| Abstract |
The solution of a unitarily transformed Dirac equation for the hydrogenic electron in zero magnetic field is investigated here. The momentum-space representation is adopted as a natural recourse. The spinor part of the transformed wavefunction in momentum space can be easily prescribed for a central potential. Hence, for the Coulomb potential, a pair of equations is obtained for the radial components in momentum space. It is shown that starting from these radial equations, one can recover the equations previously derived by Rubinowicz, Levy, and Lombardi for the problem of the Dirac hydrogen atom in momentum space. This establishes equivalence among different approaches based on the momentum representation, including the current treatment. The recovery of the equations due to Rubinowicz permits the exact eigenvalues to be written down and exact expressions to be derived for the radial components of the transformed wavefunction in momentum space. A new approach is adopted to carry out a reduction to the nonrelativistic regime and the nonrelativistic limit. At first the transformed momentum-space equation for the hydrogen atom is rewritten in terms of the hyperspherical coordinates. The zeroth-order solutions of the new equation are recovered in the limit c --> infinity where c is the speed of light. These are manifestly separable into positive- and negative-energy forms. For positive energy, these solutions have nonvanishing upper components that are two-component spinors. The latter exactly correspond to the single-component, nonrelativistic, momentum-space solutions derived by Fock. It is shown that when the upper component is corrected through first order in v(2)/c(2) but the separability is still maintained for the transformed wavefunction, one retrieves the Pauli equation in momentum space. It is also shown that for a hydrogen atom placed in a uniform magnetic field, the nonvanishing momentum-space matrix elements representing the anomalous Zeeman effect have a simple form, namely, the product of a radial integral and an angular integral. These integrals are equal to the well-known radial and angular integrals in coordinate representation. The matrix elements can be easily evaluated. (C) 2003 Wiley Periodicals, Inc. |
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