Spin Effects in the Emission of a Photon by a Two-Dimensional Hydrogen-Like Atom
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ELEMENTARY PARTICLE PHYSICS AND FIELD THEORY SPIN EFFECTS IN THE EMISSION OF A PHOTON BY A TWODIMENSIONAL HYDROGEN-LIKE ATOM V. V. Skobelev
UDC 539.12
It is shown that the result of calculation of the probability of single-photon emission by a two-dimensional hydrogen-like atom can depend on the choice of the spin operator. Specifically, if it is chosen to be the projection operator of the spin onto the plane of motion, then the result, as expected, agrees with the classical theory of emission of a three-dimensional atom, as was demonstrated by the author in one of his previous works. If, however, it is chosen as the projection operator onto the perpendicular to the plane of motion, the result does not agree with the classical theory. This puts in doubt the results presented in [S. H. Guo, X. L. Yang, F. T. Chan, Phys. Rev. A, 43, No. 3, 1197 (1991)]. Keywords: spin operator, single-photon emission, hydrogen-like atom, Lyman series.
INTRODUCTION The effect of photon emission by a hydrogen-like atom ( Ze) is a key problem both in ordinary threedimensional quantum mechanics and QED [1] and in the extension to two-dimensional space with a two-dimensional atom in the ( x, y ) plane. Interest in such low-dimensional atoms is grounded in the fundamental possibility of their experimental realization, in like manner to how Na atoms were obtained in the Bose condensate in the experiment reported in [2]. As was shown in [3–5], in a consideration of these questions, spin effects, which depend in turn on the choice of the spin operators within the framework of the Dirac theory and in an expansion over the relativistic parameter ( Z) , e 2 / c 1.137 , can play a nontrivial role in ordinary three-dimensional space. In the scheme for calculating spin effects proposed in these works, the probability of single-photon emission for an analog of the line of the Lyman series (i.e., for arbitrary Z ) is unchanged in comparison with the Schrödinger theory [6]; however, the selection rules in l can be different, although this has not yet been rigorously proved. Moreover, in this approach an interpretation of the so-called contact and spin-orbit corrections to the energy that is more intuitive in comparison with the generally accepted interpretation is also possible. This program was partly realized in [7, 8] for the two-dimensional hydrogen-like atom by the most natural choice of the spin operator as the operator projecting the spin onto the plane of motion. In analogy with [9], another choice of this operator, projecting the spin onto the z axis, is of interest. Specifically, the present paper, based on a comparison of the results of calculations based on the Dirac theory using the indicated spin operators in the Schrödinger theory, demonstrates the adequacy of the choice of the spin state [8] in comparison with adopted in [9], as was preliminary indicated in [8]. With this goal in mind, let us find the form of the spin operator equivalent to [9] within the framework of the approach developed to taking spin states into accou
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