Nonrelativistic spin 12 particle in an arbitrary non - Abelian magnetic field in two spatial dimensions

The (group and spin space) matrix Hamiltonian describing the dynamics of a nonrelativistic spin 1/2 particle moving in a static, but spatially dependent, non-Abelian magnetic field in two spatial dimensions is shown to take the form of an anticommutator of a nilpotent operator and its hermitian conjugate. Consequently, the (group space) matrix Hamiltonians for the two different spin projections form partners of a supersymmetric quantum mechanical system. The resulting supersymmetry algebra is exploited to explicitly construct the exact zero energy ground state wavefunction(s) for the system. The remaining eigenstates and eigenvalues of the two partner Hamiltonians form positive energy degenerate pairs. ∗e-mail address: [email protected] †e-mail address: [email protected] ‡e-mail address: [email protected] 1 The motion of a nonrelativistic spin 1/2 particle confined to move in a plane under the action of a magnetic field directed normal to the plane is a fundamental problem appearing in a variety of physical applications [1][3]. Previously, we examined this problem for a static magnetic field having arbitrary spatial dependence on the planar coordinates. We showed [4][5] that the model exhibited a supersymmetry [6][8] which we consequently exploited to construct the exact zero energy normalizable ground state(s) for the system. In this note, we study an analogous problem involving the planar motion of the spin 1/2 particle under the action of a non-Abelian magnetic field which is also directed normal to the plane and again having arbitrary spatial dependence on the planar coordinates. Such a configration has also been argued [9] to have relevance for various physical systems. The non-Abelian magnetic field strength is ~ B = ẑB(x, y), with a denumerating the group generators, and B = F a 12 = ∂1A a 2 − ∂2A1 + g h̄c fabcA b 1A c 2 (1) = ij(∂iA a j + g 2h̄c fabcA b iA c j). (2) The indices i, j = 1, 2 label the spatial coordinates of the plane, while g is the gauge charge and fabc are the group structure constants. Thus the fundamental representation matrices, L, satisfy [L, L] = ifabcL . It proves convenient to introduce matrix valued fields B = LB and Ai = L Ai so that B = ij(∂iAj − ig h̄c AiAj). (3) In general, the vector potential can be decomposed into transverse and longitudnal pieces as Ai = ij∂jK + ∂iC ; C = L C , K = LK. (4) We choose to work in the Coulomb gauge defined by ∂iAi = 0 and C = 0 so that Ai = ij∂jK. (5)