6335b9100de09","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"proof","assisted_by":[],"external_id":"mathph.ionization.atomic-radius","kind":"well-posed","literature_status":"open","n":"1","parents":[],"plain":"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.","posed_since":"","precise":"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.","problem_ref":null,"references":"","settled_by":"Z-independent upper and lower bounds on R(Z) for the neutral many-body ground state.","status_note":"Proved in Hartree-Fock theory (Solovej 2003); open for the Schrodinger Hamiltonian.","title":"Atomic radius bounded independently of nuclear charge","topic_ref":"139d0cc04cd3b3dbd331e014add418ba05f5b21aa4fd7ade07403261b548b505"},"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"bf2a52b11695356d77880e63de63d68f319926af8817c69f4871dbeccdaba5d3","schema":"pubphys.attested/1"},"envelope":{"attested_hash":"e677fb2722e6fb047b27df2984406e50cc13f16e612f9973ee04e84f071dadb9","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"KaMbwK9KqTKxgJ80nKyzaDOBHZt60TmoYiMbp40MGmXW2LM8DIKN53zxzWmt77-SZMD9TUmNyzQI9MJ72MSvAg"},"schema":"pubphys.envelope/1"},"record_hash":"bf2a52b11695356d77880e63de63d68f319926af8817c69f4871dbeccdaba5d3","leaf_index":1623}