The Application Of Su(3) Algebra In The Physical Model

The solution of the exact solvable physical models has always been a major concern in quantum mechanics.In the late nineteenth century,a Norwegian mathematician Sophus Lie introduced Lie algebra,since then algebra has become an effective means to study physical model.Scientists have found that algebraic method can be used to find out the algebraic structure implied in many exact solvable models.The energy eigenvalues and eigenfunctions of the system can be solved more simply by constructing the algebraic structure,thus avoiding the solution of Schrodinger equation and greatly simplifying the calculation process.Therefore,using Lie algebraic method to solve the problem of quantum mechanics has become one of the main studies in modern physics.It has been found that there are su(1,1)or su(2)algebraic structure in one-dimensional and three-dimensional quantum system.The energy eigenvalues and eigen-functions of the system can be solved successfully by constructing corresponding algebra to study these exact solvable physical models.However,although people have tried to use su(3)algebra to describe the symmetry of many physical problems,but so far,no one has systematically used su(3)algebra to discuss the Hamiltonian in detail.The main concern of this paper is to construct su(3)algebraic method and apply it to the diagonalization of Hamiltonian,and discuss the applicability in special Hamiltonian system and the three-dimensional harmonic oscillator system.First of all,express systematic Hamiltonian into the linear type of su(3)algebra.Taking advantage of this system,which has a cryptic systematical structure of dynamics,and resorting to su(3)algebra can generate translation operator with meta structure.Secondly,introduce a series of unitary transformation to eliminate the superfluous item of non-diagonal in system,and make systematic Hamiltonian reduced to the linear type of Cartan subalgebra.Then apply the representation theory of su(3)algebra to obtain the eigenvalue and eigenstate of Hamiltonian that has been diagonalizable.Eventually,apply acquired results to make some special Hamiltonian systems and three-dimensional harmonic oscillator systems to be solved,which successfully verifies the correctness of the approach that has been used in this paper.The method of su(3)algebra is a new convenient and general method for solving the Hamiltonian system with su(3)linear form,which provides a theoretical reference for the study of su(3)in the future.