二维耦合谐振子的非对易能谱

On noncommutative energy spectra in two-dimensional coupling harmonic oscillator

非对易的思想最初源于对极小尺度的时空坐标的研究,后来引起了物理学界的广泛关注,逐渐探讨了在很多领域中的非对易效应.随着非对易量子力学的建立,研究一些精确可解模型的非对易效应是非常有意义的.各类谐振子模型就是其中之一.但是在坐标与坐标,动量与动量都是不对易的情况下,很难得到耦合谐振子能谱的解析解.本文旨在研究非对易相空间中二维耦合谐振子的量子特性.首先用不变本征算符方法得到了同时包含多种耦合项的谐振子能谱的解析解.然后分析非对易参数和耦合参数对非对易能谱的影响.结果表明,由于受到非对易参数和耦合参数的影响,二维耦合谐振子的非对易能谱出现移动,能级变为非简并的.这与通常对易空间中二维谐振子能级简并(除基态外)的情况是完全不同的.

The ideas of noncommutative space originate from the research on time-space coordinate on an extremely small scale.Subsequently,the noncommutative space has gradually attracted some attention.The researchers started to explore noncommutative effect in some other fields.With the establishment of noncommutative quantum mechanics,it becomes significant to explore the noncommutative effect of exactly solvable models.The kinds of harmonic oscillators are very important and fundamental models in physics.But in noncommutative phase space,coordinate and coordinate are noncommutative,and momentum and momentum are also noncommutative.These results in the difficulty in obtaining the energy spectra of oscillators systems.In this paper the quantum properties of a two-dimensional coupling harmonic oscillator in noncommutative phase space are studied.Firstly,the Hamiltonian of the system is constructed,which includes all possible coupling types,namely,coordinate-coordinate coupling,momentum-momentum coupling,and coordinatemomentum cross-coupling.Secondly,the explicit expression of the noncommutative energy spectrum for the Hamiltonian is obtained by using the invariant eigen-operator method.In this work it is shown explicitly that the changes in the energy levels are related to the noncommutative parameters and coupling parameters.Thirdly,the effects of coupling parameters and non-commutative parameters on the energy spectra are analyzed in detail in the form of graphs.The results show that the energy levels under the influence of non-commutative parameters are non-degenerated.As the values of non-commutation parametersθandΦincrease,some energy levels increase and tend to change linearly,and other energy levels first decrease and then increase.If the limit values of the non-commutative parameters are taken as follows:θ→0 and?→0,then the noncommutative energy spectra will be consistent with the energy spectra of the two-dimensional harmonic oscillator in the commutative space in general.On the other hand,the energy levels will split under the influence of coupling parameters.Moreover,the degree to which the energy levels split can increase as the kinds of couplings in the system increase.It is found that the coordinate coupling parameterηand the momentum coupling parameterσhave the same influence on the energy levels,but the coordinate momentum cross-coupling parameterκhas less influence on the energy levels thanηandσ.Overall,the above results are completely different from those of two-dimensional oscillator in the usual commutative space,which is degenerated except for the ground state.