Eigenstates and fine structure of a hydrogenic impurity in a spherical quantum dot

The fine structure of the energy levels for a hydrogenic impurity located in the center of a spherical quantum dot is calculated using a simpler exact solution for the potential well. The results reveal that when the dot radius approaches zero, the eigenenergies are just like a free-space hydrogenic atom. When the dot radius is large enough, then the eigenenergies approach a free-space hydrogenic atom but are shifted by the confining potential. Also we find that the radial expectation values will be equal to a free-space hydrogenic atom, when the dot radius is extremely small and extremely large. Between these two situations, the radial expectation values are smaller than those of a free space because of the pressing of the confining potential. Not every dot radius influences the eigenenergy to the same degree. It is decided by the bumps of the electron’s wave function and the place of the potential well’s margin. When the margin of the well begins to push the bumps of the wave then the eigenenergy will increase more quickly. Because of the changing of the electron distribution probability, the degeneracy of the different $l$ value in a free-space hydrogenic atom is removed by the confining potential. The total-energy shifts of the fine structure of the impurity could be six times larger than the total energy shifts of a free-space atom.