Research On Non-commutative Quantum Systems In Singular Electromagnetic Fields

In this paper,the interaction between the dipole and the electromagnetic field is studied from the perspective of quantum mechanics and noncommutative quantum mechanics,and the energy eigenvalues and eigenfunctions of the dipole are calculated.We will use neutral atoms with permanent electric dipole moments or permanent magnetic moments to characterize dipoles.In the commutative quantum mechanics,firstly,we review the Landau theory,that is,the problem of energy quantization of two-dimensional electron gas in the magnetic field.Then,based on the Landau energy level,we study the interaction between the magnetic dipole and the electric field,the interaction between the electric dipole and the singular magnetic field,and the interaction between the electric dipole and the electromagnetic field.We demonstrate that only if the three conditions are satisfied the energy of a neutral atom in an electromagnetic field can be quantified in the analogy of Landau level.Finally,we also found that,in the three field-couple configurations given above,each of their stationary state Schrodinger equation can be transformed into a confluent hypergeometric equation.Therefore,according to the found solution of confluent hypergeometric equation we can get the energy eigenvalues and eigenfunctions of the dipole in different field-couple configurations.In the noncommutative quantum mechanics,we introduce the noncommutative space and the noncommutative phase space and then we use the Bopp transformation to find the Hamiltonian of electric dipole interacting with electromagnetic field in the non-commutative space and the non-commutative phase space respectively.We calculate the energy eigenvalues and the eigenfunctions of the electric dipole.Furthermore,we compare the results with those obtained from the commutative quantum mechanics.Finally,by treating the Hamiltonian modified by ? and ? as perturbation term,the noncommutative correction of the electric dipole energy level is calculated.Finally,the condition of the elimination of energy difference caused by noncommutativity is analyzed.