Supersymmetric quantum mechanics and its applications in physics

Ill ABSTRACT SUPERSYMMETRIC QUANTUM MECHANICS AND ITS APPLICATIONS IN PHYSICS ÖZER, Okan Ph. D in Engineering Physics Supervisor: Prof. Dr Bülent GÖNÜL June 2003, 95 pages Theoretical formulation of the supersymmetric quantum mechanics is re viewed and its applications to some physical problems are studied. The super- symmetric quantum mechanics (SUSYQM) is based on the factorization of the Schrödinger equation, leading to generalized operators and partner potentials. The Hamiltonian hierarchy and the factorization in the method are briefly dis cussed. It is also combined with the perturbation theory which is used for the systems that are not exactly solvable. The method is applied to the Hulthen po tential and an expression for the energy levels which gives satisfactory values for the non-zero angular momentum states of the potential is obtained. It is shown that a very general connection can be established between a class of singular potentials in iV-dimensional space through the application of a suitable transfor mation by SUSYQM. Using this way, a connection between screened Coulomb and anharmonic oscillator potentials is obtained. A general mapping procedure is described for the transformation of a differential equation with a position- dependent mass to the Shrödinger equation with a constant mass under canoni cal transformations. In the frame of SUSYQM, it is shown that the Schrödinger equation in its new form may have solutions if the original potential is solvable and shape-invariant one. Finally, It is demonstrated how the SUSYQM method can also be employed for the construction of n-parameter family of potentials which possess localized positive energy state(s) in the continuum. Key words: Supersymmetric Quantum Mechanics, Shape invariant potentials, Hamiltonian hierarchy, Supersymmetric perturbation theory, Hulthen potential, iV-dimensional space, Position-dependent mass, Bound states in continuum.