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.