Electric field of a pointlike charge in a strong magnetic field and ground state of a hydrogenlike atom

A. E. Shabad and V. V. Usov

Phys. Rev. D 77, 025001 – Published 3 January 2008

Abstract

In an external constant magnetic field, so strong that the electron Larmour length is much shorter than its Compton length, we consider the modification of the Coulomb potential of a point charge owing to the vacuum polarization. We establish a short-range component of the static interaction in the Larmour scale, expressed as a Yukawa-like law, and reveal the corresponding “photon mass” parameter. The electrostatic force regains its long-range character in the Compton scale: the tail of the potential follows an anisotropic Coulomb law, decreasing away from the charge slower along the magnetic field and faster across. In the infinite-magnetic-field limit the potential is confined to an infinitely thin string passing though the charge parallel to the external field. This is the first evidence for dimensional reduction in the photon sector of quantum electrodynamics. The one-dimensional form of the potential on the string is derived that includes a $δ$ function centered in the charge. The nonrelativistic ground-state energy of a hydrogenlike atom is found with its use and shown not to be infinite in the infinite-field limit, contrary to what was commonly accepted before, when the vacuum polarization had been ignored. These results may be useful for studying properties of matter at the surface of extremely magnetized neutron stars.

Received 25 July 2007

DOI:https://doi.org/10.1103/PhysRevD.77.025001

Authors & Affiliations

A. E. Shabad^*

P.N. Lebedev Physics Institute, Moscow, Russia

V. V. Usov^†

Center for Astrophysics, Weizmann Institute of Science, Rehovot 76100, Israel

^*shabad@lpi.ru
^†fnusov@wicc.weizmann.ac.il

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Vol. 77, Iss. 2 — 15 January 2008

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Images

Figure 1
Electron energy $- e A_{0} (x_{3}, 0)$ in the modified Coulomb potential (18) of the charge $q = Z e$ plotted along the axis $x_{3}$ passing through the charge $q$ parallel to the magnetic field, $x_{⊥} = 0$ . Thin solid lines correspond to four values of the magnetic field (from left to right): $b = 10^{6}$ , $b = 10^{5}$ , $b = 10^{4}$ , and $b = 10^{3}$ . The bold solid line is the Coulomb law $A_{0}^{C} (x_{3}, 0) = q / (4 π x_{3})$ . The thin lines approach asymptotically the bold line at both edges of the figure. The thick dashed broken line corresponds to $b = \infty$ . The abscissa represents the distance in the units $(2 m)^{- 1}$ . The ordinate represents the potential in the units $2 Z α m = Z \times 7.46 keV$ .Reuse & Permissions
Figure 2
Electron energy $- e A_{0} (0, x_{⊥})$ in the modified Coulomb potential (18) plotted along the axis $x_{⊥}$ passing through the charge $q$ transversely to the magnetic field, $x_{3} = 0$ . Conventions are the same as in Fig. 1. Thin lines approach asymptotically the solid line at the left lower edge of the figure and the short dotted lines (38) $A_{0} (0, x_{⊥}) = q / (4 π x_{⊥}^{'})$ at the upper right edge.Reuse & Permissions
Figure 3
Electron energy $- e A_{s . r .} (x_{3}, 0)$ in the short-range part Eq. (20) of the potential plotted against the (absolute value of) longitudinal distance $x_{3}$ for $x_{⊥} = 0$ (thin dashed lines from left to right correspond to $b = 10^{6}, 10^{5}, 10^{4}, 10^{3}$ ). The thick dashed broken line corresponds to $b = \infty$ . All the rest is the same as in Figs. 1, 2.Reuse & Permissions
Figure 4
Electron energy $- e A_{0} (x_{3}, x_{⊥})$ in the modified Coulomb potential (41) with $q = Z e$ plotted against the transverse coordinate $x_{⊥}$ with the longitudinal coordinate fixed at the large value $x_{3} = 10 (2 m)^{- 1}$ . The thin solid line corresponds to the magnetic field value $B = 10^{4} B_{0}$ . The bold solid line is the standard Coulomb law (5) $- e A_{0}^{C} (x_{3}, x_{⊥}) = - 2 α Z m [(2 m x_{⊥})^{2} + 100]^{- 1 / 2}$ . The thin line is indistinguishable from the anisotropic Coulomb law (38) in the scale of the drawing. The coordinate axes are the same as in Fig. 2.Reuse & Permissions
Figure 5
Electron energy $- e A_{l . r .} (x_{3}, 0)$ in the long-range part (21) of the potential for $b = \infty$ (dashed thick line) and $b = 10^{6}, 10^{5}, 3 \times 10^{4}, 10^{4}$ (dashed lines from bottom to top).Reuse & Permissions
Figure 6
Four electron energy $- e A_{0} (x_{3}, 0)$ curves in the modified potential (18) for $b = 10^{6}, 10^{5}, 3 \times 10^{4}$ , and $10^{4}$ (thin solid lines from left to right) approaching their corresponding long-range parts $- e A_{l . r .} (x_{3}, 0)$ , Eq. (21), shown in Fig. 5 (four dashed lines from bottom to top). The thick dashed broken line corresponds to the string potential $A_{0} (x_{3}, 0) |_{b = \infty}$ . Its vertical fragment symbolizes the $δ$ function (31).Reuse & Permissions

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