Modification of Coulomb law and energy levels of the hydrogen atom in a superstrong magnetic field

We obtain the following analytical formula which describes the dependence of the electric potential of a pointlike charge on the distance away from it in the direction of an external magnetic field $B$: $\mathbf{\Phi }(z)=e/|z|[1-\exp (-\sqrt{6m_e^2}|z|)+\exp (-\sqrt{(2/\pi )e^{3}B+6m_e^2}|z|)]$. The deviation from Coulomb’s law becomes essential for $B>3\pi B_{\mathrm{cr}}/\alpha =3\pi m_e^2/e^3\approx 6\times {10}^{16}\mathrm{G}$. In such superstrong fields, electrons are ultrarelativistic except those which occupy the lowest Landau level (LLL) and which have the energy ${\epsilon }_0^2=m_e^2+p_z^2$. The energy spectrum on which LLL splits in the presence of the atomic nucleus is found analytically. For $B>3\pi B_{\mathrm{cr}}/\alpha$ it differs substantially from the one obtained without accounting for the modification of the atomic potential.

Figure 1

Modification of the Coulomb potential due to the dressing of the photon propagator.

Figure 2

Potential energies of the charges $+g$ and $-g$ in $D=2$ for $g=0.5$, $m=0.1$. The black curve corresponds to $P$; the dashed (blue) curve corresponds to $\overline{P}$.

Figure 3

Relative difference of potential energies calculated with the exact and interpolating formulae for the polarization operator for $g=0.5$, $m=0.1$.

Figure 4

Modified Coulomb potential energy at $B={10}^{17}\mathrm{G}$ is shown by the dark solid (blue) curve, and its long-distance and short-distance asymptotics by the pale solid (green) and dashed (red) curves, respectively.

Figure 5

Modified Coulomb potential energy at $B={10}^{17}\mathrm{G}$. The analytical formula (approximate $\overline{P}$) shown by the dashed (blue) curve practically coincides with the result of the numerical integration (exact $P$) shown by the solid (red) curve.

Figure 6

Relative accuracy of the analytical formula for the modified Coulomb potential energy at $B={10}^{17}\mathrm{G}$.

Figure 7

Landau radius $a_H$ versus magnetic field $B$.

Figure 8

Effective potential with screening for $m=0$ [dark solid (blue) curve] and $m=-1$ (long-dashed curve), (52); simplified potential (53) (short-dashed [red] curve). The curves correspond to $B=3\times {10}^{17}\mathrm{G}$. Coulomb potential (pale solid [green]) is also shown.

Figure 9

Effective potentials according to (52) with the replacement of $z$ by $\sqrt{{\rho }^2+z^2}$ in the exponents for $m=0$ (short-dashed [brown] curve) and $m=-1$ (pale long-dashed [green] curve). For the comparison, curves which directly correspond to (52) are also shown: $m=0$ (solid [blue]) and $m=-1$ (long-dashed black). The curves correspond to $B=3\times {10}^{17}\mathrm{G}$.

Figure 10

Spectrum of hydrogen levels in the limit of infinite magnetic field. Energies are given in Rydberg units, $Ry\equiv 13.6\mathrm{eV}$.