Here, again, selleck products describes the relative motion of the electron and positron, while describes the free motion of a Ps center of gravity. Similar to (20), after simple transformations, one can obtain: (31) Repeating the calculations described above, one can derive the expression for the wave functions: (32) where . The energy of a free Ps atom in a narrow bandgap semiconductor with Kane’s dispersion law can be obtained from standard conditions: (33) As expected, the expression (33) selleck kinase inhibitor follows from (27) in the limit

case r 0 → ∞. For a clearer identification of the contribution of the SQ in a Ps energy, let us define the confinement energy as a difference between absolute values of energies of a Ps in a spherical QD and a free Ps: (34) It follows from (34) that in the limiting case r 0 → ∞, the confinement energy becomes zero, as expected. However, it becomes significant in the case of a small radius of QD. Note also that the confinement energy defined here should not be confused with the binding energy of a Ps since the latter, unlike the first, in the limiting case does not become zero. Positronium in two-dimensional QD As noted above, dimensionality reduction dramatically changes the energy of charged particles. Thus, the Coulomb

interaction between the impurity center and the electron increases significantly (up to four selleck screening library times in the ground state) [42]. Therefore, it is interesting to consider the influence of the SQ in the case of 2D interaction of the electron and positron with the nonparabolic dispersion law. Consider an electron-positron pair in an impermeable 2D circular QD with a radius R 0 (see Figure 1b). The potential energy is written as: (35) The radius of QD and effective Bohr radius of the Ps a p again play the role of the problem parameters, which

radically affect the behavior of the particle inside a 2D QD. Strong size quantization regime As it mentioned, the Coulomb interaction between the electron and positron can be neglected in this approximation. The situation is similar to the 3D case, with the only difference being that the Bessel equation is obtained for radial part of the reduced Schrödinger equation: (36) and solutions are given by the Bessel functions of the Branched chain aminotransferase first kind J m (η), where . For the electron energy, the following expression is obtained: (37) where are zeroes of the Bessel functions of the integer argument. The following result can be derived for the system total energy: (38) Here n r , m(n ′ r , m ′ ) are the radial and magnetic quantum numbers, respectively. For comparison, in the case of parabolic dispersion law for the 2D pair in a circular QD in the strong SQ regime, one can get: (39) Weak size quantization regime In this case, again, the system’s energy is caused mainly by the electron-positron Coulomb interaction, and we consider the motion of a Ps as a whole in a QD.