Abstract
A “cut-off” Coulomb potential taking into account the finite size of the nucleus is finite, and a solution of the Dirac equation can be constructed for any energy, both positive and negative. In the paper we develop an exact solution of the Dirac equation for a fixed value of the total momentum j for the whole spectrum of energies, which allows us to determine the vacuum charge and its spatial distribution. We consider nuclei with different charges Z, both Z<Zc and Z>Zc, where Z=Zc is the “critical” charge, at which the energy of the lowest discrete state reaches the boundary of the lower continuum ε=−mc2. Polarization of vacuum is determined, and the vacuum charge for several values of Z is found. For an undercritical nuclear charge, Z<Zc, the total vacuum charge appears to be zero, while for Z>Zc, the vacuum gets rearranged, and the total vacuum charge becomes equal to −2e. The vacuum charge distribution for j=1/2 for both undercritical and overcritical nuclei is calculated.
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