Tkachuk V. Supersymmetry and exactly solvable problems in quantum mechanics

The thesis concerns the study of supersymmetry in quantum mechanical problems and the use of supersymmetry as a method for finding exact solution of the eigenvalue problems. The supersymmetry of the electron in three dimensional stationary magnetic fields was considered. It is shown that supersymmetry with two, three and four supercharges can be realized in this case. We also studied the supersymmetry of the electron in both nonstationary magnetic and electric fields in a two-dimensional case. The supercharges that are the integrals of motion as well as their algebra are established. Using the obtained algebra the solutions of the nonstationary Pauli equation are generated. The Witten model of the supersymmetric quantum mechanics is generalized for the case when supersymmetric partners describe the mothin of the spin ? particle moving in magnetic field and scalar potential. We introduce the concept of shape invariant scalar and magnetic fields and it is shown that the eigenvalue problem admits exact analytical solutions when such fields are considered. We developed a new supersymmetric method for constructing quasi-exactly solvable potentials with two and three eigenstates. This method is extended for constructing conditionally-exactly solvable potentials. We also developed a systematic procedure generating a new exactly-solvable potentials which are a lower supersymmetric partners to the known exactly-solvable potentials. In the context of a two-parameter deformation of the Heisenberg algebra leading to non-zero minimal uncertainties in both position and momentum, the harmonic oscillator spectrum and eigenvectors are determined by using an extension of the techniques of conventional supersymmetric quantum mechanics combined with shape invariance under parameter scaling. Using supersymmetric quantum mechanics and shape invariance method we found exact solution of the eigenvalue problem for the relativistic Dirac oscillator with deformed Heisenberg algebra leading to isotropic nonzero minimal uncertaintiesin position. We show that there exists some intimate connection between three unconventional the Schrodinger equations based on the use of deformed canonical commutation relation, of a position-dependent effective mass or of a curved space. This occurs whenever a specific relation between the deforming function, the position-dependent mass and the (diagonal) metric tensor holds true. We illustrate these three equivalent approaches by considering a new Coulomb problem and solving it by means of supersymmetric quantum mechanics and shape invariance techniques. The supersymmetric quantum mechanics with shape invariance technique is extended for the case of the Schr?dinger equation with position-dependent mass or deformed Heisender algebra. We found three classes of potentials and corresponding functions of deformation (position-dependent mass) for which the respective eigenvalue problems can be solved exactly. We also studied quantum motion of the neutral atoms in the field of ferromagnetic wire with different types of magnetization. It is shown that ferromagnetic wire can be used for trapping and guiding neutral atoms. The possibility of experimental realization of binding sodium atoms is discussed. The quantum topological phase of electric dipole circulating around the line of magnetic charges (monopoles) is discussed. We propose to mimic the line of magnetic monopoles using a ferromagnetic wire in which magnetization is parallel to the wire and magnitude of magnetization changes linearly along the wire. The phase shift in the proposed scheme is within the reach of present-day experimental techniques and may by observed in atomic or molecular interferometry.