Schrödinger Equation with Noninteger Dimensions

ABSTRACT: Exact solutions in quantum theory play crucial roles in the application areas of the theory. For instance knowing the exact eigenvalues and eigen-functions of the Hamiltonian of the Hydrogen atom helps the Chemists to find, with a high accuracy, the energy levels of more complicated atoms like Helium and Calcium. Therefore any attempt to find an exact solvable system in quantum mechanics is remarkable. For this reason in this thesis we aim to find exactly solvable systems in quantum theory but not in integer dimensions. We consider noninteger dimensional quantum systems. The corresponding Schrödinger equation is introduced. With specific potential, an infinite well, we solve the Schrödinger equation both its angular part and radial part. The angular part admits a solution in terms of Gegenbauer polynomial functions and the radial part gives a solution in terms of the Bessel functions. Keywords: Noninteger dimensions; Schrödinger equation; Gegenbauer polynomial functions; Bessel functions. …………………………………………………………………………………………………………………………