Dynamical Algebras In The 1+1 Dirac Oscillator And The Jaynes-Cummings Model

The introduction and combination of appropriate physical models have made remarkable achievements in many different fields of physics.In this work,we study the Dirac oscillator and the Jaynes-Cummings(J-C)model.The Dirac oscillator is one of the most widely used models in classical and quantum physics.J-C model is closely related to the Dirac oscillator in quantum optics.These two models are precisely solvable quantum systems,which contain rich physical connotations and can be widely used in various fields of physics.They are commonly used models in quantum mechanics,quantum optics,laser physics and so on.At the same time,the dynamic symmetry theory,which is the classical theory of quantum physics,reveals the conservation laws of quantum systems and the corresponding dynamic symmetry in the category of group theory and Lie algebra.In this paper,we study the algebraic structure of the one-dimensional Dirac oscillator and J-C model by extending the concept of spin symmetry to the noncommutative case.The main method used in this paper is to calculate the spectrum and algebraic structure of the Dirac oscillator and J-C model by means of unitary transformation and diagonalization of the hamiltonian of the system.In the theoretical calculation,using the unitary operator in the two dimensional Dirac equation to make a unitary transformation of the orbital angular momentum,it is found that the transformed orbital angular momentum satisfies the commutation relation with the two dimensional Dirac hamiltonian,found the key reason why the dynamic symmetry of the system can not be broken by the spin-orbit coupling.we then extend the unitary operator of the 2D Dirac equation to a noncommutative case.Namely,one of the momentums in the unitary is replaced by a coordinate operator.Using this unitary,one can derive a conserved number operator and a spin of the 1D Dirac oscillator,i.e.,the shift operators which can be constructed simultaneously.With these conserved quantities and shift operators,an SO(7)4(8)algebraic structure of the system is revealed,which is said to be a dynamical symmetry in the broad sense.In addition,to some extent,the Dirac oscillator is related to the J-C model.Therefore,the algebraic solution of the J-C model in this paper is closely related to the algebraic solution of the Dirac oscillator.