Non-Commutative Corrections to the MIC-Kepler Hamiltonian

Non-commutative corrections to the MIC-Kepler System (i.e. hydrogen atom in the presence of a magnetic monopole) are computed in Cartesian and parabolic coordinates. Despite the fact that there is no simple analytic expression for non-commutative perturbative corrections to the MICKepler spectrum, there is a term that gives rise to the linear Stark effect which didn’t exist in the standard hydrogen model. Progress in the String theory has recently ignited a great deal of interest to field theories in noncommutative spaces [1]. Quantum Hall effect [2], non-commutative classical and quantum mechanics [4][12], and various non-commutative phenomenological models [3] are among these types of theories. In addition to the string theory, these models also give rise to theories describing particles with spin. While most of the research was focused on mathematical aspects of two-dimensional quantum mechanics, three-dimensional quantum mechanics also attracted some attention. The later was inspired by possible phenomenological consequences of the non-commutativity in three dimensions. In their pioneer paper Chaichian et al. [4] have found non-commutative corrections to the hydrogen atom spectrum and discovered a new channel induced by these corrections to the Lamb shift. Specific form of the vertex in non-commutative quantum field theory (NCQED) resulted in lack of similar corrections to the Stark effect. Here we consider a non-commutative MIC-Kepler model an elegant generalization of the Coulomb system. Integrable MIC-Kepler system, originally constructed by Zwanziger [13] and later rediscovered by McIntosh and Cisneros [14] is characterized by the presence of an external monopole field. Such a system is described by the Hamiltonian H0 = h̄ 2μ (i∇+ sA) + h̄ s2 2μr2 − γ r , where rotA = r r3 . (1) Its distinctive peculiarity lies in the hidden symmetry given by the following constants of motion I = h̄ 2μ [(i∇+ sA)× J − J × (i∇+ sA)] + γ r r , J = −h̄(i∇+ sA)× r + h̄sr r . (2) These constants of motion together with the Hamiltonian, form the algebra of quadratic symmetry of the Coulomb model. Operators J and I define the angular momentum and the Runge-Lenz vector for the MIC-Kepler system. For fixed negative values of energy, these constants of motion form SO(4) algebra and for positive values of energy SO(3.1) one. Due to the hidden symmetry, the MIC-Kepler problem could be factorized in some coordinate systems, e.g. in spherical and parabolic coordinates. The MICKepler system, therefore, becomes a natural generalization of the Coulomb model in the presence of Dirac’s monopole. Monopole number s satisfies the Dirac’s charge quantization rule, s = 0,±1/2,±1, . . .. The MIC-Kepler system inherits most of the properties of hydrogen atom. Its behavior however, becomes qualitatively different in a static electric field. In the case of the MIC-Kepler model, degeneracy in the energy spectrum is removed by a constant electric field. In contrast with the usual hydrogen atom introduction of such a field gives rise to non-trivial corrections to the linear Stark effect [17]. Therefore, one can expect that the non-commutative corrections yield qualitative difference between the MIC-Kepler and hydrogen atom, at least in the context of Stark effect. Our discussion we begin with a description of the MIC-Kepler system in the framework of noncommutative geometry and then, we calculate the corresponding non-commutative corrections. Noncommutativity in Cartesian coordinates can be introduced via the following non-commutative addition