Quantum-Mechanical generalization of the Thomas–Fermi model

The interaction between particles in the mean-field approximation of the many-body theory is often taken into account with the use of the semiclassical description of the particle motion. However, quantization of a part of the degrees of freedom becomes essential in certain cases. In this work, two such cases where nonlinear wave equations appear have been considered: electrons in a quantum well and excitons in a trap. In the case of indirect excitons in an annular trap, the one-dimensional Gross–Pitaevskii equation permits an analytical solution and it turns out that there can be no bound state in a one-dimensional symmetric potential well. This makes the problem qualitatively different from a similar one-body problem. In the case of electrons in a quantum well, the nonlinear integro-differential equation does not have an exact solution and the allowed energy levels have been found by the direct variational method.