13 problems

Let a negative atomic ion be described by a non-relativistic many-particle Hamiltonian whose nuclear charge varies as a parameter, and let the nuclear charge be critical when the l…

Atomic spectra provide the physical standards underlying rods and clocks. Weyl's conjecture. Atomic spectra, and hence rods and clocks, should adjust to the radius of the curvature…

An atom is being considered in the context of early quantum theory, where observed spectral lines are associated with energy quanta and transitions between atomic states. Bohr's co…

Let $I_\Psi$ be the exchange energy … where $\Psi$ is an appropriate atomic wave function in the lowest Landau band, $D$ is the Coulomb self-energy pairing, and $\rho_\Psi$ is the…

Extension conjecture. The same bound should hold for every $s\geq1$: there should exist a finite $c(s)$, with $c(s)\to\infty$ as $s\to\infty$, such that

Maximum electron number conjecture. It is conjectured that

Conjectured trial-function form. For every $n=1,2,\ldots$, this form should provide an energy accuracy of at least 3 decimal digits (non-rounded). The proposed form extends the exp…

Let $Z$ denote the nuclear charge of a helium-like system, and let $E(Z)$ be its ground-state energy. Majorana's conjecture. A polynomial of degree two in $Z$ can approximate…

Let $\rho_Z$ be the one-body density of a ground state of the many-body atomic Hamiltonian with nuclear charge $Z$ and $N=N_c(Z)$ electrons. Universality conjecture. Along subseque…

Let $Z$ denote the nuclear charge and consider the ground-state electron density of an atom in the large-$Z$ limit. On the length scale $Z^{-1}$, rescale the density around the nuc…

For an atom of nuclear charge $Z$, let $E(N,Z)$ be its ground-state energy with $N$ electrons, let $E(Z)=E(N=Z,Z)$, and define the $m$-th ionization energy by … Define the radius c…

Let $H(Z,N_e)$ be the Hamiltonian of a nucleus with charge $Z$ and $N_e$ electrons, and let … The nuclear charge $Z_{cr}$ is critical if $E(Z_{cr},N_e)=E(Z_{cr},N_e-1)$ and…

In the Thomas–Fermi description of a heavy atom with atomic number $Z$, the leading ground-state energy scales as $Z^{7/3}$. Scott's conjecture. The electrons close to the nucleus…