In this thesis, the electronic and optical properties of semiconductor nanostructures have been investigated in different conditions with different confining potentials. From the optical properties available for nanostructures, we have determined the optical absorption coefficient, refractive index, and third-order nonlinear susceptibility for a spherical quantum dot embedded in dielectric, a spherical multilayer quantum dot with impurity, a pyramidal and a cone-shaped quantum dot under applied electric fields. In all investigated quantum systems, first of all, wave functions and special energy values must be calculated. Because the analytical solution of the Schrödinger equation is not possible for most of the quantum systems with complex potential shapes under investigation, therefore numerical methods should be used, in this thesis we use the time domain finite difference method (FDTD) for this purpose. In this method, the Schrödinger equation will be converted into a diffusion equation by the imaginary time transformation technique. This method has been used to solve analytical solution problems such as the potential of a particle in a box, a simple harmonic oscillator, and a particle confined in a spherical well, and the results obtained from the analytical results have been compared and validated. The solutions obtained from the time domain finite difference method are consistent with the analytical solution with a small error percentage. In the examined structures, it can be seen that the specific values of the electronic states decrease with the increase in the size of the nanostructure, and it was also shown for the multilayer quantum dot that the electronic properties depend strongly on the thickness of the layers. Also, with the increase in the intensity of the external electric field, the levels of these values also increase. In examining the optical properties of these nanostructures, we have achieved significant results that all the results obtained fr