{"record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"f8fbfd86449587d24f3e0ff39da08f5e0ecbc5d8136e87f2947bd5283738f4a0","created":"2026-10-15T23:43:22Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["bf2a52b11695356d77880e63de63d68f319926af8817c69f4871dbeccdaba5d3","ce59fb6cd812fc05906d0c60a71c75ea22bc686bcf38e7c3ef7010834d98ee7b"],"salt":"13bb8964e0841aeadf6ec8c971863033e54542a44995a022af7f8ed4e89770c8","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":"2","parents":[],"plain":"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.","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":"bf2a52b11695356d77880e63de63d68f319926af8817c69f4871dbeccdaba5d3","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":null},"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"2e1f65164ed4411214d7efd0dccb86e82f461cb0ccfc67dccb81546a232d9388","schema":"pubphys.attested/1"},"envelope":{"attested_hash":"0adb445e384cdab7f4fbc1a93bfecefa502a60c17561cddbabbf5ac810ca7910","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"_KKRYnLRiB9zWAp-DPi_qiA04h1JjEdIz5br93rmvP3jV7SPNZ1WrMrKAEAG3ITn-5tIzYoQEcyC6BBucaW2Dg"},"schema":"pubphys.envelope/1"},"record_hash":"2e1f65164ed4411214d7efd0dccb86e82f461cb0ccfc67dccb81546a232d9388","leaf_index":3422}