Abstract
In this dissertation, using the finite difference time domain (FDTD) method, special values and special functions of quantum systems with arbitrary potentials for which no analytical solution is available have been obtained. To do this, the Schrينdinger equation is transformed into a diffusion equation by the imaginary time conversion technique. Using this method, problems with analytical solution such as particle potential in the box, simple coordinate oscillator and trapped particle in the spherical well are solved and the results are compared and validated with analytical solution solutions. The answers obtained from the finite time difference method are consistent with the analytical solution and have a small error rate. This method is a different method from the standard FDTD method for solving the Schrينdinger equation. The FDTD method can determine the energy and wave functions of the quantum systems under study. In addition to one-dimensional problems, this method is also used in two and three dimensions. After using the FDTD algorithm, this method was coded and special quantum dot values with different structures were calculated and examined. One of the most important structures simulated by this method is the quantum dot of the spherical and elliptical monolithic unit as the core-shell and the spherical multilayer quantum dot with the core-shell-well-shell structure. These structures have been selected because of quantum confinement and their effects, which have significant changes in the electronic states of the system. The obtained results show that the specific values of electron states decrease with increasing the size of the nanostructure. Also, with the addition of an electric field, the eigenvalues are affected, and as the intensity of the external electric field increases, the levels of these eigenvalues also increase. For multilayer quantum dots, it has also been shown that the electronic properties strongly depend on the thickness of th