Relativistic dynamics of positronium atom in uniform magnetic field*

Abstract

AbstractThe relativistic dynamics of the positronium atom in a uniform external magnetic field is investigated in this work. Following our previous work on a general system of two spin‐½ particles, here too we choose three distinct forms of interaction, viz., the Bethe–Salpeter interaction, the Breit interaction, and the projected Breit interaction. We also include the interaction of the anomalous magnetic moment with the external magnetic field. As the system is overall electrically neutral, the center‐of‐mass motion can be completely separated out. The residual Hamiltonian operator describes only the relative movement. This operator can be brought into a form where the one‐particle part is exactly separable into positive and negative energy states and is diagonal. However, as a marked departure from our previous methodology, the separation of the odd terms from the transformed interaction operator up to order α2, where α is the fine‐structure constant, is accomplished here by a somewhat different procedure so as to avoid those unitary transformation operators which contain the mass difference (m1−m2) in the denominators of their arguments. Because of the nature of the one‐particle operators, the odd terms of order α2 are completely removed in the present treatment. Also, no unphysical term of order α2 results from the use of the Breit interaction. The effect of the virtual electron–positron pair, which appears at second order in the fine‐structure constant, is also incorporated in the Hamiltonian. Again, the reduction of the order‐α2 pair term to a previously known simpler form is exactly achieved here. Thus we obtain three different versions of the relativistic Hamiltonian operator for a positronium atom in a uniform magnetic field, with the electrodynamical interactions taken into account through order α2. All three lead to the same energy for the same spinor wave function under the so‐called uU separation. These operators are now used to calculate the conventional nonrelativistic limit, and the same limiting operator is obtained in every case. For zero magnetic field, the Hamiltonian reduced to the nonrelativistic limit is found to be identical with the standard nonrelativistic Hamiltonian known for the positronium atom. We also obtain the nonrelativistic limit Hamiltonian for the positronum atom in the high‐field case, where the ratio of the applied magnetic field and the velocity of light B/c, or, more precisely, the Larmor frequency ωL=eB/2mc, is of the order of one atomic unit. Finally, in the Appendices, we discuss the nature of the eigenfunctions and indicate how one can calculate these functions correctly through the zeroth order in the fine‐structure constants and find the eigenvalues correct through the second order. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002