О корректности размера позитрония

Abstract

It is noted that initially the minimum size of the hydrogen atom was determined by the semiclassical method, which is still widely used today. When calculating the minimum size of a positronium atom, the method tested on the hydrogen atom was not used. Instead, in the formula for the radius, the mass was replaced by the reduced mass of positronium. There would be nothing wrong with this if the above “hydrogen” method gave the same result. However, it is not. The difference between the formula for positronium and the “hydrogen” formula is the coefficient “2”, due to the fact that the positive charge is not in the center of mass. In the literature, the minimum size of a positronium atom (Bohr radius of positronium) was determined in accordance with the two-body problem method. In the particular case under consideration, the method of the two-body problem is redundant and can be successfully replaced by solving the problem of the motion of one of the bodies. This allows you to obtain more visual and obvious solutions without excessive formalism with the reduced mass and displacement vector. The doubly refined value of the positronium radius was determined using the semiclassical method. However, positronium is a quantum object. Therefore, a quantum mechanical consideration is also presented. The textbook solutions are taken as a basis. The solution to the problem of two particles is found in the form of an equivalent solution for one of the particles moving relative to a stationary center of mass, especially since there is a reliable quantum mechanical solution for the case of a spherically symmetric force field, which assumes a stationary center of force. The latter is especially important, since the mobility of the force center (electric charge) can lead to radiation and, consequently, a change in the energy balance. It has been established that the calculated minimum radius of positronium (the distance from its center of mass at which the probability of finding an electron/positron is maximum) is equal to four, and not two Bohr radii of hydrogen, as indicated in the literature. In this case, the semiclassical and quantum solutions do not contradict each other.