Charged spin-1/2 particle in an arbitrary magnetic field in two spatial dimensions: A supersymmetric quantum mechanical system

It is shown that the 2 x 2 matrix Hamiltonian describing the dynamics of a charged spin-1/2 particle with g-factor 2 moving in an arbitrary, spatially dependent, magnetic field in two spatial dimensions can be written as the anticommutator of a nilpotent operator and its Hermitian conjugate. Consequ ently, the Hamiltonians for the two different spin projections form partner s of a supersymmetric quantum mechanical system. The resulting supersymmetr y algebra can then be exploited to explicitly construct the exact zero ener gy ground state wave function for the system. Module this ground state, the remainder of the eigenstates and eigenvalues of the two partner Hamiltonia ns form positive energy degenerate pairs. We also construct the spatially a symptotic form of the magnetic field which produces a finite magnetic flux and associated zero energy normalizable ground state wave function.