On noncommutative energy spectra in two-dimensional coupling harmonic oscillator
Authors: Gou Li-Dan ,   School of Physics, Changchun University of Science and Technology, Changchun 130022, China
Published: September 2021

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 coordinate-momentum 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 parameters <inline-formula><tex-math id="M1">\begin{document}$ \theta $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20210092_M1.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20210092_M1.png"/></alternatives></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ \phi $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20210092_M2.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20210092_M2.png"/></alternatives></inline-formula> increase, some energy levels increase and tend to change linearly, and other energy levels first decrease and then increase. If the limit values of the non-commutative parameters are taken as follows: <inline-formula><tex-math id="M3">\begin{document}$ \theta \to 0 $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20210092_M3.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20210092_M3.png"/></alternatives></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ \phi \to 0 $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20210092_M4.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20210092_M4.png"/></alternatives></inline-formula>, then the noncommutative energy spectra will be consistent with the energy spectra of the two-dimensional harmonic oscillator in the commutative space in general. On the other hand, the energy levels will split under the influence of coupling parameters. Moreover, the degree to which the energy levels split can increase as the kinds of couplings in the system increase. It is found that the coordinate coupling parameter <inline-formula><tex-math id="M5">\begin{document}$ \eta $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20210092_M5.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20210092_M5.png"/></alternatives></inline-formula> and the momentum coupling parameter <inline-formula><tex-math id="M6">\begin{document}$ \sigma $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20210092_M6.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20210092_M6.png"/></alternatives></inline-formula> have the same influence on the energy levels, but the coordinate momentum cross-coupling parameter <inline-formula><tex-math id="M7">\begin{document}$ \kappa $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20210092_M7.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20210092_M7.png"/></alternatives></inline-formula> has less influence on the energy levels than <inline-formula><tex-math id="M8">\begin{document}$ \eta $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20210092_M8.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20210092_M8.png"/></alternatives></inline-formula> and <inline-formula><tex-math id="M9">\begin{document}$ \sigma $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20210092_M9.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="20-20210092_M9.png"/></alternatives></inline-formula>. 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.