Energy Level Quantization in the 1D Quantum Well in Case of Instantaneous Stationary State with the Non-Relativistic Wall and Particle Motion

Abstract

The paper considers the problem of finding energy levels in the 1D quantum well in case of its width alteration at the nonrelativistic rate. According to the reviewed literature, the exact solution is known only in the case of nonrelativistic motion of the 1D quantum well wall at the constant rate. It is shown that motion with the constant rate is physically unrealizable. Therefore, it is necessary to find at least small areas of the Schrodinger equation solution for a wider range of nonrelativistic alterations in the 1D quantum well width. Analysis results presented in the study show existence of such areas. The found areas correspond to the instantaneous stationary states satisfying the Bohr quantization condition. In this case, the Dirichlet condition is also satisfied on the moving wall. It means that in this case energy of the level with the n number also becomes a function of the k second quantum number, which takes into account dynamic alteration in the 1D quantum well width. Variants were found of the k second quantum number spectrum and of the quantum level spectrum in various cases of the wall continuous motion with zero initial speed and finite acceleration. Within the framework of the analysis used, formulas were obtained to change the difference between energies of the two arbitrary levels. An analysis was made for the boundaries of the wall speed and the 1D quantum well width in considering the nonrelativistic problem. The obtained results and their possible applications are under discussion, including analysis of the problems related to nanotechnology