XXVII Workshop on Geometric Methods in Physics 29.06-05.07.2008

Hossein Panahi

Tow-dimensional quantum Hamiltonians with shape invariance symmetry

It is shown that the Casimir operator associated with the $ U(1)$ Lie derivative defined on $ S^2 = SU(2)/U(1)$ base manifold, can be interpreted as Hamiltonians of pair of scalar particle and scalar anti-particle with opposite charge over $S^2$ manifold in the presence of magnetic monopole located at its origin and an external electric field. Using the $SU(2)$ representation, the spectra of these Hamiltonians have been obtained.It is also proved that these Hamitonians are isospectral and having the shape invariance symmetry, i.e. they are supersymmetric partner of each other. Also the Dirac's quantization of magnetic charge comes very naturally from the finieness of the $SU(2)$ representation.